Fifth Conference on Optimization Techniques. Rome 1973: Part 1 (Lecture Notes in Computer Science)

Editorial Board D. Gries • P. Brinch Hansen • C. Moter • G. Seegmfiller • N. Wirth Prof. Dr. R. Conti Istituto di Mate...

Author: Roberto Conti | Antonio Ruberti

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Editorial Board D. Gries • P. Brinch Hansen • C. Moter • G. Seegmfiller • N. Wirth

Prof. Dr. R. Conti Istituto di Matematica "Ulisse Dini" Universit5 di Firenze Viale Morgagni 67/A 1-50134 Firenze/Italia Prof. Dr. Antonio Ruberti Istituto di Automatica Universit5 di Roma Via Eudossiana 18 1-00184 Roma/Italia

AMS Subject Classifications (1970): 49-02, 49A40, 49B20, 49B35, 49B40, 49C05, 49DXX, 65K05, 68A25, 68A45, 90-02, 90C05, 90C20, 90C30, 90C50, 90C99, 90D05, 90D25, 93-02, 93B05, 93B10, 93B20, 93B30, 93B35, 93C20, 93E05, 93E20

49C10, 90C10, 90D99, 93B99,

ISBN 3-540-06583-0 Springer-Verlag Berlin , Heidelberg ' New York ISBN 0-387-06583-0 Springer-Verlag New York • Heidelberg - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks, Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, tt~e amount o(the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin. Heidelberg i973. Library of Congress Catalog Card Number 73-20818. Printed in Germany. Offsetprinting and bookbinding: Julius Beltz, Hemsbach/Bergstr.

PREFACE

T h e s e P r o c e e d i n g s a r e b a s e d on the p a p e r s p r e s e n t e d at the 5th I F I P C o n f e r e n c e on O p t i m i z a t i o n T e c h n i q u e s h e l d in R o m e , l~Iay 7-11, 1973. T h e C o n f e r e n c e w a s s p o n s o r e d by the I F I P T e c h n i c a l C o m m i t t e e on O p t i m i z a t i o n ( T C - 7 ) and by the C o n s i g l i o N a z i o n a l e d e l l e R i c e r c h e (Italian N a t i o n a l R e s e a r c h

Council). The

Conference

was

and their application Major

emphasis

including:

devoted

to recent advances

to modelling,

identification

of the Conference

Environmental

An interesting

was

Systems,

feature

recent application

Soeio-economic

Systems,

was

Proceedings

the papers

in which

are divided

the methodological

those dealing with various The R. Conti,

International A. Ruberti

W. J. Karplus L. S. Pontryagin

Previously

(USA),

application

Program

J. L. Lions

optimization

aspects

Systems.

of specialists

engineering.

In the first are collected

are emphasized;

in the second

areas. of the Conference

Fe de Veubeke

(France),

E. Rofman

of systems

volumes.

Committee

(Italy) Chairmen,

(USSR),

published

into two

areas,

Biological

the participation

both in control theory and in the field of application The

techniques

and control of large systems.

on the most

of the Conference

in optimization

G. Marehuk

(Argentina),

consisted

(Belgium), (USSR),

of:

E. Goto (Japan), C. Oleeh

J. Stoer (FRG),

(Poland),

J.H. Westcott

(UK).

conferences:

Colloquium

on Methods of Optimization. (Lecture Notes in Mathematics,

Held in Novosibirsk/USSR, Vol. 112)

Symposium

on Optimization. Held in Nice, June 1969. (Lecture Notes in Mathematics, Vol. 132)

Computing

Methods in Optimization Problems. (Lecture Notes in Operation Research

June

1968.

Held in San Remo, September and Mathematical Economics,

1968. Vol. 14)

TABLE OF CONTENTS

SYSTEM

MODELLING

AND

IDENTIFICATION

I d e n t i f i c a t i o n of S y s t e m s S u b j e c t t o R a n d o m State D i s t u r b a n c e A. V. B a l a k r i s h n a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

A d a p t i v e C o m p a r t i m e n t a l S t r u c t u r e s in B i o l o g y and S o c i e t y R. R. M o h l e r , W. D. S m i t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

On the O p t i m a l Size of S y s t e m Model M. Z. Dajani .................................................

33

Information-theoretic Methods for Modelling and Analysing Large Systems R. E. Rink ...................................................

37

A New

Criterion for Modelling Systems L. W. Taylor, Jr .............................................

46

Stochastic Extension and Functional Restrictions of Ill-posed Estimation Problems E. Mosca ...................................................

57

Regression Operator Application to Some A. Szymanski

69

in Infinite Dimensional Vector Spaces and Its Identification Problems ...............................................

An Approach to Identification and Optimization in Quality Control W. Runggaldier, G.R. Jacur ................................... On Optimal Estimation Y. A. Rosanov ~

and Innovation

Processes

Identification de Domaines J. C e a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

T h e M o d e l l i n g of E d i b l e Oil F a t M i x t u r e s J. O. G r a y , J . A . A i n s l e y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DISTRIBUTED

83

92 103

SYSTEMS

F r e e B o u n d a r y P r o b l e m s and I m p u l s e C o n t r o l J. L. L i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

A C o n v e x P r o g r a m m i n g Method in H i l b e r t S p a c e and Its A p p l i c a t i o n s to O p t i m a l C o n t r o l of S y s t e m s D e s c r i b e d by P a r a b o l i c E q u a t i o n s K. M a l a n o w s k i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

~ p a p e r not r e c e i v e d

VI

About

Some Free C. Baiocchi

Boundary Problems Connected with Hydraulics .................................................

I 37

M4thode de D4composition Appliqu~e au Contr61e Optimal de Systbmes Distribu4s A. Bensoussan, R. Glowinski, J.L. Lions .......................

14 !

Approximation of Optimal Control Problems of Systems Described by Boundary-value Mixed Problems of Dirichlet-Neumann Type P. Colli Franzone ............................................

] 52

Control

GAME

of Parabolic P. K. C. Wang

Systems with Boundary Conditions Involving ................................................

Time-Delays ] 65

THEORY

Characterization of Cones of Functions Isomorphic to Cones of Convex Functions J. -P. Aubin .................................................

174

Necessary Conditions and Sufficient Conditions for Pareto Optimality in a Multicriterion Perturbed System J. -L. Goffin, A. Haurie .......................................

] 84

A Unified Theory Differential Games C. Marchal

194

About

PATTERN

of Deterministic

Two-Players

Zero-Sum

..................................................

Optimality of Time of Pursuit M. S. Nikol'skii ...............................................

202

RECOGNITION

Algebraic Automata E. Astesiano,

and Optimal Solutions in Pattern Recognition G. Costa ........................................

A New Feature Selection Procedure for Pattern Recognition Based Supervised Learning J. Kittler .................................................... On Recognition Descriptions S. T y s z k o

A Classification G. Walch

of High ~

Deformed

Patterns

by Means

206 on 218

of Multilevel

...................................................

Problem in Medical Radioscintigraphy ......................................................

The Dynamic Clusters Method and Optimization in Non-Hierarchical Clustering E. Diday .....................................................

~paper not received

250

241

Vll

OPTIMAL

CONTROL

A Maximum Principle for General Constrained Optimal Control Problems - An Epsilon Technique Approach J . W. M e r s k y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

O p t i m a l C o n t r o l of S y s t e m s G o v e r n e d b y V a r i a t i o n a l I n e q u a l i t i e s J. P . Y v o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265

O n D e t e r m i n i n g t h e S u b m a n i f o l d s of S t a t e S p a c e W h e r e t h e O p t i m a l Value Surface Has an Infinite Derivative H. L. S t a l f o r d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

276

C o n t r o l of A f f i n e S y s t e m s w i t h M e m o r y M. C. D e l f o u r , S . K . M i t t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

292

C o m p u t a t i o n a l M e t h o d s in H i l b e r t S p a c e f o r O p t i m a l C o n t r o l P r o b l e m s with Delays A. P . W i e r z b i c k i , A . H a t k o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

304

S u f f i c i e n t C o n d i t i o n s of O p t i m a l i t y f o r C o n t i n g e n t E q u a t i o n s V. I. B l a g o d a t s k i h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

V a r i a t i o n a l A p p r o x i m a t i o n s of S o m e O p t i m a l C o n t r o l P r o b l e m s T. Z o l e z z i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329

Norm P e r t u r b a t i o n o:[ S u p r e m u m Problems J. Baranger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

333

O n T w o Conjectures about the Closed-Loop Time-Optimal Control P. Brunovsk~ ................................................

341

C o u p l i n g of S t a t e V a r i a b l e s in t h e O p t i m a l L o w T h r u s t O r b i t a l Transfer Problem R. H e n r i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

O p t i m i z a t i o n of t h e A m m o n i a O x i d a t i o n P r o c e s s U s e d i n t h e M a n u f a c t u r e of N i t r i c A c i d P. U r o n e n , E. K i u k a a n n i e m i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

360

STOCHASTIC

CONTROL

S t o c h a s t i c C o n t r o l w i t h a t M o s t D e n u m e r a b l e N u m b e r of C o r r e c t i o n s J. Z a b c z y k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

370

D e s i g n of O p t i m a l I n c o m p l e t e S t a t e F e e d b a c k C o n t r o l l e r s f o r L a r g e Linear Constant Systems W.J. Naeije, P. Valk, O.H. Bosgra ............................

375

C o n t r o l of a N o n L i n e a r S t o c h a s t i c B o u n d a r y V a l u e P r o b l e m J. P . K e r n e v e z , J. P . Q u a d r a t , M. V i o t . . . . . . . . . . . . . . . . . . . . . . . . .

389

An A l g o r i t h m to E s t i m a t e S u b - O p t i m a l P r e s e n t V a l u e s f o r U n i c h a i n Markov Processes with Alternative Reward Structures S. D a s G u p t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399

VIII MATHEMATICAL Some Penalty

PROGRAMMING

Recent Developments in Nonlinear Programming G. Zoutendijk ................................................

407

Methods and Augmented Lagrangians in Nonlinear Programming R. T. Rockafellar .............................................

4t8

On INF-Compaet Mathematical Programs R. J. -B. Wets ................................................ Nonconvex Quadratic Programs, Integer Linear Programs F. Giannessi, E. Tomasin

Linear

Complementarity

Problems,

426 and 437

.....................................

A Widely Convergent Minimization Algorithm with Quadratic Termination Property G. Treccani .................................................

450

A Heuristic Approach E. Biondi, P.C.

460

A New

to Combinatorial Optimization Problems Palermo ......................................

Solution for the General L. B. Kov~cs

Set Covering

Problem 471

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Theoretical Prediction of the Input-Output Table E. Klafszky .................................................. An

484

Improved Algorithm for Pseudo-Boolean Programming S.Walukiewicz, L. SZomifiski, M. Faner .........................

493

Numerical Algorithms for Global Extremum Search J. Evtushenko ................................................

NUMERICAL

505

METHODS

Generalized Sequential Gradient-Restoration to Problems with Bounded Control, Bounded Rate of the State A. Miele, J.N. Damoulakis ~

Algorith with Applications State, and Bounded Time-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gradient Techniques for Computation of Stationary Points E. K. Blum ...................................................

509

Parameterization and Graphic Aid in Gradient Methods J. -P. Peltier .................................................

517

Les Algorithmes de Coordination dans la M~thode Mixte d'Optimisation Deux Niveaux G. Grateloup, A° Titli, T. Lef~vre ..............................

~paper

not received

& 526

IX

Applications of Decomposition and Multi-Level Techniques to the Optimization of Distributed Parameter Systems Ph. Cambon, L. Le Letty .....................................

538

Attempt to Solve a Combinatorial of Extension-Reduction E. Spedicato, G. Tagliabue

554

Problem

in the Continuum

by a Method

....................................

Contents

of Part

(Lecture

Notes

URBAN Some

AND

in Computer

SOCIETY

Science,

Vol.

4)

SYSTEMS

Aspects of Urban Systems of Relevance to Optimization Techniques D. Bayliss ..................................................

Selection of Optimal S. Czamanski Optimal An

II

Industrial Clusters for Regional Development ................................................

Investment S. Giulianelli,

9

Policies in Transportation Networks A. La Bella .....................................

On-Line Optimization Procedure C. J. Macleod, A. J. AI-Khalili

for an Urban Traffic System .................................

Hierarchical Strategies for the On-Line Control of Urban Road Signals M. G. Singh ..................................................

I

22 31

Traffic

Application of Optimization Approach to the Problem of Land Use Plan Design K. C. Sinha, A. J. Hartmann ....................................

42

60

Some Optimization Problems in the Analysis of Urban and Municipal Systems E. J. Beltrami ~ ................................................ Combinatorial Optimization and Preference Pattern Aggregation J. M. Blin, A. B. Whinston ......................................

73

A Microsimulation Model of the Health Care System in the United States: The Role of the Physician Services Sector D. E. Yett, L. Drabek, M.D. Intriligator, L.J. Kimbell ............

85

COMPUTER

AND

COMMUNICATION

NETWORKS

A Model for F i n i t e Storage M e s s a g e Switching Networks F" B o r g o n o v o , L. F r a t t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

On C o n s t r a i n e d D i a m e t e r and M e d i u m O p t i m a l Spanning T r e e s F. Maffioli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110

S i m u l a t i o n T e c h n i q u e s f o r the Study of M o d u l a t e d C o m m u n i c a t i o n Channels J. K. Skwirzynski .............................................

118

~ p a p e r not r e c e i v e d

XII

Gestion Optimale Virtuelle E. Gelenbe,

d'un Ordinateur D. Potier,

Multiprogramme

A. Brandwajn,

~ M@moire

J. Lenfant

................

State-Space Approach in Problem-Solving Optimization A. S. Vincentelli, M. Somalvico ................................

132 144

SYSTEMS

ENVIRONMENTAL Perturbation Theory G. I. Marchuk

and the Statement of Inverse Problems ...............................................

159

A Model for the Evaluation of Alternative Policies for Atmospheric Pollutant Source Emissions R. Aguilar, L. F. G. de Cevallos, P. G. de Cos, F. G6mez-Pallete, G. Martfnez Sanchez .........................................

167

Mathematical Modelling of a Nordic Hydrological System, and the Use of a Simplified Run-Off Model in the Stochastic Optimal Control of a Hydroelectrical Power System M. Fjeld, S. L. Meyer, S. Aam ................................

179

A Two-Dimensional C. Chignoli,

203

Model for the Lagoon of Venice R. Rabagliati ....................................

Sea Level Prediction Models for Venice A. Artegiani, A. Giommoni, A. Goldmann, P. Sguazzero, A. Tomasin .................................................

213

Optimal Estuary Control Theory W. Hullett

222

Aeration:

An Application

of Distributed

Parameter

..................................................

Interactive Simulation Program for Water Flood Routing Systems F. Greco, L. Panattoni ....................................... An

Automatic River Planning E. Martino, B. Simeone,

Operating System (ARPOS) T. Toffoli ............................

On the Optimal Control on an Infinite Planning Horizon of Consumption, Pollution, Population, and Natural Resource Use A. Haurie, M.P. Polls, P. Yansouni ...........................

ECONOMIC Limited

231 241

251

MODELS

Role of Entropy in Information Economics J. Marschak .................................................

On a Dual Control Approach to the Pricing Policies of a Trading Specialist M. Aoki .....................................................

264

272

XIII

Problems of Optimal Investments with Finite Lifetime Capital B. Nicoletti, L. Mariani ...................................... Some

Economic Models M. J. H. Mogridge

Utilization of Heuristics J. Christoffersen,

283

of Markets ............................................

295

in Manufacturing Planning and Optimization P. Falster, E. Suonsivu, B. Sv~irdson ..........

303

Economic Simulation of a Small Chemical Plant G. Burgess, G. L. Wells ....................................... An

313

Optimal Growth Model for the Hungarian National Economy I. Ligeti .....................................................

BIOLOGICAL

324

SYSTEMS

On Optimization of Health Care Systems J. If. Milsum ................................................ Theoretical and Operational the Circulatory System B. Abbiati, R. Fumero

Problems

in Driving

, F. M. Monteveechi,

a Physical C. Parrella

335

Model

of

..........

347

Modelling, Simulation, Identification and Optimal Control of Large Biochemical Systems J. P. Kernevez ..............................................

3S7

Mod@lisation du Transfert Gazeux Pulmonaire et Caleul Automatique de la Capacit@ de Diffusion D. Silvie, H. Robin~ C. Boulenguez .............................

366

A Model for the Generation of the Muscle Force During Saccadie Eye Movements A. Cerutti, F. Peterlongo, R. Sehmid ~ ......................... On Some Models of the Muscle Spindle C. Badi, G. Borgonovo, L. Divieti

paper

not received

.............................

378

IDENTIFICATION OF SYSTEMS SUBJECT TO ~OM STATE DISTURBANCE by A.V. Balakrishnan Department of System Science University of California, Los Angeles* ABSTRACT A theory of identification of a class of linear systems --lumped and distributed -in the presence of state or process noise. As a specific application~the problem of of identifying aircraft stability -- control derivatives in turbulence is considered, and results obtained on actual flight data are included. INTRODUCTION The conscious use of mathematical models in system optimization has been growing rapidly in recent years. Perhaps the most spectacular exan~le is the "world model" of Forrester [i]. If it is assumed that a model, before it can be used to predict the future, must first be verified on currently available data, then we are faced with the problem of "system identification" -- of estimating unknown parameters in the model based on obserw~d data. There is a large engineering literature (see [2]) on the subject of system identification and parameter estimation; much of it is however, in the nature of a collection of techniques with little or no precise mathematical framework. Often the authors make ad hoc simplifications for the avowed reason that the otherwise the mathematics involved is too complex. Whereas such a point of view may be harmless when theory plays a secondary role, as in the design of a physical system which can and is finally tested in the laboratory before the design is finalized, the situation is quite different in the identification problem where the end result is a set of numbers and we have really no means of being certain of their accuracy. Surely this would argue for extreme care in the mathematical formulation and analysis, and make the identification problem basically a mathematical and computational one. In this paper we study the problem of identifying a linear system with time continuous description (rather than the discrete-time as in the bulk of the engineering literature), and furthermore taking into account unknown inputs (modelled as random state disturbance), usually omitted for reasons of mathematical complexity. Such a formulation turns out to be actually necessary in the practical problem of estimating afrcraft parameters from flight data in turbulence, see [3]. More specifically, we can state the problem as follows: The observation y(t;~), (in the usual notation, ~ denotes the "sample point" corresponding to the random process) has the form: y(t;~) = s(t;~) + n!(t;~)

(i.i)

where nl(t;~) is the unaw~idable measurement errors modelled as a white Gaussian process (as a reasonable idealization of a band-limited process of band-width large compared to that of the process s(t;~)). The system is assumed to be linear in its response to the (known) input u(t), and to the unknown ("state") disturbance, assumed to be a physically realizable Gaussian process, and hence we can write: t t s(t;~) = ~ o B(t-S) u(s)ds + ~ o F(t-s) n2(s;~)ds

(1.2)

Research supported in part under Grant No. 73-2492, Applied Mathematics Division, AFOSR U.S. Air Force.

where F(.) is determined from the spectral density of the disturbance process. We have omitted initial conditions, as we may, since we shall assume that the system is stable, that in fact: ilB(t) lj + llF(t) ll = 0(exp -kt),

k > 0

(i.3)

and allow for long observation time (only asymptotic properties of estimates are considered). We may also assume that the measurement noise in (i.I) is independent of the disturbance. The identification problem is that of estimating unknown parameters in the functions B(.), and F(.), from the observation y(s,~), 0 s s S T, and the given known input u(s), 0 s s ~ T. Both nl(t,e) and n2(t,~) are white Gaussian with unit spectral density (matrix), and are independent of each other. It should be noted that instrument errors such as calibration error, bias error, timedelays etc., in the observation y(t;~) are supposed to be known and corrected for. Although seemingly simple, this formulation we hasten to point out, includes all known identification (and control) problems for linear systems. For example, the most commonly stated version: ½(t) = A x(t) + B u(t) + F nl(t;~)

(1.4)

y(t) = C x(t) + n2(t;~)

(1.5)

is obviously of this form; in fact we readily see that: F(t) = C e At F B(t) = C e At B and that (1.3) is satisfied if A is stable. The main simplification occurring in this example is of course that fact that the Laplace transforms of F(.) and B(.) are rational. In particular, (1.4) and (1.5) represent a "lumped parameter system," governed by ordinary differential equations. Our model includes also "distributed parameter" systems, governed by partial differential equations. For example, let G be a region in three-space dimensions with boundary F. Then we may consider a structure of the form (we omit the details): 8f = ~ a ~--~

$2f ij ~x i ~x. + n(t,.,~) 3

(1.6)

f(t,~) on F = u(t,.) y(t) = C f(t,.) + nl(t,~)

(1.7)

where C is a linear transformation with finite dimensional range. "Solving" (1.6), we see that (1.7) can again be expressed in the terms (i.i), (1.2). What is important to note is that the formulation (i.I), (1.2) requires only the specification (within unknown parameters) of the functions B(.) and F(.). Yet another example is one in which the state space is finite dimensional but the state noise does not have a rational spectrum. A practical instance of this is the problem of identifying aircraft parameters in turbulence characterized by the Von Karman spectrum. In other words, the Laplace transform of B(.) is rational, but: 2 I o

e2~ift F(t)dt

f4 k (a+f2) 17/6

Although we shall not go into it here, we wish to note that it is also possible to formulate stochastic control problems in terms of (i.i), (1.2). Thus we may seek to minimize T fE[Qs(t;m), o

s(t;~)] dt

subject to constraints on the control n(t), which is required to be adapted to the observation process y(t,~). See [4] for more on this. Estimation Theory Let @ denote the vector of unknown parameters that we need to identify. We shall assume that there is a true value go (in^other words, no "model" error), and that it lies in a known interval, I(0). Let e T denote an estimate of 0o based on y(s), 0 ~ s s T. We shall be interested only in estimates with the following two properties: (i) (ii)

asymptotically unbiased: consistent:

E(~T) + 0o

@T converges in a suitable sense to O ° as T + infinity

We shall show that it is possible to find an estimate with these properties by invoking the classical method of "maximum likelihood." For this we can proceed along one of two [essentially equivalent, as we shall show] points of view concerning the noise process nl(t;~). In the first (the Ito or Wiener process point of view), we write (i.i) in the integrated version: t Y(t;~) = I s(o;~)d~ + Wl(t;e) (2.1) Jo where Wl(t;~) is a Wiener process on C(0,T), the Banach space of continuous functions on [O,T]. Then for each fixed @ (assuming or "given" 0) the process Y(t;~), 0 ~ t ! T with sample functions in C[0,T], induces a measure thereon which is absolutely continuous wit~ respect to the Wiener measure (induced by Wl(t,e)). Moreover we can then calculate the corresponding Radon-Nikodym derivative, which is then our "likelihood" functional, denoted p(Y;O;T), and the maximum likelihood estimate 0T is the value of @ which yields the maximum in 1.0 (or, actually in a subinterval). The ca$culation of the derivative p(Y;O;T) can be accomplished by one of two ways. The first method is to use the Krein factorization technique (see his monograph [5]). Thus let tL re(t;8) = 2, B(t-s) u(s)ds (2.2) % 2

and let R(@;t;s) = E[(s(t;~) - m(t))(s(s;<0) - re(s))*] E denoting expected value. Rf=g;

(2.3)

Consider the operator T

g(t) =

f

R(t;s) f(s)ds

(2.4)

o mapping L2(0,T) into itself. Then R is non-negative definite and trace-class. Let I denote the identity operator. Then, following the procedure invented by Krein [5], we can "factorize" (I+R) -I as: (I+R) -I = ( I - ~ )

(I-~)

(2.5)

whereof is a uniquely determined Volterra operator, such that furthermore, ( ~ + ~ * ) is trace-class (nuclear). The probabilistic interpretation is that the process: t Z(t;~) = Y(t;~) -

f

Jo where

9(s;e)ds

(2.6)

t Y(t;co) = Y(t;co) -

~o m(s;@)ds

(2.7)

t y(t;w) =

L(t;s) dY(s;~),

(2.8)

o L(t;s) being the kernel corresponding to ~ (and of course depends on 9), is a Wiener process, Moreover from this fundamental result, we can readily deduce that (cf [8]) : I T T =

+

m(t) ll

dt - 2 [ ~

[m(t)

-o

(2.9)

+ y(t;0J), dY(t;W)]] where the integral: T f [y(t;~), dY(t~0~)] o

(2.10)

is to be calculated as an Ito integral. This is fine~ except that there is a fundamental practical difficulty in calculating (2.10) in that the observation is never actually such that W!(t;~) is a Wiener process. More accurately, one should model nl(t;~) as a band-limited process, of band large compared with that of s(t;e). On the other hand, if we do this, then we cannot calculate the likelihood functional. Hence we may proceed in the following approximate way: we assume in theory that Wl(t;e) which is band-limited, but of large bandwidth, so that, following [6], and exploiting the fact that ( ~ F + ~ * ) is trace-class, we have the approximations: t ~(t~) ~

f

L ( t ; s ) ( y ( s ; ~ ) - m(s;@))ds o

T

T

7~ Ira(t,@), dY(t;~)] ~

and the crucial approximation [see T

£

[re(t;0), Y(t;~)]dt

[6]J: T

[9(t ;~) , dY(t ;~) ] =

e [y(t;~), y(t;~)]dt - ~1 Tr. ( ~ + ~ )

(2.11)

o

Using these, we can finally write: T

T

1)fily(t;~)+m(t;o)ll2dt-2fo[(m(t;O)

p(Y;@;T) = exp(- ~

+ y(t;~)), y(t;~)] dt + Tr. ( ~ +

f~*)

(2.12)

We shall next (briefly) show that we obtain the same expression for the likelihood functional, using a slightly different point of view, which may be somewhat closer to reality. Thus we take the view that nl(t;~) is of bandwidth sufficiently large that we can assume it to be "white noise"-nl(t;~) for each ~ is an element of L2(0,T) with Gauss measure defined on the (field of) cylinder sets. Such a measure cannot be countably additive on the Borel sets. Denoting the measure by PG, we have that ('~eak distribution ~')

exp i[nl(.~),h]

dPG = exp -

IIhll2

for every h in L2(0,T). Our basic sample point ~ can now be taken to be in the associated product Hilbert space H to accommodate nl(t,~) and n2(t;~) , with Gauss measure thereon. Then we can rewrite (I.i), (1.2) as: y(t;m) = m(t;0) + s(t;~) + G n(t;o~) t

(2.13)

s(t;~) =

fF(t-s)D o

(2.14)

n(t;~) =

Ln(t;w)J

GG* = I ;

DGe = O; DD

n(sl0~)ds

where

Then with ~

= I (Identity matrix in appropriate dimensions)

defined as before, we can readily see that t

(2.15) again defines white noise:

for h in H. Allowing for finitely additive measures, we can again show that the measure induced by y(t;~) is absolutely continuous with respect to the Gauss measure induced by nl(t;~), and that the derivative can be expressed:

+ Tr. ( ~ + ~

.] )

(2.16)

which upon simplification is readily seen to yield the same expression as (2.12). The main feature of (2.16) is that it is the formal direct analogue of what is obtained in the case when the observation consists of a finite number of points.

(See [7]), Next we shall show how to obtain ~ by exploiting the state space representation, again in the generality of non-rational transforms of B(.) and F(.) STATE SPACE REPRESENTATION When the Laplace transforms of B(.), and F(.) are rational we can obtain the (finite-dimensional) state representation (as (1.4), (1.5)) for (i.i), (1.2) by a well-known procedure. Moreover, the determination of c~ using this representation turns out to nothing more than recursive filtering, with the corresponding likelihood functional expressions as given in [8]. To be specific, we shall now assume that: u(t) is p-by-i B(.) is n-by-p F(.) is n-by-m

Then the dimension of n(t;~) in (2.13), (2.14) is (n/m-by-i), D is m-by-n+m, G is n-by-n+m. We shall work with the form (2.13), (2.14). Let ~ d e n o t e the Hilbert space L2(0,~) n of n-by-one functions. In addition to (1.3), we also assume that:

(3.1)

!IB(t) II + lIF(t) II = 0(exp -kt), k > 0 dot representing derivatives. have:

This is probably stronger than we need.

Then we

Theorem Under conditions (1.3) and (3.1), there exist linear transformations B, F, mapping Ep and Em into ~ such that t

X(t ;0~) =

/

T(t-S)(m U(S)~-FD n(s ;~)I as

Jo

-

(3.2)

i

being the generalized solution of : x(t;W) = A x(t;w) + B u(t) + FD n(t;~)

(3.3)

where A is the operator with: domain of A = [fEJ~,

f absolutely continuous and derivative f ' ~ . ~ ] ,

Af = f' and T(t) is the semigroup generated by A. can be expressed:

Moreover (1.2), or equivalently (2o14),

y(t;~) = C x(t;~) + G n(t;~)

(3.4)

where C is a linear bounded transformation mapping j ~ i n t o

E n.

Here B can be taken

as: B u ~ (-L) ~

B(s+nL)u

,

ueE P

O F w ~ (-L)

~

F(s+ne)w

o where L is

a fixed

positive Ch =

gi

,

wEE

m

number, and then C is defined L

by:

/ o h(s)ds ' hg~%o

Proof Although the genesis of this representation was by a long and tedious route, it can be verified quite quickly. First of all, the conditions (1.3) and (3.1) yield the convergence in the norm of ~Y{ of both series: B(s+nL)u

and

o

~ o

Next, it is immediately verified that CT(t)B = B(t)u,

t >- 0

CT(t)F w = F(t)w, t >_ 0

#(s+nL)w

The interested reader will find it useful to rewrite (3.3) as a nonhomogeneous first order partial differential equation. Note that the state space we have obtained need have no relation to the original state space, if any. The state space we have obtained is reduced, and the controllable part coincides with the controllable part of the original state space. See [9]. The controllable part is finite dimensional if and only if the Laplace transforms or B(.) and F(.) are rational. Using this representation, we shall next provide a (constructive, "real-time") characterization of (~. In fact, formally, the final results can be written exactly as in the case where the state space is finite dimensional. The proofs however will be quite different. First let us fix @, and let t y(t;~) = y(t;~) -re(t;0); x(t;~) = x(t;~) - fT(t-s)Bu(s)ds Note first of all that

t

x(t;~) =

fT(t-s)

FD n(s;m)ds

(3.5)

and hence E(]l~(t;~)ll

2) < =

0!

t ST

0-<

s -< t _ <

Let q (t;~) denote the mapping: n(t;w)

mapping ~ into L2(0,t) n.

=

~(s;~),

T

Then we can calculate the conditional expectation

E or, equivalently, the minimizing (linear bounded) operator

- L

2)

over the class of linear bounded transformations L mapping L 2 ( 0 , t ) i n t o (i0) it is known that the minimizing operator is given by

;YC.

From

Let Rx(t;s) = E(x(t;60)~(s;~)*) Then we can see that:

(3.6)

Lo(t) = Rl2(t) (I + R22 (t)) -I where

t Rl2(t)f = h;

h =

f O Rx(t;s) C* f(s)ds t

R22(t)f = g;

g(s) =

C Rx(S;~ ) C* f(o)d~, 0 < s < t.

Note furthermore that Rx(t;s) = T(t-s) Rx(S;S)

for

s < t

(3.7)

We can draw all the results we need from these. ~(t;~)

- c ~(t;~)

First let us define:

= Zo(t;~)

(3.8)

and let (3.9)

so that in particular we have: R22(t) Using (3.9)

in

(3.8)we

- R22(t)

G(t) = G(t) = R22(t) - G(t) R22(t)

have: t

Zo(t;m) = Y(t;~) -

t

(3o10)

t

f o R22(t;s) Y(S;~)ds + f o R22(t;s)ds JoPG(t;s;~),(~;~)d~

where we denote the kernel corresponding to G(t) by G(t;s;~) , and exploiting (3.10) we h=ve: t Zo(t;~) = y(t;~) -

jofG(t;t;s) y(s;~)ds

and comparing with the Krein factorization (Appendix II in [8], for example), we obtain that t t j~O # G(t;t;s) y(s;~0)ds = C x(t;~) =

f

L(t;S) y(s;~)ds

o

with L(t;s) given by (2.8), nhus characterizing ~ , and hence also: z ° (t ;~)

=

z (t

;~)

Next let us express x(t;~) in terms of z(s;~)~ 0 -< s _< t. For this, let us note that we may regard • as mapping L2(0,t) into itself (being a Volterra operator) and interpreting the adjoint ( ~ t) correspondingly, we have: (i - ~£t)

(I -.~)

=

I+R22(t)

and using this we can write: x(t;co) = a x ( t ; s )

c

(s;~) -

G(c~;s;c0 z ( e ; w ) d

where we have used the fact that G(O;~;s)* = G(o;s;o). =

ft

x(t;s) C *z(s;~)ds

o

ds

We can rewrite this as:

- foe (f~ Rx(t;s) C * G(o;s;O)d sl z(O;~0)do i (3.12)

From: x(t;00) = it follows that~

jo Rx(t;s)C *

(s;00)-

(3.tl)

G(t;s;O) y(O;00)d ds

t Rx(t;s)C*CKx(S;t)ds -

=

o t

t

fRx(t;s)C*G(t;s;~)ds o

fCRx(~;t)d~ o

(3.13)

Now using the right side of (3.10) we have: t CRx(S;C~)C

*

-

G(t;s;T) CRx(T;~)C dT

fo

*

= G(t ;s;~)

and hence: t G(t;s;t) = CRx(s;t)C

-

f

G(t;s;T) CRx(T ;t)C dT

o

Substituting this we have: t oRx

t

t ;s)C* G(t;sl;t)ds =

Rx(t;s)C* CRx(S;t)C ds o t

t

- /Rx(t;s)

/ o C ~ G ( t ; s ; T ) C R ( T ; t ) C dT ds

o

= E(i(t~) ~(t;~))c* using (3.13).

(3.14)

But letting

P(t) = E(IE~(t~)

x(t~ )112)

we h a v e f r o m ( 3 . 1 2 ) , u s i n g ( 3 . 1 4 ) and ( 3 . 7 ) t h a t t

x(t;~) = / T ( t - s ) o

P(s)C* z(s;~)ds

(3.15)

which is the expression in terms of z(. ;~0) we sought. It should be noted that the line of reasoning for this proof is quite different from the usual (e.g. [8]) in that no Martingale theory is invoked. Now (3.15) in turn yields that t E(x(t;~) x(t;~)*) =

/ o T(t-s)P(s)C~CP(s)T(t-s)~ds

and hence we have that: t

t

P(t) = / o T(t-s)F F * T(t-s) ~ d s - / T(t-s)P (s)C* CP(s)T(t-s) ds ~o and hence for x,y in the domain of A

we have:

[P(t)x,y] = [e(t)A*x,y] + [P(t)x, A y] + [FF x,y] - [e(t)C CP(t)x,y] with P(0) = 0.

Also from:

(3.16)

10

t o we have that, denoting by K the operator: t Kf = g;

CT(t-s)P(s)C

O~

f(s)ds = g(t),

t
mapping L2(0,T) n into itself, C~(.;m) = (! - K)-IK y(.;~)

(3.17)

where K is a Volterra operator of Hilbert-Schmidt type. The operator P(s) is clearly non-negative definite and nuclear. From (3.17) we have that

~f=

(I - K)-IK

and hence

T Trace ( ~ +

2*)

= Tr. (K + K*) = 2

/o CP(s)C*ds

(3.18)

Finally, we can also see that ~(t;~) is the generalized solution of ~(t;~) - A ~(t;~) = P(t)C*(y(t;~) being the recursive form sought.

(3.19)

- C~(t;~))

Finally we note that we can write:

(3.20)

z(t;~) = y(t;~) - C x(t;~) where t X(t;~) = x(t;e) +

/ O T(t-s) B u(s)ds

and x(t;0~) is readily verified to be the generalized solution of ~(t;c~) -A ~ ( t ; ~ )

= P(t)C

y(t;c0)

+ Bu(t)

- C~(t;~)

(3.21)

We h a v e t h u s o b t a i n e d t h e q u a n t i t i e s i n ( 2 . 1 6 ) a s s o l u t i o n s o f d i f f e r e n t i a l equations, solvable in real time. Thus ( 2 . 1 6 ) c a n h e w r i t t e n , u s i n g ( 3 . 2 0 ) , ( 3 . 2 1 ) , and ( 3 . 7 ) :

p(Y;0;T) = exp - ~

[Ci(t;~), Ci(t;~)]dt - 2

[y(t;~), Ci(t;~)ldt T

+ 2 Tr.

f

CP(t)C*dt

(3.22)

Jo COMPUTATION OF THE ESTIMATE Let us next examine the computation of the estimate of the unknown parameter vector. First it is necessary to study some properties of the functional defined by (2.9), or, in the form in which we shall actually use it, (3.22). We shall use p(y;e;T) to indicate the functional generally, and p(Y;0;T) to indicate the value when we use the actual observation, in which case it will be called the "likelihood functional." Let ~ denote the sample space L2(0,T)n, and PG the Gauss measure thereon. When we have

11

~

p(y;8;T) dPG = i

so that using q(y;e;T) = Log p(y;8;T) (so that q(Y;0;T) is the log-likelihood functional), we have ~

(V0q(Y;8;T))

P(Y;8;T) dPG = 0

(since the dependence on 8 is sufficiently smooth).

In particular then:

E(V 8 q(Y;0o;T)) = 0 Similarly (analogous to the familiar classical arguments): 22

38. 30, 3 i

q(y;0;T) p(y;0;T) dPG

= - ~1%

q(Y;8;T)) ( ~ q ( Y ; 0 ; T ) )

where {8 i} denote the components of 8.

These relations

P(Y;8;T)dPG

In particular then:

are important to us, because our technique consists

in miximizing

the log likelihood functional in the known interval in which 8o lies; or, more accurately, we seek a root of the gradient of the log likelihood functional: V e q(Y;8;T) = 0

(4.2)

using in particular the Newton-Raphson technique, hut avoiding the use of second derivatives, thanks basically to (4.1). First we note that, from (3.22), we have: 30i3 q(8;g;T) = (-i)

y(t;~) - Ci(t;w), ~--~C~(t;~)l

dt

T + Tr.

~ o

CP(t)C*dt

(4.3)

i

Let M(O;T) denote the matrix with components mij defined by: T mij (8 ;T) =

3 c~(t;0) ~c£(t;0)

dt

(4.4)

Then our basic "identifiability condition" is that M(0 o) = Limit (l/T) E(M(0o;T)) (4.5) T÷~ be positive definite; to check this, in practice, since 8o is unknown, we must verify that this is satisfied for every 8 in the interval in which 8o lies. (The

12

positive definiteness of M(@ o) assures it in a sufficiently small neighborhood of Oo). Our basic algorithm then is: @n+l = @n + (M(en ;T) /T )-I (v0 q(Y;en;T)/T) The asserted positive definiteness also assures the positive definiteness of M(@n;T)/T for all large enough T in a sufficiently small neighborhood of @o" these aspects see [8]° Let us also note here that E

~

$

(4.6)

For

(4.7)

which is most easily verified by using the Ito-integral version of the likelihood functional, (2.9). Assuming that T !im

~

[u(t), u(t+s) ldt

T-+co

exists and is continuous in s, we can proceed to evaluate M(0) as in [8]. Reference may also be made to [8] for the asymptotic (ergodic) properties of unbiasedness and consistency, for the case where the system is finite dimensional (Laplace transforms of F(.) and B(.) are rational). The general case considered here can be treated in similar fashion, although space does not permit inclusion here. EXPERIENCE ON ACTUAL DATA We turn now to indicate briefly the results obtained using our computational technique on actual (as opposed to simulated) data. The dynamic system considered arises from the longitudinal mode perturbation equations for an aircraft in windgust (turbulence) (Rediess-Iliff-Taylor, see [3]). This is a system where only a "rational" approximation of the spectrum of turbulence is used, so that the total system is finite dimensional. Leaving the many essential details to the comprehensive work of lliff [Ii], the state space formulation of the problem is as follows: (see also [3]): x(t) = A x(t) + B u(t) + F n2(t) V(t) = C x(t) + D u(t) + G nl(t) where nl(.), and n2(.) are independent white Gaussian, and the matrices in the equations have the form:

A =

Z1

0

i

Z1

0

0

i

0

MI

0

M3

MI

0

0

0

i000

[z4] ~01 F = d

LO J 20vJ

13

0

0

1

0

0

1

0

0

IOM 3

lOMl-VZ I 0

C =

IOMI-~Z I

g

g

32k 1 kI

k1

0

0 D =

I° ] 10M4-vZ4 g 0

g = acceleration G = diag.

[.0005,

due to gravity .0001,

.01, .00001]

The lettered entries are unknown, except for ~, which is known the turbulence power is an unknown parameter.

(=1670).

Note that

Figure 1 shows the complete time history of the observation v(t) (four components) subdivided into various regions for later identification, as well as the input timehistory. Estimates were computed over the various subregions each by three methods: Method I:

Method II: Method III:

Neglecting the measurement noise on the angle of attack measurement (v4) and following the corresponding maximal likelihood technique developed in [3]. This is reasonable for this particular example at high turbulence levels. This is the method developed herein,

see also

[7].

This was a "check" method, in which the turbulence was ignored completely in the model.

The results are summarized in Figure 2. Sample means and variances of the estimates obtained over the different data-regions are shown, along with the wind-tunnel values as well estimates obtained on other turbulence-free (smooth air) flights. It can be seen that Method II yields the most consistent estimates. It also turns out that Method II is the least in computational time -- the estimates converging in fewer iterations. It can also be seen that ignoring the turbulence leads to the worst results. For more discussion see [ii]. The remaining figures indicate the nature of the "fit" obtained using the estimated coefficients to the observed data. Figure 3 shows the close agreement provided by Method II. Figures 4 and 5 indicate how much worse the agre~nent is on the same stretch of data if the turbulence is not accounted for.

14

v~

O.00RAD/SEC -0. ~0-

0o~07 V2 O.

00-~

RAD -n_~n-

v3 g's

v. RAD

RAD

Figure 1.

Total Jetstar Turbulence Time History Showing Intervals of Each Maneuver

15

O

SMOOTH A I R

[]

METHOD I

A

METHOD III

METHOD II

-2

Z1

f ii

tx WIND TUNNEL PREDICTION

r~

-1

0

i

i

i

i

i

i

i

i

i

i

I

i

i

i

i

i

i

I

I

-20

M1

~10

0 -4

M3

-2

0 -.3

Z4

T

- ,1

I

.1

I

I

.J-

-20

M4

-10

I MOOTH A I R

I METHOD 1

I METHOD I1

METHOD I11 WIND T U N N E L PREDICTION

Figure 2.

Means and Standard Deviations for Five Methods of Estimating Coefficients

16

v~

RAD/SEC

%

:

g'$

!

ii

v48 o," RAD

,

l

]

~J______ '0

I

I

i

I

I

10

20

30

40

50

T, SEC Figure 3.

Comparison of Flight Data from Maneuver ABCD and the Estimated Data -- Method II

17

v~ B RAD/SEC

v2

RAD

v3

g's q..

v,

RAD

~

I

I

I

t

t

I

,,l_,

2

4

6

8

10

12

14

T, SEC Figure 4.

Comparisonof Flight Data from Maneuver E and Estimated Data - Method II

18

- - 0 ,

~I

FLIGHT ESTIMATED

.................

DEG/SEC

v~ 8! d E)EG

8T

.,.1 v3

T o

g's 8

v4

r

8 u

t

~1_

I

0

2

,

L

I

4

6

I

I

I

I

8

10

12

14

T, SEC Figure 5.

Comparison of Flight Data from Maneuver E and the Estimated Data Obtained by Method III

19

REFERENCES i.

Forrester, J.W.:

"World Dynamics," Wright-Allen Press, 1971.

2.

Proceedings of the Second IFAC Symposium on Identification and Process Parameter Estimation, Prague, Czechoslovakia, June 1970.

3.

Balakrishnan, A.V.: Identification and Adaptive Control: An Application to Flight Control Systems, Journal of Optimization Theory and Applications, March 1972.

4.

Balakrishnan, A.V.: Identification and Adaptive Control of Non-dynamic Systems, Proceedings of the Third IFAC Symposium on Identification and Process Parameter Estimation, June 1973.

5.

Gohberg, I., and Krein, M°G°:

6.

Balakrishnan, A.V.: On the Approximation of Ito Integrals using Band-limited Processes (to be published, SIAM Journal on Control, February 1974).

7.

Balakrishnan, A.V.: Modelling and Identification Theory: A Flight Control Application, in "Theory and Applications of Variable Structure Systems," Academic Press, 1972.

8.

Balakrishnan, A.V.: Stochastic Differential Systems, Vol. 84, Lecture Notes on Economics and Mathematical Systems, Springer-Verlag, 1973.

9.

Balakrishnan, A.V.: and Signal Theory,"

Volterra Operators, AMS Translations, 1970.

"System Theory and Stochastic Optimization," in "Network Peter Peregrinns Ltd., 1972.

i0.

Balakrishnan, A.V.: Introduction to Optimization Theory in a Hilbert Space, Lecture Notes on Operations Research and Mathematical Systems, Vol. 42, Springer Verlag, 1971.

ii°

Iliff, K.W.: Identification and Stochastic Control, An Application to Flight Control in Turbulence, UCLA Ph.D Dissertation, 1973.

ADAPTIVE COMPARTIMENTAL STRUCTURES IN BIOLOGY AND SOCIETY

R.R. Mohler and W.D. Smith

INTRODUCTION Many dynamical processes~ which are relevant to man, his biochemistry and the society of which he is a part, are modelled conveniently as adaptive control systems with appropriate compartmental structures. It is assumed here that the systems of concern may be described or accurately approximated by precise mathematical relationships. Such equations are derived generally from established physical laws, intuition, or experimental data. If this is not the case, it may be possible to use a linguistic approach such as suggested by Zadeh for the development of a parallel modeling and identification theory. Even with the assumption that the system may be quantified conventionally, hoE ever, it is not possible to say that the necessary theory of modelling and identification has been developed. In more cases than not, linear system models, despite their popularity in the literature, fail because of their rigid structure. Still, as is shown here, certain results from linear system theory may be utilized some ~imes to derive effective models of complex adaptive processes. This convenience is due to the linear-in-state structure of the bilinear model which is suggested in this paper. Some bilinear system theory and its application to dynamic modeling has been developed and much of this work is summarized by Mohler. The proceedings of the first U.S.- Italy Seminar on Variable Structure Systems, which was an outgrowth of common bilinear system research, further establishes the base for modeling and identification as they relate to adaptive processes in general ISee Mohler and Ruberti I• Biological and societal processes, like those generally found throughout nature, are distributed as well as adaptive. Still such processes are usually ameanable to discretization according to compartments. For example, biological compartments may be formed for anatomical convenience, and societal compartments according to ge! graphical and governmental convenience. Suppose a single substance X is distributed among n compartments between which it diffuses naturally at a rate described by co~ Servation equations of the form

R.R. Mohler is with the Department of Electrical and Computer Engineering Oregon State University, Corvallis,Oregon 97331.USA.(P~esently visiting University of Rome) W.D. Smith is with the University of New Mexico Medical School, Albuquerque, New Mexico 87106. USA.

The research reported here is supported by NSF Grant GK 33249

21

dx. dti =

n.

n

~ ~'" i ~ki + ~ia - ~ai + Pi - di lJ - k ~j=l

,

(I)

i = l,...,n, where x. is the concentration of substance X in the ith compartment, ~i" is the flux of X from the jth to the ith compartment, #. is the flux of X from theJenvironment la of uniform concentration x a to the ith compartment, Pi is the rate of production and d i is the rate of destruction of X in the ith compartment. The primed summation denotes deletion of the ith term. Compartmental volumes are assumed constant. The flux to the it:h compartment from the jth compartment is described by ~ij = Pij xj

,

(2)

where i, j=l,...,n,aandPij is an exchange parameter which may be constant or may be a multiplicative control. Additive control may be synthesized through the net produc tion of substance (Pi-di) which is zero for a eonversative system. (For brevlty,any n o n m a nipulative terms have been neglected). It is readily seen from Eqs. (i) and (2) that a collection of terms leads to a bilinear system of the followin~ form: dx dt - Ax +

m

~ Bk u k x + Cu k=l

,

(3)

where xcRn is the state vector; ueRTM is the control vector; A, Bk and C are nxn, nxn and nxm matrices respectively. Again, x is composed of compartmental quantities of substance X; additive control may arise from net production, and multiplicative control from those exchange parameters which are manipulated. While, additive and multi plicative controls are independent variables for the type of process considered here, in general they might not be. Therefore, it is assumed here that zeros appear in the appropriate positions of Bk and C to maintain independent additive and multiplicative control variables where necessary. Those exchange parameters which can not be manipulated result in constant coefficients of the A matrix. Here, it is assumed that compartmental capacities do not change significantly. While the process may include transport of more than one substance,no generality is lost in utilizing the above form of equations. In the human body such processes are controlled by what is called homeostasis.

BIOLOGICAL PROCESSES A single cell may represent a convenient compartment for modeling a more compl! cated organism. Nutrients are generated for cell growth and cell division by biochemical processes within the cell itself and by transfer across the cell membrane. Enzymes effect control by manipulation of membrane permeability and by triggering the formation of proteins and nucleic acids and their precursors. In addition to the natural diffusion from compartments of high concentration to those of low concentration, there exists an active transport from low concentration to high concentration with the necessary energy supplied by cellular metabolism. Such active transport as derived by Hill and Kedem is described by a nonlinear function (division of summations) of compartmental densities. The process may be approxima-

22

ted, however, by a bilinear equation ISee Mohler I . The development considered here has assumed that the process involves mass tran sport. Again the concept is more general and in physiology alone includes numerous processes such as body fluid and electrolyte balance ISee Alvil,thermal regulation IMilsuml, kinetics of material ingested or excreted from the body ISnyderl, control of carbon dioxide in the lungs IGrodinsl as well as kinetics of metabolities in cell suspensions ISheppard and Householder 1

Tracer experimentation For biological processes it is usually impossible to measure the necessary quantities directly. Frequently, it is possible, however, to estimate compartmental system parameters by observing the behavior of a tracer, such as a radioisotope or a dye, which has been injected or ingested into the system ISheppard and Householder I • If it is assumed that the tracer in every compartment is distributed uniformly through substance X, that labeled and unlabeled substances behave identically, and that the tracer does not appreciably affect the compartmental system behavior, then the tracer system with specific compartmental activity ai(t) is described by n

d(xiai) at

n ak - [ ~ (t) a i + ~ (t)a. (t) ~ " ~ik (t) ~qi ia la k=l q=! -~

o al

i

i

a. + f.(t) 1

,

<4)

I

i = 1,...,n where ai(t) is specific activity of tracer in the ith compartment, aid(t) is the sp~ cific activity of substance in influx ~ia(t), and fi(t) is the influx of tracer which is inserted directly into compartment i. Normally, tracer experiments are conducted with the compartmental process in "equilibrium" (+), and it is further assumed that the process is closed, ~ai =~ia =0, and conservative, Pi =di = O, i=l~...,n. Let tracer be inserted directly into p compartments with its behavior measured in q compartments. Then, with the above assumptions da -=-d t and

-I S a+X d P f

w = Qa

(5)

,

(6)

where asR n is the specific activity vector; X d is the diagonal matrix of elements xi,...,Xn; P is an input matrix with unity elements in each row corresponding to a compartment which is accessible for tracer insertion and with zero elements elsewhere; f is the p-vector of inserted total activity fluxes, and therefore Xd IP f is the vector of inserted specific activity fluxes; w is the output q-vector of observed compartmental specific activities. Also, S = [sij] with sij = ~ij/xi and

,

j =l,...,n, i # j

n sii = _ i__ x i k~= I ~ik °

(+) I.e., with x i = $ i j = S a i = $ i a = P i =di =0; i,j = I ..... n,i#j, but possible net flux between compartments and with environment.

23

If the tracer enters the system by natural flux routes from p separate external sources of labeled stubstance, each with its own tracer specific activity ai, i=l,...,p,

then d_~a = Sa+ dt

and

w = Qa .

Xd I ~ad P a

(7)

(S)

Here ~ad is a diagonal matrix of elements ~ia "'" ~na' ~ is a p-dimensional tracer source activity vector, and P is the n×p output matrix with a single unity in each row, i, for which aia # O. In practice, the subject might be placed in a radioactive bath, and its absorption of tracer monitored in a so-called "soak up" experiment. In "wash out" experiments, the subject is saturated to a given specific activity of tracer, and wash out is observed in a tracer free environment. In either case, the presence of all compart_ ments can not be detected unless the tracer system, Eqs. (7) and (8), is completely controllable and completely observable. Obviously, the tracer system is linear, and may be analyzed by classical linear system theory. For example, the conventional rank tests for controllability and observability may be applied to either tracer system to see that sufficient accessibility is available to identify the minimum realization of the system. For the time va riant case, rank tests may still be applied to two convenient time-variant test matrices to show that the model is a minimal realization ISee D'Angelo I . Standard procedures may then be used to synthesize the minimal realization from the input-output observations. These, inturn, dictate the necessary tracer experiments.

Adaptation and Control Cells, perhaps the most basic of biological compartments, multiply in a manner similar to that of populations of biological species IMohlerl. This process of cellu lar fission also may be likened to that of nuclear fission and the generation of radioactive particles IMohler and Shen I. Cells may multiply in a controlled manner to synthesize an organ or an organism in the evolution of life which must adapt to its environment. Just as the cellular processes are regulated in the body by homeostasis, so are the combinations of these cells working together to perform integrated specific tasks in the functioning of a living organism or human being. In the regulation of body temperature, for example, heat (additive control) is produced internally by basal metabolism and heat is lost by means of respiration,excretion, evaporation from skin and radiation from skin. Transfer of heat between com partments (such as skin, muscle and core of various internal organs) is regulated by means of vasomotor control of blood vessels which effectively alters conductivity of heat transfer as a bilinear control mechanism. The control policy in this case is established by the hypothalamus in the brain based on temperature feedback information. It is convenient to examine a two-compartment model of thermoregulation since its behavior can be studied in a state plane. Skin and core are the compartments selected since temperatures in these compartments are most meaningfult to system behavior and they are monitored by the central nervous system for regulation of temperature. ISee Mohler I

Water Balance Water, comprising on the average about sixty percent of the body weight, must be carefully regulated as part of the overall body process. About two-thirds of body wa

24

ter is located within the cells, and the remainder is in the so-called extracellular compartment. The extracellular compartment or intravasculer compartment includes plasma, the interstitual compartment in which tissue cells are bathed and the transcellular compartment. In some cases there is a distinct anatomical boundary, such as wall membrane, but for the interstitual compartment such boundaries are not so definitely defined. The transcellular compartment, which includes digestive secretions, is separated from plasma by a continuous layer of epithelial cells and capillary men brane. Tracer experiments are used to obtain the distribution of water throughout these compartments. The intracellular compartment can not be so isolated, and must be accounted for by taking differences. Drinking, eating, skin absorption and respiratory absorption are all mechanisms by which water enters the body's extracellular compartment. Oxidative water, a metabolic end product, varies with diet and metabolic rate, and becomes an influx to the intracellular compartment. Effluxes from the extracellular compartment to the enviro~ ment include urine, fecal water, skin evaporation and respiration. Water balance is regulated by homeostasis through pressure forces and membrane permeabilities. Feedback control again is established through the hypothalamus. An amount of water equal to two-thirds of the blood volume is exchanged between the blood and the extravascular fluid each minute. The kidney with its thousands of filtration passages called nephrons is the key component in body water balance. It processes blood plasma through its nephrons to produce urine. The water loss in each nephron is regulated by the action of the anti diuretic hormone (ADH) on the tubule and the collecting duct permeability IPittsl. Glomerular filtration rate(GFR)~which is collected at the arterial end of the nephron, and the renal plasma flow (RPF) may be regulated to control water loss. RPF is a function of arteriolar resistance in the kidney which is controlled by constrictor muscles, again parametric control. While details of the process are quite complicated, water excretion in urine may be approximated by a bilinear system with efflux of form ~u = Uk (c g +bgWb)

,

(9)

where control uk is a function of permeability (determined by ADH), arteriolar resi~ tance and osmotic pressure driving water from renal tubules; Wb, body water, is asum of compartmental amounts of water which in turn may be state variables. Cg , bg const. Though to a much lesser degree, water balance is regulated by parametric control through the skin and through the lungs. Vasomotor control of blood flow and hormone regulated permeability control water losses. Vasomotor constriction reduces hydrostatic pressure and thus reduces water transfer to the skin, and also by dropping skin temperature reduces avaporation. In the lungs the net loss is regulated parametrically by ventilation rate and vasomotor control of respiratory blood vessels. Normally, the skin and the lungs play a much lesser role in controlling water balance of mammals than do the kidneys. But under adverse conditions they can be significant. Water loss through skin is extremely complicated and still not very well understood. It is transported in three stages: exchange between the blood and the dermal connective tissue, exchange between the dermis and the epidermis, and finally excha~ ge between the epidermis and the surroundings. The exchange takes place by filtration and diffusion~ The force which induces capillary filtrarion is described by the Starling Hypothesis° This hypothesis states that the net force driving water out of a capillary is equal to the hydrostatic pressure difference minus the osmotic pressure difference with both differences taken between the blood and tissue IRuch and

25

Patton, p. 6251 . This results in a net outward force in the capillaries at the arterioral end and a net im~ard force at the venous end. Water diffusion across capillary walls is regulated by an effective permeability, and it further diffuses through layers of skin fibers and through duct openings. Then, water loss between skin and surrounding air is usually assumed to be proportional to the difference between saturated vapor pressure at skin temperature and the vapor pressure of the air IChew, p. 881 • Consequently, it can be shown that net water loss through the skin is approximately described during equilibrium operation by ~sa = Ps(Wd-We)

= ks(WePse-

rp

sa

) ,

where Ps is effective permeability, w d is dermis water concentration, w e is epidermis water concentration, Pse is saturated vapor pressure of skin, Psa is saturated vapor pressure of air, and r is relative humidity. Again, the role of parametric adaptive control is apparent. This skin water loss may then be further related to mo re basic mammalian control functions as follows: Ps (t)Ywb(t) + ksrPsa -rp

p (t) s ~sa = ks

+k ks3 ks2

( i + Wb(t )

sa (I0)

s

ksl)Us(t)

where Wb(t ) is amount of body water, us(t ) is portion of blood flowing to skin, and ksi (i = 1,2,3) are constants. Here, a bilinear model could be used as an approximation, again more accurate than a linear model. A similarly complex relationship may be derived for parametric control of water loss through the lungs. In the lungs, it can be shown that ventilation rate is a parametric control variable. Some animals can considerably alter their body water by control of breathing. For convenience, tracer experuments were conducted on a wild house mouse in a controlled environment in order to obtain a simple model for its water balance. A simplified single compartment which is used merely to demonstrate the procedure is shown in fig. io Obviously, a more meaningful model would include more compartments Such as for intracellular water, extracellular water and storage water. Equilibrium acclimation graphs averaged over several animals of drinking influx, food-water influx, urine efflux and net exchange by skin and lungs are provided by fig. 2 IHainesl. The solid curve in fig. 2 (a) connects experimental values of drinking water influxes given at various levels of water deprivation. The dashed curve connects the calculated values of water influx from food. It is found that this influx for the unconstrained mouse is nearly proportional to body mass. Urinary and fecal daily water losses are charted by dash-dot curves on fig. 2 (b). The solid curve joins experimental data for net evaporative water effluxes at different drinking levels. As drinking water is reduced, it shows that urinary-fecal and evaporative losses are decreased. Fig. 2 (b) further breaks down evaporative losses through skin (short dashes) and through evaporation (long dashes). A typical long-term tracer wash out record for an equilibrium (ad libitum) animal is shown in fig. 3. After tritiated water injection, periodic samples of body water were analyzed for radioactivity. Note that the radioactive decay of tritium (half life of 12.5 years) has negligible effect on the experiment. Obviously, the de

26

cline can be closely fitted by a single negative exponential over the time interval shown. From the single compartmental equation for body water change dw b

d--~ = ~in - #out

(ii)

and the specific tracer activity change da

~in

dt

wb

a

,

(12)

it is seen that in equilibrium the negative slope of the tracer experiment graph, fig. 3, in magnitude is inversely proportional to the quantity of body water. If the specific activity plotted in fig. 3, were plotted from separate experimental data for evaporate and plasma specific activities, their graphs would have the same slope as shown in fig. 3, but each would have its own level. This can be ex plained by a two compartment model of: (I) internal water in equilibrium with blood plasma, and (2) extravaseular fluid in skin and lungs affected by evaporative exchan ge. SOCIETAL PROCESSES There have been numerous special journal issues, books, conferences and comprehensive reports devoted to such major areas of society as urban dynamics, public sy~ tems, health care, transportation and economics. Economics, with Keynes as a pioneer researcher, probably has received most of the concentration. Again an adaptive stru~ ture, such as that of the bilinear system, is necessary to satisfy well-developed i~ tuitive concepts which in many cases have been substantiated experimentally. As noted by Runyan, gross simplifications to a very complicated system, in many instances, has hindered rather than helped interdisciplinary work in economics. The adaptive bilinear structure offers a significant improvement in this respect, and parametric control by economic investment is a good example of its use. For population transfer models, arrival rates of different classes are equal to appropriate population class terms multiplied by corresponding migration attractiveness multipliers. Again~ to yield a bilinear model. In socioeconomics, adaptive control may be synthesized through such quantities as legislation, land zoning, taxes and capital investment.

Transportation Systems Transportation represents one particularly interesting socioeconomic process. It may be desired to develop a model in order to study the transportation needs of a na tion, state or region. A significant part of this study might include an analysis of transportation effectiveness to regulate socioeconomic development. It is apparent from historical developments of railroad lines, freeways, air service and shipping that industrial development and socioeconomic development in general follow the tran sportation networks° A systems overview of the transportation process is shown by fig. 4~ This block diagram shows the major interactions and feedback loops of demography, economics, transportation planning and transportation network dynamics. The simplicity of this functional diagram is somewhat deceiving, since each block represents a major system in its own right, numerous interconnections and feedback loops. Fortunately, this study need only concern itself with transportation re-

27

lated socioeconomic and environmental processes. While land-use and pollution are not shown as separate subsystems in fig. 4, they are inherently part of the transportation planning/controller and the transportation dynamics subsystems. Land-use planning plays a most significant role in the generation of transportation policies which are shown as inputs to the network dynamics. Th~ transportation "plant" dynamics, which includes people and commodity multimodal transportation, is composed of three major components, viz., trip generation, distribution and assignment (including modal choice). The trip generation portion establishes a relationship between trip end-volume and such socioeconomic factors as land use, demography and economics. Demographic in formation utilized here includes total population, school enrollment and number of household units in a zone of interest. Useful economic data includes total employment, average income, automobile ownership and retail sales. Hosford in a Missouri study showed that only a few basic factors are necessary to estimate trip generation with an effective trip attraction factor entering the process as a parametric control. The trip-distribution model assigns trips from the generation model to competing destination zones or compartments. Three widely used adaptive models are: (I) the Fratar model, (2) the intervening opportunities model, and (3) the gravity model IHeanne and Pyers I. The Fratar model expands existing highway travel patterns by a growth factor for each origin and destination zone. The future trips are then loaded on to transportation networks according to some assignment technique based on travel time, convenien ce, etc. The intervening opportunities model is an exponential probability model based on the simple relation that a trip will terminate in a region or compartment is equal to the probability that this region contains an acceptable destination, times the probability that an acceptable destination closer to the origin of the trip has not been found. The gravity model, perhaps the most popular trip distribution model, is analogous to Newton's gravitational law of masses. Consequently, the traffic between zones or compartments is proportional to the product of the compartmental populations and inversely proportional to the square of the distance between the compar~ ments of interest. Frequently, the gravity model is further adapted to a particular system by means of some parametric multiplier or attraction factor. The traffic-assignment model allocates trip interchanges to a specific transpo~ tation system. This includes modal selection as well as network assignment. Most traffic assignment models are based on a desired quality of minimum path length. But more realistic models should include convenience, comfort, cost, safety, etc. Greenshields derived a mathematical expression for such a quality of traffic flow. Transportation models of this kind may be used to formulate development plans for a statewide system which must be integrated with similar national models. Like the national model which is an expansion of the Northeast Corridor study, statewide studies rmast include well formed goals, diverse socioeconomic systems, numerous tran sportation modes and geometrical nodes and different concepts of quality of transpo! tation as they relate to an overall quality of life. Five state-wide transportation studies of a superficial nature are compared by Hazen. REFERENCES Alvi, Z.M., "Predictive Aspects of Monitored Medical Data", Ph.D. dissertation, School of Engineering, University of California, Los Angeles, 1968. Chew, R.M., "Water Metabolism of Mammals", Physiological Mammalogy, vol. II (Eds. W.V.Mayer and R.G.VanGelder), Academic Press, New York, 1965.

28

D'Angelo, H., Linear Time-Varying Systems: Analysis and Synthesis, Allyn and Bacon, Boston, 1970. Greenshields, B.D., ~The Measurement of Highway Traffic Performance", Traffic Engineer, 39, 26-30, 1969. Grodins, F.S~, Control Theory and Biological Systems, Columbia Press, New York,1963. Haines, H., Personal Communication, University of Oklahoma. Hazen, P.Io, T'A Comparative Analysis of Statewide Transportation Studies", Highway Research Record, 401, 39-54, 1972o Heanne, K.E. and Pyers~ C.E., "A Compartative Evaluation of Trip Distribution Proce dures s', Highway Research Record, 114, 20-50, 1965. Hill, T.Lo and Kedem, 0., "Studies in Irreversible Thermodynamics III. Models for Steady-state and Active Transport across Membranes". J. Theoret. Biology IO, 399-441, 1966. Hosford, J.E., "Development of a Statewide Traffic Model for the State of Missouri", Final Report of Project No. 2794-P, Midwest Research Institute, Kansas Cit~ Missouri, 1966. Keynes, J.M.~ General Theory of Employment, Interest and Money, Harcourt, Brace and Co., New York, 1935. Mohler, R.R., Bilinear Control Processes with Applications to Engineering, Ecology and Medicine, Academic Press, New York, to be published. Moh!er, R.R. and Ruberti, A., Eds., Theory and Application of Variable Structure Sy~ tems, Academic Press, New York, 19720 Mohler, R.R. and Shen, C+N., Optimal Control of Nuclear Reactors, Academic Press, New York+ 1970. Milsum, J.H., Biological Control Systems Analysis, McGraw Hill, New York, 1966. Pitts, R.F+, Physiology of the Kidney and Body Fluids, Year Book Medical Publishers, Chicago, 1963. Ruch, T.C. and Patton, H.D+, Physiology and Biophysics 19th edition, W.B. Saunders, Philadelphia, 1965. Runyan, H.M., "Cybernetics of Economic Systems +', IEEE Transactions Systems, Man and Cybernetics, SMC-I, 8-17, 1971. Snyder, W.S., et alo, "Urinary Excretion of Tritium following Exposure of Man to HTO-a two Exponential Model", Phys. Med. Biology 13, 547-559, 1968. Sheppard, C.W. and Householder, A.S., "The Mathematical Basis of the Interpretation of Tracer Experiments in Closed Steady-state Systems", J. Applied Physics, 22, 5!O~520, 1951. Zadeh, L.A., "Outline of a New Approach to the Analysis of Complex Systems and Decision Processes", IEEE Transactions Systems, Man and Cybernetics, SMC-3, 28-44, 1973.

29

INFLUXES ! Drinking Water~ . ~ . ~ Water From Food-----_.J~-~Water Skin Absorption------"-~ jM'Content Respiratory Absorption~ WB

Fig. I.

EFFLUXES ~.fUrinary + Fecal Water ~ S k i n Evaporation "'~" Respiratory Evaporation

A Simplified Water Exchange Model for the Experimental Wild House Mouse.

30

:3-(~ ~-. I

~

~

Drinking Water

- - - ~ - - Water from Food

~

(Metabolic+Hygroscopic)

~z_ilk...-----.........~,,

20=C, 50=/= relative humidity

;, o~-

,2~

,5 ,h

FRACTION OF AD LIBITUM DRINKING WATER o. Influxes 3i~

~---0---* ~ . ---~L---L~-,~ =~~-ak---

--~" -~

Urinary+ Fecal Water Net Evaporative Water (Skin + Respiratory) Net Skin Evaporation Net Respiratory Evaporation O C, 5OYo relative humidity

u_ "J tl.

°'0

LlJ

O-

I/4 178' FRACTION OF AD LIBITUM DRINKING WATER b. Efftuxes

I~/Z

Fig. 2.

~Equilibrium" Water Fluxes Versus Water Deprivation in the Wild House Mouse.

31

~ 0.60t

~~o~o] ~_° -~o.2o] ~o.,o 1 o.o01 0.061 0

Fig. 3.

~ , i , i 2 3 4 5 6 TIME AFTER THO INJECTION

7 (days)

Specific Activity Versus Time After THO Injection in an Ad Libitum Wild House Mouse (Semilog Grid).

j 8

f 9

Note:

__

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I

SOCIAL IPREFERENCE5

HOUSING

TRADE, INDUSTRY

//

s. MODEL

~o |

CONTROLLER i

Ii

/ !

j

1 I

.....

~~ "

RTHWEST 8~ ~ - ~ - ]

pAc~FIc

TRANSP '~ -"~ TRANSP, -EMAND >'~TRANSPORTATION ' P O L I C ~ 4

POPULATION LEVEL, CLASSIFICATION & DISTRIBUTION

q i I

t ARTERY, TRIP CONVENIENCE, I COST & PERFORMANCE)

TRANSPORTATION

Fig~ 4

Interconnection levels of Dynamic Socioeconomic & Transportation Systems

Arrows generally represent multivariable outputs. E.go, TransPortation policies include components of land use, legislation, taxation, investment in transportation for rapid transi~ highways, airports and seaports Transportation . Outputs include trips/modal artery, trip convenience, safety and cost, pollution (CO, NOx, HC), etc.

.

BIRTH RATE,~ . . . . r ~ ~ DEATH R A T q D E M O G R A P H I C MIGRATION--: -->-i MODEL RATE ~ ,

ON THE

OPTIMAL

SIZE

Mohamed

OF S Y S T E M

MODEL

Z. D a j a n i

University

of L o u v a i n

Louvain-la-Neuw~,

Belgium

ABSTRACT

The p r o b l e m sidered. model,

of f i n d i n g

By e x p l i c i t l y

one

is able

ger p r o g r a m m i n g

the best

synthesizing

to t r a n s f o r m

Finding

may

"best"

accuracy,

conflicting

[i],

Note

that

increase

of the

JG

of s y s t e m

model

inte-

has

occupied Since model

of b e t t e r

the a t t e n -

complicating is the

opti-

accuracy

and

model

(inter

is not

but

affected

and n u m b e r

general

Dewly,

produced by the

its

by

the

choice

of o p e r a t o r s ) .

input-output

to some

of fin-

or i n v e r s e l y

singularly

is also

measurable

subject

incorporates,

complexity,

connections,

with

model

that

on the p r o b l e m

pairs

it is

quantitative

per-

criterion

to p r o c e e d

it as

time

+ Jcontrol one

has

advances.

initial

:~ D e f i n i t i o n

system

state

of s y m b o l s

+

Jcomplexity

to start

from

-

a rough

JAccuracy model

and

(i)~

improve

Let

= f(x,u;~,t) be the

is con-

a constrained

"best"

demands

is f o c u s e d

complexity

process

a best

that

and the

of m o d e l

dimension

method

= JEstimator

In o r d e r upon

attention

optimization

a physical find

model

of a p r o p o s e d

[3].

paper,

model's

of a r e a l i z a t i o n

formance

[2],

dimension

accuracy.

to

into

cost

of i n v e s t i g a t o r s .

its

the

For

is a d i l e m m a

number

the

quantitatively

desired

model

increasing

brief

the b e s t

system

INTRODUCTION

between

models

In this

problem

increase

compromise

simpler

ding

the

of an e v e r

the m o d e l mal

this

for the

complexity

problem.

i.

tion

size the

are

;

~

: (s×l)

;

model

of s - o r d e r

given

in the

list

~

: (m×l)

with

known

(2) initial

of s y m b o l s .

eondi-

34

tions lue +

~(0),

of

and

s

s = {, a n d

and

-

being that

operators

determined

as p e r

+

{

-

of

in

the

a changable u = [

such

that

(i) are s y m b o l i c

application,

}

variable.

~ e +{'}

Find

JG

the

optimal

is m i n i m i z e d .

and t h e i r

exact

va-

The

nature

is

e.g.,

{

,

}

•

(3)

= e-{'}

2. A N A L Y S I S

Defining integration per

unit

Cstat e

the

of s u b m o d e l s

(p.u.)

: p.u.

Ccontrol

cost

gle

Cgai n

set

: p~u.

Ceontrol

complexity

: p.u.

of d e p l o y i n g of m a p p i n g

closed

loop

: p.u.

cost

cost

gain : p.u.

Assuming J

complexity

:

an a r b i t r a r y 2S

is the

explicit

riables

s

and

g. At this

following

(estimate)

state into

;

a sin-

; energy

Riccati gain)

per

control

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channel

to e v a l u a t e

; the

;

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different

states

gain.;

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set

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g. T h e r e f o r e ~

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prevailing positive

output,

energy

(4)

as a f u n c t i o n

costs

and

noting

that

forms

in the

difference

of the

intercomplexity

Jaccuracy

will

literature,

between

the

index

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i.e.,

process

va-

chosen the

output

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Jcomplexity

of p.u.

it is w o r t h

with

as the

× C g a i n + 2g × C i n t e r c o n n e c t i o n

a chosen

sum of the

and the m o d e l

state

number

expression

point~

in a c c o r d a n c e weighted

a single

x C s t a t e + (mxs)

+ s(s-l)

the

(estimator)

one

of

to d e f i n e

model

(estimator

a unit

needs

Jeomplexity

a single

of i n p u t

cost

with

one

cost,

:

control

for s o l v i n g

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of s u b c o s t s

cost

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of the m o d e l

]

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; Yi-Ymi

l

for the input

same

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.

(5)

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it

is not

of the m o d e l

explicitly

order,

s.

evident,

Jaccuracy

is an

implicit

35

The are

remaining

of the

familiar

The m o s t tryagin veal

maximum

ciple

Ideally the

group of

principle.

and

first

step.

are

then

next

that

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JG

to a p p l y

application

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is a n a l y t i c a l l y

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only

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an a l g o r i t h m whose

treated

(essentially output

the m i n i m u m

of

is the

where

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[~T~]

is a s i n g u l a r

the

integer

entries

are

the

~

the

results

of

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is o p t i of the

is d i s c u s s e d e.g.

[4].

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programming algoA . v a l u e of s = s.

such

that

:

constraint)

the m o d e l

order

is a c h a r a c t e r i s t i c

delayed

both

integer

optimal

(inequality

representing

matrix.

that

DEFINITION

Jcomplexity

i $ s 5 S ~: ,

process

prin-

optimization

JM

using

for

optimizing

first

in l i t e r a t u r e , an

re-

stochastic

In the p r e s e n t

in w h i c h ,

optimization

is w i d e l y

will

extension.

simultaneous

is o p t i m i z e d

Pon-

of the m a x i m u m

a useful

formidable.

is a d o p t e d

the

optimization

by s i m u l t a n e o u s l y

however

3. THE P R O B L E M

tem w h o s e

seems

to h a n d l e

is c e r t a i n l y

to o p t i m i z e

procedure

is p r e s e n t e d

Find

type.

direct

inability

interdependent,

paper

of

section

rithm)

system.

of p r o b l e m s

Js.c.

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its

point

* Jestimation

JM ~ J c o m p l e x i t y + J a c c u r a c y , a n d the A Js.c. = J e s t i m a t i o n + J c o n t r o l - It is true

a suboptimal and

Jcontrol

cost,

cost,

mized,

at this

However,

indicates

type

of c o s t s

JM

ment

and

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modeling

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approach

configuration

to this

indices

quadratic-weighted-integral

natural

no r e s u l t

non-fixed

two p e r f o r m a n c e

input

and

output

(6)

, at w h i c h matrix

of the

case

of the same

sys-

system.

O

For

details

see A s t r S m

As t y p i c a l optimal

model

ditlons

most

of

size

and E y k h o f f

integer ~

of which

programming

is not are

[i].

trivial

justified

problems

the

and r e q u i r e s

in o u r p r e s e n t

solution to m e e t

for the

certain

formulation

[5].

con-

36

4.

It is h o p e d in the

that

engineering

by e x p l i c i t l y lization

this

effort

is t r a n s f o r m e d

and

its

into

extension

of this

principle

or the

presentation

problem

including

CONCLUSION

of

finding

will

stimulate

the m o s t

a complexity

cost

versatility.

The m o d e l

a constrained

methodology calculus

that

integer

is the

feasible

system

the m o d e l

identification

programming

to

interest

reflects

generalization

of v a r i a t i o n s

deeper

systems

model

problem

problem.

of the with

rea-

Natural

maximum

not

fixed

con-

figuration.

5.

BIBLIOGRAPHY

O

[i] A s t r 6 m ,

K.J.~

Automatica, [2]

[3]

Woodside~

[4]

C.M.

Vol.

Sage~

A.P.,

York~

Special

Graves,

7, pp.

Melsa,

identification

- A survey.

1971. order

of l i n e a r

systems.

Automa-

1971.

J.L.

System

identification.

Academic

Press,

1972. on l i n e a r - q u a d r a t i e - G a u s s i a n

Control, and

Vol.

16,

Wolfe~

McGraw-Hill,

x

: (s×l)

model

u

: (m×l)

system

: (q×r)

model

: the

System

of the

727-733,

model

state

random

ai,b i : t i m e - v a r y i n g

LIST

OF

IEEE

Trans.

1971.

P. R e c e n t New

problem.

advances

York,

in M a t h e m a t i c a l

pro-

1963.

SYMBOLS

vector

control

size

Dec.

Inc.,

8.

s

P.

123-162~

Estimation

and

L.G.,

srammin$~

Eykhoff,

7, pp.

issue

Automatic [5]

Vole

tica,

New

and

vector

parameter

(dimension

weighting

matrix.

of s t a t e

vector~

x)

coefficients

Acknowledgement°

The

author

wishes

particular

his

University

of L o u v a i n

to t h a n k

colleagues

the

at the

for t h e i r

Faculty

Unit6

of A p p l i e d

Science

de M 6 c a n i q u e . A p p l i q u 6 e

continuous

support.

and

in

of the

on

INFORMATION-THEORETIC METHODS FOR MODELLING AND ANALYSINGLARGE SYSTEMS Raymond E. Rink Department of Electrical and Computer Engineering Oregon State University Corvallis, Oregon Introduction The pairwise, causal interactions between components and between variables in a complex system may be viewed as processes of communication.

That is, for the

values of one variable to affect the values of another, there must be a direct path (channel) which conveys information to the latter about the values of the former. This point of view can provide some useful methods for modelling and analysing large systems,

as will be outlined in the following sections.

Transmission Model of a System The first step in modelling a real system involves choosing a suitable set of stat6 variables (if the model is to be Markovian) and identifying the direct interactions between them. The statement that "the value of variable x at time t has a causal effect on the value of variable y at time t + I" implies that x(t) and y(t+l) are not statistically independent, over the ensemble of all possible time series generated by the system.

This hypothesis can be tested by measuring the transmission (mutual infor-

mation) between x(t) and T(t+l), given some data.

If the system is stationary and

ergodic, this transmission, denoted* T(x,y'ly), can be measured from a single timeseries of data.

It is easily shown that T(x,y'ly) is zero if and only if x(t) and

y(t+l) are statistically independent.

This idea was apparently first discussed by

Watanabe [I], and was called "interdependence analysis" by him.

Conant [2] later

showed how the measured transmissions may be used to group the system variables into strongly connected subsystems. The transmission is defined as T(x,y'[y) = H(y'[y) - H(y'Ix,y ) where the H's are entropy functions.

If x and y are discrete variables (or quan-

tized continuous variables) with L possible levels each, the entropies can be written as

L L H(y'ly) = - ~ ~ Py,,y(i,j)logPy, ly(ilj) i=l j=l

and

H(y' Ix,y) = -

j[ k[ PY''Y'X(i'j'k)l°gPY'lY'x(ilj'k)'

where Py, ly(ilj) is

*y' = y(t+l) is, in most dynamic systems, most strongly dependent on y(t) itself. This dependence is eliminated by conditioning the transmission on y, leaving a more sensitive measure of interaction.

38

the probability that y~ will take on its ith value, given that y had its jth value, etc.

These probabilities are, of course, not known without prior knowledge and must

be estimated from frequency data.

If N values of each of the variables are avail-

able, then the estimated transmission is L

L

T(x,y~iY) = - ~

^

I

~ ~ y, , (i,j) N

log

i=l j=l ij

k

N

N , (i,j) ~ y 'y N

N

As N becomes very large, T will approach T if the data are statistically well^ behaved° In practice, N>>L 3 usually implies that T = T. One weakness of the method is that the converse to the statement of the previous paragraph does not hold, i.e. it is not true that statistical dependence necessarily implies a direct, causal effect of x(t) on y(t+l).

It may be that a third

variable influences both x(t) and y(t+l) in such a way that the apparent transmission T(x,y'ly)

is spurious.

There is, however, the following simple procedure

which can be used, provided that enough data are available, to identify those interactions that are direct and causal. Suppose that there are n state variables {Xl,X 2, ..... xn} in a minimal representation of the system, and that the causal actions upon x I of all other (n-l) variables are to be determined.

The first step is to measure all the palrwise trans-

missions and to find the i I for which takes its maximum value.

~(Xil,Xl, lXl )

This identifies a^variable Xil, which is tentatively

assumed to have a causal action on x I.

If T(Xil,Xl'IX I) is, in fact, spurious, then

H(Xl'l{xili~il}) = H(Xl'l{xi}). In other words, the addition of x i

to the set of conditioning variables does not

produce a further decrease in the ~onditional entropy of x

if x~ is not directly ~I is directly connected to x I and there is noise i

connected to x I. Whereas, if x. 11 anywhere in the system, the value of x i

at time t is not uniquely determined by

the values of the other (n-l) variableslat time t, and the conditional entropy of X1

at time t+l will be further reduced by the addition of Xil to the conditioning

set.

This will have to be checked at the end of the procedure. Assuming, then, x.

to be causal, the second step is to remeasure the trans-

I

missions between the ot~er (n-2) variables and i 2 which maximizes

X1

conditioned on Xil

to find the

~(xi2,Xl, lXl,Xil)"

If this maximum is not zero, it identlfies a variable x. which is tentatively 12 taken to have a causal action on x I that is not mediated by x i . Again, the condi1 tional entropies of x I with and without x.l in the conditioning set will have to be checked at the end of the procedure, in order to verify this assumption.

39

The third step is to remeasure the transmissions

between the remaining

(n-3)

variables and x I conditioned on both Xll. and xi2 , and to identify the xi3 which has the largest,

etc.

The process continues

maining conditional

transmissions

for s I steps with Sl~(n-l) , until the re-

are all zero.

Of course,

found to be zero at any step need not be remeasured, to be noncausal

any transmission

that is

for that variable is then known

for x I.

At the end of the procedure

it is necessary to check whether the net transmis-

^

sions

T(x i J ,Xl'IXl,Xi~Xi2 ^ ~ ~ ..... x.lj-I ,x ij+l ,. .... x i Sl ) = H(Xl'|Xl,Xil,Xi2,^ ..... x. ,x. , ..... x i lj-i lj+l sI - H(Xl'IX~,X. ,x . . . . . . . . . . . . . x i ) ± l I 12 s1

are all nonzero for j=l,2, ..... Sl, thus confirming that the s I identified variables are indeed causal for x I.

If the net transmission

is zero for any j, then that x i

is not causal and may be deleted from the subset, which, transmissions

to be re-evaluated

for the remaining

is zero, that variable must also be deleted, etc.

in turn, requires the net j

(Sl-I) variables.

is a set of (sl-r I) variables with nonzero net transmissions is repeated for each of the state variables, model of the system.

Obviously,

to x I.

If this process

the overall result is the transmission

It can be shown diagrammatically

the arrows are labeled with the "strength"

If one of these

The final result of the process

with nodes and arrows, where

of the net transmission.

s = max{s.} cannot be very large if this procedure

is to be effec-

1

tire, given a finite amount of data and computing time. which come from an (s+l)-dimensional dimension are encountered

distribution with L discrete levels along each

in the course of the computation,

mates of these probabilities ables.

Conditional probabilities

and good frequency esti-

would require N>>L s+l values for each of the state vari-

The number of computer operations

is similarly large.

This does not prevent the method from being used for large-scale ever, provided that the interconneetions n could be several hundred,

are relatively sparse.

systems,

how-

The state-dimension

and, if the maximum number of interconnections

s which

impinge on any state variable were not more than, say, ten, the method could be used with L = 4 or L = 8.

Many large-scale systems are of this type, where variables

interact directly only with neighboring variables, tially distributed socio-economic

as in a diffusion process.

Spa-

and ecological systems in general would have this

property. Functional

Form of the Dependencies

The transmission modelling approach described perficial similarity

to correlation methods,

function models of linear systems. however,

in the last section has some su-

especially

since they assume a linear form of relationship

the transmission model refers to statistical assumption of any particular

to the partial-coherence-

Those methods are really much more restricted,

functional form.

dependence

between variables, whereas in general, without prior

40

Of course~

the functional forms must still be specified, if the model is to be

used for extensive analysis.

The huge number of conditional transition probabili-

ties that comprise a purely stochastic model of a large system may well be unmanageable for simulation or optimization studies. One approach to the needed simplification is to assume that the state variables are governed continuously by a dynamic system of equations, each having a ratemultiplier structure of the type used by Forrester, e.g. sl-r I

sl-r 1

x I = Xl[(al+nl)j~l flj(Xij ) - (bl+ml)j=iH glj(Xij )]"

(i)

Here the rate of accumulation of x I is homogeneous of first degree in Xl, as is the case in ecological and socio-economic systems where birth and death rates for species; growth and decay rates for commodities and capital, etc., all are proportional to the existing level of the variable. The factors (al+n I) and (bl+m I) include the "normal" exponential growth and decay parameters a I and b!, i.e. the average exponents that would be found in the absence of feedback from other variables.

They also include the "noise ~' fluctua-

tions nl(t) and ml(t) that may occur in the exponent due to randomness in the generation or degeneration process for the species or commodity. The factors f~. (x.) or (normally only one of them would vary from • 3 I. glj (xi~) L 3 3 unity for a particular x. ) represent the modulation in a parameter value that is 1 caused by feedback from a~other variable in the system. In cases where the system is a cybernetic one, these multipliers can perhaps be interpreted as endogenous control variables whose values are being optimized in some sense.

Thus, the general

problem of identification of flj or glj would involve finding an appropriate performance index and then synthesizing flj or glj as a function of xi. , i.e. the dual optimization problem.

We will consider here a special case, but on~ which is fre-

quently plausible; namely the "bang-bang '~ control law flj(Xi.) = 1 + ~ 2 3

Sgn(xi. - ~'I.)" 3 3

(2)

Here xi. is the :'normal~ value of x.l., and (l±dlj/2)

is the modifying factor that

is appl~ed to the normal growth parameter al, depending on whether x.l.> ~ . favorable or unfavorable to the growth of x I.

is

The only parameter to 3be d~termined

is dlj, the "depth of modulation," which will be seen to be related to the transmission.

We may remark in passing that, if (Hf i ) is considered to he a composite

control variable, the dynamic system is of the

J bilinear type, a class whose

excellent controllability and performance qualities have been analyzed elsewhere [3,4,5]. Consider the difference equation y(t+l) = y(t) + y(t)[a+n(t)]

d [i + ~ Sgn(x(t) - x)].

(3)

41

This is equivalent to the equation z(t+l) ~ y(t+l) y(t) " - 1 = a[l + ..n(t) ...a

] [i + ~d

Sgn(x(t) - x)].

(4)

This transformation would correspond, in an actual identification process, to taking the data {y(t)}, forming the ratios of successive values, and subtracting unity. The precise relationship between T(x,z') and d depends on the statistics of x(t) and n(t).

If the data {x} come from a stationary process, the "normal" value

could be taken as that value which is exceeded by one-half the data. equivalent to assuming that Sgn(x(t)-x) is binary with probabilities regardless of the distribution of x(t) itself.

This is

(1/2, 1/2),

Also, in many cases of large scale

systems (e.g. socio-economic) the fluctuations n(t) represent the aggregate decision noise of a collection of individuals, hence will tend to be normally distributed. Under these particular assumptions, a Monte Carlo simulation was used to deterA mine the transmission T(x,z') for various values of c and d, where c/2 is the standard deviation of n(t)/a. tors and z' computed.

Values of x and n were generated by random number genera-

The simulated "data" {x} and {z} were then scaled and sorted

into L=I6 quantum intervals and the entropies calculated from the frequency estimates of probability. The resulting transmission values are plotted in Figure i, as functions of d/2 with c/2 as parameter.

Clearly, the transmission value alone is not enough to de-

termine d if c is also ~known, as would usually be the case with real data. The residual entropy H(z'Ix ) provides the entra information that is needed, ^

and it is plotted in Figure 2.

If the measured entropy (e.g. H(z'Ix ) = 1.85 nats)

is drawn as a horizontal line on Figure 2. the intersections of that line with the parametric family provide several pairs of values (c/2, d/2) which, when transferred to Figure i, can be fitted to a curve as shown.

The intersection of this curve with

the horizontal line corresponding to the measured value of transmission (e.g.

^

T = .563) gives the values of c/2 and d/2 (e.g. c/2=0.25 and d/2=0.435). For the multivariable case where, for example, the interactions of (sl-rl) variables {xi.} with x I are to be modelled by equations of the form of (i) and (2), the data can 3be transformed as above to yield {Zl}.

If the standard deviations of

m I and n I are known, the corresponding curves of Figure i can be used directly to find the value of dlj/2 for each measured net transmission T(xi. , Xl'). J If, on the other hand, the standard deviation of nl, say, is unknown, but it is known that ml=O and that all interactions are of the type flj rather than glj" then Figures i and 2 can be used together to estimate the dlj/2 values.

(This case

may occur, for example, in socio-economic systems where birth rates and capital generation rates result from individual decisions and have, in the aggregate, both causal and random fluctuations, whereas death rates and depreciation rates do not.) The procedure involves simply taking the interactions one by one, treating the other

42

0.7" T(x,z')

0.6

/

0.5

\ \ I

0.]

0.4

0.3

0.2

0.i

0

-----+ 0.I

figure i.

~{(z'

~

0.2

1

0.3

i

0.4

0 ~.6

0.5

t

0,7

0.8

d/2

Transmission as a function of d/2, with c/2 as parameter.

I~)~

2.2

2.0

i. 8 =

3

c~0

4

0.1

1.6

1.4

1.2

1.0

d/2 0

figure 2.

Residual Entropy as a function of d/2, with c/2 as parameter

43

factors as a resultant, normally-distributed "noise ~ factor (a+nl)j~k ~ flj (xij) 5 (a+~Ik) with residual entropy ^

^

^

H(Zl'IXik) = H(zI') - T(Xik, Zl). Here nlk has a complicated expansion in terms of n I and the functions

dlj 2

Sgn (xi.

- ~i),

j ¢ k,

3 3 and the only justification for treating it as having a normal distribution lies in the tendency toward normality for functions of many stochastic variables, i.e. the law of large numbers. In the more general case where both types of interactions flj and glj are known to exist and the standard deviations of n i and m.l are unknown, more general procedures than those developed here will be needed. Information and Perform~ance Quite apart from the parametric type of model discussed in the last section, there are some direct inferences that can be made about the potential performance of a large system, based on the transmission diagram.

This is possible because

there is, in any feedback control system, an intimate relationship between the amount of information available to the controller and the error performance of the system.

In fact, it can he shown [6] that if the error entropy is taken as the

measure of performance for a linear system, the reduction in error entropy achieved by feedback control is bounded by the information available to the controller. Each state variable in a large system may be imagined to have a local controller which is receiving information about the values of certain other variables over separate communication channels,

corresponding to those paths in the

transmission diagram which impinge on the state variable in question.

These chan-

nels represent a certain composite channel capacity ("information-gathering capacity" would be a more descriptive term) that is available to the local controller. This composite local information capacity is not, however~ simply the sum of the impinging transmission rates.

The latter must be smaller than capacity if the

channel reliability is to be adequate.

Although Shannon's Coding Theorem shows

that error-free transmission can be achieved at rates approaching the channel capacity, this can only be achieved at the expense of increasingly long time-delays for encoding and decoding the data, and arbitrarily long delays cannot be tolerated in real-time control.

On the other hand, the total transmission rate will not be un-

necessarily small, either, since control effectiveness increases with the amount of reliable data that is used. and capacity.

Thus, there is some optimal transmission between zero

44

It has been shown [7] that, if the plant and controller are linear and certain other conditions are met, the optimal feedback transmission rate R satisfied the equation 2R = nE(R), where n is the number of state variables being transmitted and E(R) is the channel reliability function. capacity~

This equation always has a unique solution between zero and

Conversely~ if the transmission rate and n are known, the channel capa-

city can be found as C(R,n). Thus, the composite local capacity C 1 available for the control of state variable x I can be estimated by calculating the total net transmission rate sl-r I R1=¥

Z

T(xi.,x l'Ixil ...... x.

, x .....

x.

)

j=l 3 /j-i lj+l ISl-r 1 where T is the sampling (coding) time interval, and then calculating C 1 = C(Kl,Sl-rl)The total information capacity in the system is then n C:ZC.. t i=l

1

Consider now the potential performance of the system in the extreme case where the total information-gathering capacity C t is centralized, i.e. where all ~'local control" of state variables is replaced by a centralized controller which has the objective of optimizing a single performance index for the system as a whole. If that performance index is the mean square state error and if the inherent "plant" mechanisms of the system are linear (not an unreasonable assumption for many socio-economic processes like population, capital, etc., as mentioned previously), then the optimum form of control law is linear state feedback and the separation theorem applies,

in this case it can be shown [7,8] that the total mean

square state error (m.s.e.) is bounded by E[ 11x(t)-Xoptimumil 2 ] ! N2 (T) i~2~i (T).=

~+nF,(Cnt',CTt)' r )

e 2 (T)

(5)

where the summation gives the m.s.e, due to estimation error, i.e. limited information, and e2(T) is the m. Soe. resulting from the controller's limitations alone. N2(T) is the mean square state disturbance that would be caused by the exogenous noise acting on the uncontrolled system over the sample interval T and y(T) is the square of the largest eigenvalue of the state transition matrix of the uncontrolled system.

The %i(T) are the squares of the largest eigenvalues of matrices ~i which

govern the estimation-induced state error dynamics according to the expression Ax(t) =

~ ~i[x(t-iT) - x(t-iT)], i=2

where x(t-iT) is the real-time controller's estimate of the state vector of time (t-iT),

The function F is defined by

45

F(n,Ct,T) = (2A+k) exp [-2R~ CtT/n] where A is an algebraic function of n, Ct, and T~ k is a parameter between zero and one which measures the relative entropy power of the exogenous noise source compared to that of a white Gaussian source, and Rt* is the solution of the equation 2Rt* = nE(Rt* ). Several important conclusions can be drawn from the bound (5). For given state dimension n and controller gain matrix, the m.s.e, is a function of only Ct, the total information capacity, and T, the sampling-codlng time.

There exists the

possibility of information instability if i - y(T)'F(n,Ct,T) is not greater than zero.

In particular, if llm T+0 [2A(n'Ct'T) + k] > i,

then T must be greater than zero for the system to be informationally stable [8]. This result differs from the case of ordinary sampled-data control, where only e2(T) is present in the m.s.e, and the optimum sampling interval T is always zero. Thus, for given n,Ct, and optimum central controller, there exists a Top t ~ 0 which minimizes the bound (5). The relationship between these minimum values and Ct provides the link between the potential performance of a large-scale system and its transmission diagram.

REFERENCES l,

Watanabe, S., IRE Transactions on Information Theory, PGIT-4, 84 (1954).

2.

Conant, R., "Detecting Subsystems of a Complex System, '~ IEEE Transactions o_~n Systems, Man, and Cybernetics, pp. 550-553, (1972).

3.

Rink, R. E. and R. R. Mohler, "Completely Controllable Bilinear Systems," SIAM Journal on Control, Vol. 6, 477-487 (1968).

4.

Mohler, R. R. and R. E. Rink, "Control with a Multipllcative Mode, ~' Journal of Basic EngineerlnE, Vol. 91, 201-206 (1968).

5.

Mohler, R. R., Bilinear Control Processes, with Application t__ooEngineering, Ecology, @nd Medicine, Academic Press, N.Y. (1973).

6.

Wiedemann, H. L., "Entropy Analysis of Feedback Control Systems," in Advances in Control Systems, C. T. Leondes, Ed., Academic Press, N.Y., Vol. 7 (1969).

7.

Rink, R. E., "Optimal Utilization of Fixed-Capacity Channels in Feedback Control," Automatica, Vol. 9, 251-255 (March 1973).

8.

Rink, R. E., "Coded-Data Control of Linear Systems," (to be published).

A NEW CRITERION FOR MODELING SYSTEMS By Lawrence W. Taylor~ Jr. NASA Langley Research Center Hampton~ Virginia

The problem of modeling systems continues to receive considerable attention because of its importance and because of the difficulties involved. A wealth of information has accumulated in the technical literature on the subject of systems identification and parameter estimation; references i through 5 are offered as s~mmaries. The problem that has received most attention is one in which the form of the system dynamics is known 3 input and noisy output data are available~ and only the values of the unknown model parameters are sought which optimize a likelihood function or the mean-square response error. It would seem with the variety of estin~tes~ the aigorithms~ the error bounds~ and the convergence proofs~ and the numerical examples that exist for these systems identification problems, that any difficulties in obtaining accurate estimates of the unknown parameter values should be a thing of the past. Unfortunately~ difficulties continue to confront the analyst. Perhaps the most important reason for difficulties in modeling systems is that we do not know the form of the system dynamics. Consequently, the analyst must try a number of candidate forms and obtain estimates for each. He is then confronted with the problem of choosing one of them with little or no basis on which to base his selection. It is tempting to use a model with many parameters since it will fit the measured response error best. Unfortunately, it is often the case that a simpler model would be better for predicting the system's response because the fewer number of unknown parameters could be estimated with greater accuracy. It is this problem of the analyst that this paper addresses and offers a criterion which can be used to select the best candidate model. Specifically~ a numerical example will be used to illustrate the notions expressed in references 4~ 6, and 7.

SYSTEMB IDENTIFICATION

WITH MODEL FORMAT KNOWN

The systems identification problem of determining the parameters of a linear, constant-coefficient model will he considered for several types of estimates. It will be shown that maximum-likelihood estimates can be identical to those which minimize the mean-square response error. The subject algorithm~ therefore, can be used to obtain a variety of similar estimates. Attention is also given to the calculation of the gradient that is involved in the algorithm and to the Cramer-Rao bound which indicates the variance of the estimates. Problem Statement The problem considered is that of determining the values of certain model parameters which are best with regard to a particular criterion, if the input and noisy measurements of the response of a linear, constant-coefficient system are given. The system to be identified is defined b y t h e following equations: i = Ax + Bu

y = Fx + Fu + b

z = y + n

where x, u, y~ b, n~ z are the state, control, calculated response, bias, noise 3 and measured response, respectively. The unknown parameters will form a vector c. The matrices A, B, F, and G and the vectors b and x(O) are functions of c. Minimum Response Error Estimate One criterion that is often used in systems identification is the mean-squ~re difference between the measured response and that given by the model. A cost function which is proportional to the mean-square error can be written as

47

N

(zi - Yi)~Dl(zi " Yi)

J : [ i=l

where D 1 is a weighting matrix and a time integral. The estimate of c

i is a time index. is then

The summation approximates

c=ARG c

where

ARG MIN

means that vector

c

which minimizes the cost function

Linearize the calculated response vector c

y

H.

with respect to the unknown parameter

Yi = Yi 0 + VcYi(C - CO) where of y

Yio is the nominal response calculated by using Co~ Vcy i with respect to c~ and c O is the nominal c vector.

Substituting Yi into the expression for which minimizes J yields

J

is the gradient

and solving for the value of

c

If this relationship is applied iteratively to update the calculated nominal response and its gradient with respect to the unknown parameter vector~ the minimumresponse error estimate ~ will result. The method has been called quasilinearization and modified Newton-Raphson. The latter seems more appropriate since the Newton-Raphson (ref. l) method would give

Ck+ 1 = ck +

J

cJ

where N

VcJk : -2 Z

(VcYik)~Dl(zi - Yik )

i=l N

N

Vc2Jk = 2 ~ (VcYik)TDiVcYik - 2 ~ (Vc2yik)TDl(Zi_ Yik ) i=l i=l The second term of Vc2Jk diminishes as the response error (z i - Yik) diminishes. The modified Newton-Raphson method is identical to quasi-linearization if the term is neglected. Maximum Conditional Likelihood Estimate Another criterion that is often used is to select of the measured response when c is given

: ARG MAX{P(zlc)} e

c

to maximize the likelihood

48

where -i

P(zlc)

=

~

(~)N(I~)/21MIIN/2

exP I" g

i=l

(zi " yi)TMI

(zi " Yi

The maximum conditional likelihood estimate of the unknown parameters will be the set of values of c which maximize P(zlc)~ if it is recognized that y is a function of c. If it is noted that the max_imization of P ~ I c ) occurs for the value of c which minimizes the exponent~ the maximum conditional likelihood estimate is the ~ same as that which minimizes the mean-square response error provided the weighting matrix D 1 equals Ml'l. Maximum Unconditional Likelihood (Bayesian) Estimate The unconditional probability density function of

z

can be expressed as

P(z) = P(zlelP(e) The probability of c relates to the a priori information available for before use is made of the measurements z. The unconditional probability of

z

c

is then N

1

exp~

P z) : ( )[( =) mC )l/21mlNl2LM21112

T

-1

~ (zi-Yl) M1 (zi'Yi)

7 i=l

. !2(~ - °o)~M2"i(° " c°~ Again~ the expression is maximized by minimizing the exponent. gradient with respect to c equal to zero and solving yields ^ ^ ck+l = Ck +

T -I ( ~ Y i k > M1 VCYik + M2"

%i:l -

M

Setting the

-1 (vcTyikl~M1

(z i - Yik)

:i

2 -lt^ ~ek - c0 )I

An identical estimate would have resulted if a weighted sum of mean-square response error and the mean-square difference of c and its a priori value are minimized~ provided the weighting matrices used equaled the appropriate inverse error covariance matrices. Consequently, the same algorithm can be used for both estimates

a = ~a ~ i ~ l (zi - Yi)~-i (zi-Yi) +(c- Co)~M2-1(c- co Variance of the Estimates A very important aspect of systems identification is the quality of the estimates. Since the estimates themselves can be considered to be random numbers~ it

49

is useful to consider their variance~ or specifically their error covariance matrix. The Cramer-Rao bound (ref. l) provides an estimate of the error covarlance matrix.

E~-

Ctrue)(~-Ctrue)~

= ~, (VcYi)TMl-~cYi + M 2=I

If the a priori information is not used#

M2 -1

will be the null m t r i x .

Importance of Testin6 Results The importance of testing a model once estimates of the unknown parameters have been made~ cannot be overemphasized. The test can be to predict response measurements not included as part of the data used in obtaining the model parameters. If the fit errorj J, is greater for this test than for a similar test using a model with fewer unknown parameters, it would indicate that the data base is insufficient for the more ccmplicated model. The testing should be repeated and the number of model parameters reduced until the fit error for the test is greater for a simpler model. The testing procedure should be used even if the model format is "known" since the data base can easily lack the quantity and the quality for estimating all of the model parameters with sufficient accuracy. Reference 4 provides an example of a model which failed a test of predicting the system's response. Figure 1 shows how the fit error varied for both the data used to determine the model parameters and that of the test~ versus the number of model parameters. The lo~er curve shows that as additional unknown parameters are included, the fit error decreases. When the models were tested by predicting independent response measurements ~ however~ the upper curve shows the models having more tmknownparameters performed more poorly than simpler models. Fortunately, more data were available but had not been used in the interest of reducing the computational effort. When four times the data were used to determine the parameters of the same models~ the results shown in figure 2 were more successful. The fit error for the predicted response tests continued to decrease as the number of unknown model parameters increased~ thereby validating the more ccmplicated models. Unless such tests are performed~ the analyst is not assured of the validity of his model. Unfortunately~ reserving part of the data for testing is costly. A means of making such tests without requiring independent data is discussed in a later section of the paper.

SY~

IDENTIFICATION WITH MODEL FORMAT UNKNOWN

One never knows in an absolute sense what the model format is of any physical system because it is impossible to obtain complete isolation. Even if the model format was known with certainty, it is necessary to test one's results against a simpler model~ a point made in an earlier section. Regardless of the causej the requirement is the same: a number of candidate models must be compared on a basis which reflects the performance of each in achieving the modelts intended use. For the purpose of this discussion~ it will be assumed that the model's intended use is to predict the response of a system and that a meaningful measure of the model's performance is a weighted mean-square error. A Criterion for Com~arin6 Candidate Models Let us continue the discussion of the problem stated earlier using the same notation. The weighted mean-squ~re response error w h i c h ~ s minimized by the minimum response error estimate was

50

N

J = L (z i - yi)TDl(Zi - yi ) i=l Let us denote the weighted mean-square response error which corresponds to testing the model's performance in predicting the system's response as N

= ~

(zli - yi)TDl(zli - yi )

i=l where z I is measured response data that is not part of z which is used to determine the model parameters. For canvenience, we cansider the inputj uj to be identical in both eases. The criterion suggested for comparing candidate models is the expected value of j1. If it is possible to express the expected value~ E{J1}~ in terms not involving actual data for zl~ then a considerable saving in data can be made and improved estimates will result from being able to use all available data for estimating. Let us examine first the expected value of the fit error with respect to the data used to determine estimates of the unknown parameters. We can express the response error as z

Y = Ytrue + n - y ~ Ytrue + n - Ytrue - c-VY(~ - Ctrue) = n - V y ( c^ - Ctrue )

assuming the response can be !inearized with respect to the model parameters over the range in which they are in error. The expected value of the fit error,

= Ei

E{JI} , becomes

" Yl)%l(zi - Yl

= E~ ~n-

(VcYi)(~-

Ctrue)~TDl~n - (VcYi)(~ - ctl~le

~i=l Expanding we get

:

_

=l

+ E

Dl(VeYi)(o

-

~i=l

(a T

Ctrue)(VcYi) Dl(VcYi)(c - ctr u

5i If a maximum likelihood estimate is used, or if 'the minimum mean-square response error estimate is used with a weighting equal to the measurement error covariauce matrix 3 then we can write D I = M -I

= Ctrue +

=l

J [%=1

J

Again, linearization is assumed and it is noted that zi - Ytrue i = n i After substituting we get

E{J}=E(7.niTM-ini ) _2EpTQ-) where N

i=l N

i=l Next, let us examine e~ected fit error E{J} of a model used to predict response measurements, zl, which are independent of the data z, used to determine the estimates of the model. We can again express the expected fit error as

E(~)

= E(i=~1 nliTDlnli~ -2E~i=~ 1 nliTDiVcYi( ~ _ Ctrue~

+ E~I=~1 (~T_ cTru~(VcYi)TD1VcYi(~. Ctrue~ Note that the onl~v difference between the above expression and that obtained earlier for E{J} is that the noise vector is nI instead of n. The same expression can be used for ~ as before since it is the estimate of c based on the data, z, that is desired.

52

~ = Ctrue +|i=~l

(VcYi)TM-1VcY

(VcYi)TM-ln

Substituting the above expression for @j and M-I for Di we get

-

~E

nliTM-iVcYi Q-I i=~l(VCYi)TM-Inj

M

~i=l + E
where P and Q are defined as before. Since the noise vector~ nI and n 3 are uncorrelated~ that is = 0 for all i ~ J then the second term is zero. Since the noise vectors n and nI have the same covariance matrix~ M~ we sam write

E
= TRACE = TRACE

n

E
= TRACE[N9 = N • MZ where N is the number of time samples and MZ quantities. Also E

is the number of measurement

niTM-lni ~ = N "

j

We

can

E[J I} :E(J} Examining the second term

ki=l

+2E {P~Q'~}

53

After taking the trace of the scalar and reordering the vectors we get

E ~ P~Q'~

~ ~P~CE

Vyi Q-1 ~=I

(VYj)~ j=l

Because the noise is uncorrelated at unlike times, the term simplifies to

E{pTQ-~)

T-1 njn i M iVcYiQ (Vcyj )TM"

= TRACE j=l

Finally, we have that the expected fit error for the case of testing the model's prediction of the system's response as

E [ jl}

= j + 2 TRACE

= I VcYiQ'I J=l (VcYjlTM"

Since it is available, the actual fit error, J, is used instead of its expected value. The intent of the new criterion, E{Jl}, is that it be used instead of J in determining the level of model complexity that is best. A n Applieatic~ of the New Modelin6 Criterion The control input and response time histories of the numerical example used to demonstrate the use of the new modeling criterion are shown in figure 3. The dynamics resemble that of the lateral-directional modes of an airplane and are described in detail in reference 9. The level of noise that has been added to the calculated time histories is also indicated in figure 3. The sampling rate was lO samples per second. The unknown parameters in the A, B 3 and b matrices can number as many as lO, 9, and 4, respectively. Only 16 parameters have values other than zero. Five sets of calculated responses to which noise was added, were analyzed using the algorithm of reference 9. The analysis was repeated, allowing a different number of parameters to be determined by the algorithm. The fit error, J, in each case was averaged for the five sets of responses, and plotted as the lower curve of figure 4 as a function of the number of unknown parameters allowed to vary. As the number of unknown parameters was increased, terms were always added and never substituted. Because of this it can be argued that the fit error, J, should be monotonically decreasing as the number of unknown parameters increases. It can be seen in figure 4, however 3 that a point is reached beyond which J increases. The reason for the increase is because there is a point reached where convergence is a problem because of the essentially redundant unkno~-n parameters. Although some of the five cases for 23 unknown parameters showed a decrease in J, others did not. The analyst might be tempted to settle for the model having 19 unknown parameters since the calculated response best fits the "measured response." It would be a mistake to do so; one that is often made. In addressing the question "Which of the candidate models best predict system response?" a test ~as applied. The results of the test were then compared to the modeling criterion in the following way. For each set of unknown parameters determined, the corresponding calculated response was compared to the true response to which independent noise was added. The resulting fit errors were again averaged and plotted in figure 4 as the upper curve. The test performed simulates the practice of reserving actual response data for model testing purposes only. As a

54

result of these tests it can be seen that a model with only i0 unknown parameters is best in terms of predicting system response. This is six parameters fewer than the "true model." That is to say s there was less error in setting six parameters to zero than the error resulting from trying to determine more than l0 unknown parameters. The number of parameters that best predict system response will increase if additional data are used or if the noise is decreased. The purpose of the new modeling criterion is to eliminate the need of independent response measurements comparing the model candidates~ thereby allowing all of the response data to be used in determining the unknown parameters. The new criterion was calculated as part of the system's identification algorithm and the results are shown in figure 5. The results of testing the candidate models given in the previous figure are included in figure 5 for comparison. The minimum in the curve for the new criterion occurs at l0 unknown parameters as was the case for the previous test results. This indicates that the ne~ criterion can be used instead of testing the model candidates against independent response data. The Problem of Too Many C ~ d i d a t e Models Although it is a great help to have a criterion for comparing candidate models, there remains a problem of an excessive number of candidate models. A simple calculation illustrates the enormous number of candidate models that result from the combinations of unknown parameters that can be used for the simple example used to illustrate the modeling criterion. The total number of possible candidate models exceeds 8 million. Even though it is possible to greatly reduce the amount of calculation effort by neglecting changes in the gradient of the response with respect to the unknown parameters, Vcy , solving a set of simultaneous algebraic equations would still be required for each model. Consequently, the calculations involved for 8 million candidate models becomes an economic consideration. It is estimated that testing all of the 8 million candidate models would require about one hour on a CDC-6600 computer. As the maximum number of unknown parameters increases~ the number of possible candidate models rapidly becomes astronomical. In practice~ the analyst has enough understanding of the dynamics of the system being modeled to know to some degree which terms are primary and which are less important. It would be valuable, however, if one did not have to rely on the analyst's judgment. The problem of searching for the best candidate model, therefore~ remains a formidable problem worthy of attention.

REFERENCES

1.

Balakrishnan, A. V.: Co~nunication Theory. McGraw-Hill Book Company, C. 1968. IFAC Symposium 1967 on the Problems of Identification in Automatic Control Systems (Prague, Czechoslovakia)j June 12-17~ 1967.

2.

Cuenod, M.~ and Sage, A. P.: Comparison of Some Methods Used for Process Identification. IFAC Symposium 1967 on the Problems of Identification in Automatic Control Systems (Prague~ Czechoslovakia), June 12-17, 1967.

3.

Eykhoff, P.: Process Parameter and State Estin~tion. IFAC Symposium 1967 on the Problems of Identification in Automatic Control Systems (Prague, Czechoslovakia), J~me 12-17, 1967.

4.

Taylor, Lawrence W., Jr.: Nonlinear Time-Domain Models of Human Controllers. Hawaii International Conference on System Sciences (Honolulu~ Hawaii), January 29-31, 1968.

55

5.

Taylor~ Lawrence W.j Jr. 3 and liiff, Kenneth W.: Systems Identification Using a Modified Newton-Raphson Method - A Fortran Program. NASA TN D-6734, May 1972.

6.

Taylor, Lawrence W., Jr.: How Complex Should a Model Be? 1970 Joint Autc~atic Control Conference (Atlanta~ Georgia), June 24-26j 1970.

7.

Akaike, H. : Statistical Predictor Identification. Vol. 22, 1970.

Ann. Inst. Statist. Math.,

56

Test data-....,..

-- 0

Test d o l o - ~ ~____~

Fit error

odel data

J

J

o o

I 0

i0

!

l

I

l

20

30

40

50

An example o f i n s u f f i c i e n t

data/parameters, T = 60 seconds.

I

I

I

IO 20 30 40 NUMBERof UNKNOWNPARAMETERS

I 50

Figure 2. An example of sufficient data/ parameters. Pilot tracking case. T = 240 seconds°

NUMBER~ UNKNOWNPARAMETERS Flgure l .

I 0

Pilot tracking case, 40

Figure 3. Response histories for example problem. Lateral-directlonal case.

-,oo

Unitsin Degrees,Degreesper 5ec

-8

\

8al0~ 60

.......

¢

-t0

-6 L

Time- sec

0 ,~

-

9

sec

Intm°

Minimum

% Fit error

Time

i

Testdata

Fit error

J

J

~ o d e l data

0 0

~%~MInlmum

i al ~L.~ | 5 I0 15 20 NUMBERof UNKNOWNPARAMETERS

I 25

Figure 4. Comparison of fit error using model and test data, Lateraldirectional case.

0 0

!

I

I

I

I

5

I0

15

20

25

NUMBERof UNKNOWNPARAMETERS Figure 5. Comparison of fit error using test data and new criterion. Lateraldirectional case.

STOCHASTIC EXTENSION AND FUNCTIONAL RESTRICTIONS OF ILL-POSED ESTIMATION PROBLEMS Edoardo Mosca Facolt~ di Ingegneria UniversitA di Firenze

Abstract - In Theorem 1 it is shown that under mild conditions the minimumvariance smoothed-estimation of a Gaussian process y(t) with covariance Ry(t,~ from noise-corrupted measurements z(t)=y(t)+n(t), t~T, with n(t) Gaussian with covariance Rn(t,~) , is equivalent to a purely deterministic optimization problem, namely, findingi~ H (Ry) that minimizes the functional

L (ylz) o IIyll 2 Here <°/'>R

Ry "

denotes izmer-product of the reproducing kernel Hilbert space (RKHS)~H).

Theorems 2 and 3 deal with the more general situation where y is a set of linear measurements on an unknown function w(~ ), ~ A. The above stochastic-deterministic equivalence provides valuable insight and important consequences for modelling. An example shows how this equivalence allows one to make use of Ealman-Bucy filtering in a purely deterministic problem of smoothing. 1. Introduction The present paper evolves from some ideas of the author (Mosca 1970 a, b, 1971 a, b, 1972 a, b, c), and is very close in spirit with some recent works by Kimeldorf and Wahba (1970 , 1971), and Parzen (]971). It deals with a problem that often occurs in experimental work, for instance, in indirect-sensing experiments. Roughly speaking, the problem consists of reconstructing auunknown function or signal w from a set of linear measurements on w and from the knowledge of the class to which the signal belongs. Specifically, it is assumed that: (al). The unknown w is an element of a complex Hilbert space ~ o f functions mapping their domain A into the p-fold Cartesian preduut Cp of the complex numbers C. The inner product o f ~ i s denoted by ~ / o ~ , and the corresponding norm by I~'l~. In most applications, either ~ = L 2 ( A ) , where A is a Lebesgue-meas~urable subset of the r-fold Cartesian product R r of the real line R, or ~ = 1 2 ( A ) , where A denotes a subset of the set of all integers. (a2). A linear measurement on an element w E ~ near functional

equals the value of a continuous li-

where, by the Riesz representation theorem (Naylcr and Sell, 1971), x ~ i s presentator of the functional.

the re-

58

Under the above assumptions, the problem can be stated as follows. Problem I. Given the observations

equal to the sum of an additive noise n(t) and a set of linear measurements on w e

with

i X{'/~)/

~ ~ ~ }

a known family of functions in ~ ,

find a suitable estimate of the unknown w~X. [ ] With reference to Problem I some further assumptions

are needed:

(a3). The observation set T is a Lebesque-measurable subset of the q-fold Cartesian product R q of the real line R. (a4). The additive noise n is a sample function from a zero-mean stochastic process with covariance kernel

Example I.

Let u be the input to an n-dimensional linear system with continuous

coefficients

starting from the initial state X(b~)=_Xo . By observing the system output y over T= [te, tl] , it is desired to reconstruct the initial state ~ and the system input over the same interval T. The variation of constants f~rmula (Brockett, 1970 ) gives imme-

= <X_o,~/(~,eo)ce)_ _ where:

_~(t,~ ) i~

~ + < ~,k(%)

the state-transi~io~ ~atri~ f o r / ( ~ ) _-

C,o,t~ " t s Ct,e~2

AC~)_X(~) ;

is the system impulse-response; ~(t) the unit step function; E n is the Euclidean nspace; and L2[to~ tl] is the space of all real-valued square-inte~rable functions on Its, tl]. If one defines [ as the direct sum (Ealmos, 1957)

and 7

59

the observed output y can be expressed in terms of the an.~o~m

as in (5)o I ~ I f ~ i s infinite-dimensional and T a continuous observation set, under fairly general conditions it has been shown (Mosca, 1970 a, b, 1971 a), that no linear unbiased estimate of w E ~ exists. This means that there is no wE~.such that ~ where y(t) is the minimum-variance ~b

linear estimate of the value y(t) taken on by y at t ~ T. Following Hadamard's terminolog~ (Hadamard, 1952), Problem I is referred to as an il!-posed estimation problemo The chief goal of this paper is to compare the two main approaches that have been devised to deal with the class of ill-posed problems described above. The first approach (Franklin, 1970 ' Turchin, Kozlov, and Malkevich, 1971), called hereafter the stochastic-extension approach, consists of reinterpreting the unknown w as a sample function from a random process with zero-mean and covariance kernel R w defined on A x A. Thus, known methods of statistical inference can be used to get a minimum-variance linear estimate of w(~ ), ~ A . The second approach (Mosca, 1971 b, 1972~u)oalled hereafter the functional-restriction approach, consists of constraining the unknown w to a linear Subset o f ~ o f well-behaved functions ~ o n A~ completely characterized by an Hermitian nonnegative-definite kernel K x o n A x A. In functional analysis this linear set of functions is denoted by H(EX~ , and called the reproduoin~ kernel Hilbert space (RKHS) with reproducing kernel (RK) EX. 2. Stochastic Extensions For the sake of simplicity, hereafter ~ i s assumed to be equal to L2(A ). This makes the discussion somewhat less abstract, and the notation less cumbersome. However, it should be clear that the conclusions that will be obtained are valid for more general Hilbert spaces ~ . Let Rw be the covariance kernel of the zero mean process w

The stochastic nature of w induces a randomness on y. in fact, because of (I) , y turns out to be a zero-mean stochastic process with covariance kernel

Eq. (2) can be rigorously justified by using Fubini-Tonelli theorem (Royden t 1968 ) in conjunction with some auxiliary assumption on Rwl for instant%the finiteness of its trace

60

Assuming the processes w and n uncorrelated

the crosscorrelation between w and z is simply

Using a form ala (Parzen~ 19671 that gives the minimum-variance linear estimate in terms of a RKHS inner-product j, in the present instance one has

It is of some interest to note that the image of ~ under the transformation (I) mapping functions on A into functions on T coincides with the minimttm-variance linear estimate ~ of y. In fact~

~(K) X'(~,,{)oI~ = <~(-'¢)/

ISAx('4.;t')~(~.I,~)X'(~4~T')I~4~°(4 ~,+~'m.

(3)

From now on, w and n are assumed to be Gaussian. Most of the results that fo!low~ however~ can also be reinterpreted for non-Gaussian processes under the usual provision of translating strict-semse concepts into their corresponding wide-sense versions. The following theorem~ whose proof can be found in the Appendix, is the key result that allows one to relate the stochastic-extension approach and the functionalrestriction approach. Theorem I. Let the two probability measures induced on the space of sample functions on T by the kernels Ry+R n an~ R n be equivalent. Then the minimum-variance estimate ~(t), t ~ T, of y(t) based on z is with probability one the element in H(Ry) minimizing the functional

1 over all y ~ H(Ry) Jff the two probability measures corresponding to Ry+R n and Rn are strongly equivalent in the H~jek's sense.

I Throughout this paper the following notational convention is adopted. Given an Hermitian nennegative-definite kernel P, H(P) will denote the unique corresponding R/~TS, 4" ' ° the inner product of H(P), and IIf llp

61 In

(4) the functional

equals minus the natural log of the probability density funotional~ Radon-Nikodym derivative or likelihood ratio, of the probability meastLre induced on the space of sample functions by the process with~respect to the probability measure corresponding to the process ~ ~(t)=n(t)l t T ~ ~ Notice also that Theorem I gives a necessary and sufficient condition for to be~in H(Ry). This condition r namely strong equivalence, if2

~(~)

=

~C~-~)

,

i~e~ n is white r is simply

T ~. Functional Restrictions L e t ~ b e again an arbitrary functional Hilbert space r and r be the linear transformation mapping~into functions y on T according to (~). Let ~ [ C ] be the range of ~ . Next, let ~ C ~ b e the space of all minimum-norm solutions~of (%)r when y runs o v e r ~ [ P ~.~can also be equivalently defined as the closed s1~bspace of~spanned by the t-indexed family of functions~x(., t)r t ~ T ~ T h e n r the restrict-

ionT

of P t o ~ ,

has the properties listed in the following Lemma (Mosca 197Oa , b, 1971 a, b, 1972a ) i) The range o f ~ o~incides with REHS

K~(e,~) <x(.,~),×~.~e)~ =

H(Ky) with RK given by

~ t,~ ~T.

(6)

i i ) The transformation ~ : ~ - - ~ H ( b ) establishes a con~menoe or isometric isomorphism between~and H(Ey), i.e.

~,~ ~

21f n is white, the discussion can be made mathemically rigorous by using the notion of a generalized process (Gelfand and Vilenkin r 1964 ). On a more qualitative basis, one can think of H(~) as the pseudo-REHS L2(T ).

62

iii)

The inverse transformation~-1: H(Ky)-~)~of ~ is given by

where the right-hand side of this equation defines the preimage of y under ~ i n the norm-topology o f ~ Properties i)-~ii) have naturally direct relation with the problem of the present paper i f ~ i s identified with L2(A)o With these premises, we are now ready to describe the functional -restriction approach through the steps that follow. H(~)~.

I. Choose an Hermitian no~_ueg~tive-definite kernel i ~ Thus,as is proved in the Appendix, for any ~

such a way that

(87

'~(+,) = < X, x~o,t))Kx where

From the Lemma it follows immediately that the transformation defined by (8) establishes a congruence between H(K)0 and RKHS H(Ky) with RK

AA Moreover, if y and ~ a r e related as in (8)I

2. Among all y on the hypersphere

find the one~ call it ~¢ ~ minimizing the functional

d e f i n e d by ( 5 ) o 3. Increase the radius~ until large variations ~

produce small variations

of v'

Denote the function y obtained in this way by y. 4. The preimage of ~ under the transformation (8)

is the functional-restrict-

tion estimate ~ of ~.

Notice, that, by introducing the Lagrange multiplier ~ ~ steps 2 and 3 are e q ~ valent to minimizing the functional

63

with suitable ~ , among all y ~ H ( ~ ) . To this end, it has already been shown (Mosca, 1972~that an iterative algorithm of the steepest-descent type can be used to compute ~ and ~v iff the two probability measures corresponding to Ky + R n and R n are strongly equivalent. Finally, it is to be noticed, that, regardless of the interpretation that led us to (13), the problem of minimizing the functional (13) is the classical problem of smoothin~ in applied mathematics. 4" Equivalence between Stochastic Extensions and Functional Restrictions Comparing (13) with (4), and taking into account the basic role played in both cases by the condition of strong equivalence, one can conclude that if this condition is fulfilled and

then the stochastic-extension (minimum-variance) estimate ~ equals the functionalrestriction estimate ~. In practice, the comparison between the two approaches is even simpler for in the stochastic-extension approach Ry and Rn are specified up to multiplicative positive constants. This means that in practical instances ~ has to be found byminimizing

instead of (4). As in (13), here ~ must be estimated adaptively from the actual received data z(t), t ~ T. L ~ (y I z) and F ~ (yl z) are now of the same form, and therefore the next theorem follows. Theorem 2 (Equivalenc e between ~ and ~). Let Ky be the RK used in the functionalrestriction estimation of y, and Ry the covariance kernel used in the stochastic-extension estimation of y. Let Ky = Ry, and the two probability measures corresponding to Ry + R n strongly equivalent. Then the functional-restriction estimate of y equals with probability one the stochastic-extension estimate of y

Next theorem concerns the estimation of ~ . Th@orem 3 (Equivalence between ~ and ~ ) . Let K ~ b e the RK used in the functionalrestriction estimation o f ~ , and R w be the covariance kernel used in the stochasticextenslon estimation of w. Let

K~ : Rw

(~5)

H(~<X) = ~

(16).

be such that

64

Finally, let the two probability measures corresponding to R + Rn and Rn be strongJ ly equivalent. Then, the functional-restriction estimates equal with probability one the corresponding stochastic-extension estimates, namely,

Proof. The first of (17) follows as a corol]ary of Theorem I. The second of (17) can be proved by noting that (8), being a congruence, is I-I between H ( K ~ ) and H(Ky), and that, being ~ in H(K~),

where the last equality was proved in (3). ~. Conclusions. The equivalence that has been proved to exist between functional restrictions and stochastic extensions in estimation, provides valuable insight and important consequences for modelling. To those more inclined to accept deterministic rather than stochastic formulations, the above results tell that the covariance kernel R w used in the estimation of w can be chosen on the grounds of the analytic properties of the REHS H(Rw). The opposite is evidently also true. More importantly, one can make direct use of efficient algorithms developed within the stochastic framework to find functional-restriction estimates. E ~ m p l e 2 (Smogthin~). Let z(t), t ~ T = [to, t l ] , be an observed p~ocess. The problem is to find the function y on T with absolutely continuous first derivative and with ]

=

o

minimizing the functional T

T

T h i s i s a c o m p l e t e l y d e t e r m i n i s t i c problem o f smoothing, t h e f u n c t i o n a l t o be m i n i mized b e i n g a compromise between t h e smoothness o f t h e e s t i m a t e and f i d e l i t y t o t h e data° However, i n o r d e r t o make use o f known a l g o r i t h m s , l i k e Kalman-Buoy f i l t e r i n g 1 d e v e l o p e d a n d u s u a l l y a p p l i e d i n a ptu~el7 s ~ o o h a s t i o c o n t e x t , we w i l l r e f o r m u l a t e

the problem in a convenient way. First~ note that the first integral at the right-hand side of the last equation can be interpreted as the probability density functional ~(y I~) corresponding to the process

with y a given deterministic signal and n a zero-mean white Gaussian process with covariance

65

Second~ consider the set'of all signals y generated as outputs on T of the finite-dimensional linear system ~r

_x (~o)

driven by inputs u E L2(T):

o

~ o a n also be seen to be the class of a l l f ~ c t i o n s on T with absolutely continuous f i r s t d e r i v a t i v e t with second deri~rative s ~ r e - i n t e ~ a b l e on TI and such that y(to) = ~ ( t o ) : O. Therefore the d e e i r e d y that solves the smoothing problem must be an element of ~ . Each y E ~ has the representation T

,here ~(~ ) -~ <.1(~ ) i s the impulse response o f ~ , T~erefore, acccrdin~ to pa~ i ) of the Lemma, ~ is the R~S with R~

s~, where A denotes minimum. Moreover, according to part i i ) of the Lemma, there must be a closed s~bspace ~ of L2(T) congruent with H(~y), i . e . , f o r each y ~ H ( ~ ) there is a unique u ~ ~such that

11~ II~(T ) But, bein~ ~(t)--u(t), i t must aleo be

--

II ~ 11~

Whence the conclusion is

The initial problem is therefore equivalent to finding the element y ~ H(Ky) that minimizes the functional

L($1~.)= ~(~tz)+ ll~lt~f.

66 Being the condition of strong equivalence TrK • ~ fulfilled, according to Theorem 2, this can be looked at as a stochastic-exteYnsion estimation problem. For this problem we already know [compare 62)and 00)]that the finite-dimensional linear system above driven by white noise generates at its output a process with cov~riance ~ . Therefore~ the smoothed estimate we are looking for can be computed by first Kalman-Bucy filtering the observations z(t), t 6 T, as in the figure below, and then smoothing the output~of the Kalma~u-Bucy filter (K~ilath and Frost, 1968 , Rhodes, 1971)1

,

I

white noise

~)

I 1 I

IA ~

.,~

white noise

x((;o) = o

i I

Equivalent ~tochastic Model

r

t

Kalman-Bucyl ~ filter ~-----~ ~

X(~o)= o APPENDI X Proof of Theorem I. The assumed equivalence between the two probability measures induced by Ry + R n and Rn can be expressed, as Kalliampur and 0odaira (1963) showed, by the fact that Ry can be represented by the expansion

where

~

~Z

~ ~ is ~ orthonorm~l se~s~ce of f=ot~ans ~ converges pointwise to Ry(t~qY) on T x T. Now let orthonormal sequence in H(Rn). Thus

~<~), a=d the f=otion~l series (~U { ~ be a complete

Therefore v using the two previous expansions,

From this expansion, o n e sees that { ~ o r t h o n o r m ~ l sequence i n ~(R~ + ~n)" ~ i n g

~j

~" ~ ~ ( ~ } use o f ( 3 ) , we get

is a complete

67 where the second equality holds by continuity of the inner product, and ~j~ ~)(4+~j )v~ ~'(')~ y ~ R ~ are independent, identically distributed, Gaussi~n random variables with zero-mean and unit variance. We are now ready to show that ~ H(Ry) with pr. I ~ R y

+ Rn and Rn are strongly equivalent.

In fact,

E There fore

~=l i*:x i

i,ij,1__j

i ''~ "~+~i

>i

1"=i 1+~ i

f=.t

In the above line, the first equivalence is a consequence of Kolmogorcw 0-I Law (Lo~ve, 1963), whereas the second equivalence can be proved by using the positivity of the ~i's. The condition =~4 ~'<~I has been..called,by H~jek (1962) strong equivalence between the probability measures corresponalng to Ry + Rn and Rn. The fact that under the strong equivalence condition ~ minimizes (4) follows easily.[] Proof of (8). If ~

H(KX), by the reproducing property of H ( K ~ ), one has

Kx Hence

A

68

REFERENCES

BROCK~T,

R.W., 1970 , Finlte Dimensional Linear Systems (New York: Wiley)

FRANKLIN, J.N., 1970 ' J. Math. Analysi s Applic., 31, 682-716. GELFAND, I.M., and VIL~KIN, N.Ya., 1964 ' Generalized Functions, Vol. 4 (New York: Academic); HADAM~D, J.~ 1952, Lectures on Canchy's Problem inLinear Partial Differential Equations (New York: Dover) HAJEK, J., 1962 , Czech Math. J., 87, 404-444HALMOS, P.R., 1957, Introduction to H ilbert Spage (New York: Chelsea Publishing Co.) KAiLATH, T. and FROST, P., 1968 , I.E.E.E. Trans. Autqmatic Control, 13, 655-660. KALLIAMPUR, G. mud OODAIRA, H., 1963, Time Series AnalEsis, M. Rosenblatt, Ed. (New York: Wiley) KIMELDORF, G.S. and WA~LBA, G., 1970 , The Annals of ~ Math. Statlst~cs, 1970, 41, 245502; 1971, J. of Math. Analysis Ap~llg., I, 82-95. LOEVE, M., 1963 , Probabilit T h e ,

(princeton, N.J.: Van Nostrand)

MOSCA, E., 1970 a, Preprints 2nd P r a t e IFAC S,vmp. Idsntification and Process Parameter Estimation, Paper 1.1, June (Prague: Academia); 1970 b, Functional Analytic Foundations of System EnKineering, Eng. Summer Conf. Notes, The University of Michigan, Ann Arbor, July; 1971 a, I.E.E.E. Trans. Inf, ThegrE, 17, 686-696; 1971 b, Proc~ I.E.E.E. Decision and Control Conf., December; 1972 a, I.E.E.E. Trans. Automatic Control, 17, 459-465; 1972 b, Int. J. Systems Science, 357-374; 1972 c, I.E.E.E. Trans. Inf. Theory, 18, 481-487. NAYLOR, A.W. and SELL, G.R., 1971, Linear 0perator TheorE in Engineering and Science (New York: Holt, Rinehart, and Winston) PARZEN, E., 1967, Time Series Analysls £aPers (San Francisco: Holden-Day); 1971, Proc. Fifth Princeton Conf. Inf. Sciences and system.9.s, March. RHODES, I.B., 1971, I.E.E.E. Trans. Automat±c Control, 16, 688-706 ROYDEN, H.L., 1968, Real Analysis (New York: Macmillan) TURCHIN, V.F.~ KOZLOV, V.P., and MALKEVICH, M.S., 1971 , Soviet Physics Uspekhi (English translation), $3, 681-703.

REGRESSION

OPERATOR

IN I N F I N I T E D I M E N S I O N A L

VECTOR

SPACES

AND

ITS

A P P L I C A T I O N T O SOME; I D E N T I F I C A T I O N P R O B L E M S

Andrzej

Szymanski

Institut of Mathematics University

of Lodz

POLAND

I n t r o d u c t i o n T h e paper is an attempt to transfer the notion of r e g r e s s i o n and its properties k n o w n for numerical r a n d o m variables

into the theory of

Banach valued r a n d o m variables and to apply that theory to some ident i f i c a t i o n problems. A s s u m i n g that the trajectories of stochastic processes r e a l i z e d in the system are elements of a separable Banach space, we may regard t h e m as Banach valued r a n d o m variables and the problem of identification can be t r a n s f e r e d into the domain of the infinite dimentional p r o b a b i l i t y theory. On the basis of the g e n e r a l i z e d R a d o n - N i k o d y m t h e o r e m given by R i e f f e l for the B a n a c h valued r a n d o m variables, conditional

the d e f i n i t i o n of the

expectation of a r a n d o m variable with respect to any

field has been f o r m u l a t e d in the paper and its existence and uniqueness have been proved. The K o l m o g o r o f f t h e o r e m on the form of the conditional e x p e c t a t i o n has also been transfered. That makes

it possible

to formulate the d e f i n i t i o n of the r e g r e s s i o n operator for the m e n t i o n e d r a n d o m variables. The p r o b l e m of i d e n t i f i c a t i o n can be reduced to the p r o b l e m of finding the r e g r e s s i o n operator on the basis of the realizations stochastic processes

of

in the input and output of the system.

Part II of the paper refers to some properties of the r e g r e s s i o n operator in a separable H i l b e r t space. On the basis of the given here a p p r o x i m a t i o n t h e o r e m an a p p r o x i m a t i v e a l g o r i t h m has b e e n found for linear operators and one example given.

70

1. T h e

R e gr

e s s ion

O per

a t or

in

a

B a n a ch

Space 1.1. B asi c concepts and d e f i n i t i o n s . Let us denote by ~ - f i e l d of subsets

(~ , ~ ,

p) a p r o b a b i l i t y space, w h e r e ~

of the set ~

is some

and p is normed c o m p l e t e m e a s u r e de-

fined on ~ . Let X be a s e p a r a b l e B a n a c h space over the field of real numbers. D E F I N I T I O N 1.1. By the X - v a l u e d r a n d o m v a r i a b l e ~ we m e a n a strongly measurable

(with r e s p e c t to ~

) m a p p i n g of the s p a c e ~

into the space

X. D E F I N I T I O N 1.2. A n element of a B a n a c h space X defined as follows

is called the m e a n value or the e x p e c t a t i o n E ~ dom variable. The integral

is m e a n t

of the X - v a l u e d ran-

in the sense of Bochner.

We a s s u m e m o r e ever that all the r a n d o m v a r i a b l e s c o n s i d e r e d in this paper are Bochner

1.2. C o n d i t i o n a l

integrable.

ex~qtation

of an X - v a l u e d r a n d o m v a r i a b l e w i t h res-

pect to a ~-field. D E F I N I T I O N 1 °3. A n fies,

~-measurable

for every A ~

r a n d o m v a r i a b l e E (~I~) w h i c h satis-

~ the r e l a t i o n s h i p

is c a l l e d the c o n d i t i o n a l e x p e c t a t i o n of X - v a l u e d r a n d o m v a r i a b l e w i t h r e s p e c t to a ~-field ~ ¢ ~ . The e x i s t e n c e and u n i q u e n e s s the r a n d o m v a r i a b l e s

of the c o n d i t i o n a l e x p e c t a t i o n for

c o n s i d e r e d h e r e follows from the g e n e r a l i z a t i o n

of the c l a s s i c a l R a d o n - N i k o d y m t h e o r e m for an X - v a l u e d measures, shed by N. R i e f f e l

publ~

([5], p. 466)

A n a l o g o u s l y as for real r a n d o m v a r i a b l e s we h a v e the following: P R O P E R T Y 1.1. If ~ = ~ + ~

, then for any ~-field ~ r 2 t h e e q u a l i t y

71

holds. PROPERTY 1.2. If a random variable I-field

~,

is measurable with respect to a

then

1.3. Conditional expectation with respect to a random variable and concept of regression operator. Let ~ and q be measurable m a p p i n ~ of the set ~ Banach spaces X 1 and X 2 respectively. Let ~ i

into separable

(i=i,2) denote the I-field

of borelian subsets of the space X . With the use of the mapping ~ we l transpose the measure from the I - f i e l d , i n t o the ~-field ~i" acee~ ting for every set B ~

1 p~(B) = p(~-IB)

Let us define the

~-fiel£

~ - i ~ I as follows

It follows from the above remarks that ~ - l ~ ~ . DEFINITION !.4. The conditional expectation of the random variable with respect to the

I-field

~ - i % 1 is called the conditional expec-

tation of the random variable q with respect to the random variable and denoted by E ( ~ ; ~ ) . We have

For the conditional expectation E ( ~ ] ~ )

the following theorem an !

logous to that of Kolmogoroff is true THEOREM i.i. If ~ is Xl-Valued random variable and q is a X2-valued random variable, then there exists a borelian mapping ~ : X I - ~ a measurable mapping with respect to the ~-IB ~ ~l

~-field ~ l

X 2 i.e.

and such that

for B~ ~ 2 , which satisfies the relationship

72

for almost all ~ lence

0~

being determining u n i q u e l y precisely to equiva-

(in the induced measure The K o l m o g o r o f f

p~).

theorem for X - v a l u e d

ved by use of the R a d o n - N i k o d y m - R i e f f e l classical

case

DEFINITION

random variables

may be pro-

theorem analogously

as in the

([6]).

1o5o A b o r e l i a n mapping M: X I - ~ X 2 satisfying the relation-

ship M [~(~)] :E(ql ~ ) (~) is called

t h e

r e g r e s s i o n

for almost all ~ , n o p e r a t o r

of the X2-va-

lued r a n d o m variable ~ with respect to the X l - V a l u e d r a n d o m variable The existence

and uniqueness

from the g e n e r a l i z e d K o l m o g o r o f f 1.4. I d e n t i f i c a t i o n

operator

follows

theorem.

theorem.

It is easy to verify that, riables~

of the r e g r e s s i o n

for m e n t i o n e d

in this paper random va-

we have following

P R O P E R T Y 1.3. If Banach valued r a n d o m variables

~ and ~ are indepen-

dent in the sense of [4], than

holds with p r o b a b m l i t y THEOREM

1.2.

i.

(Identification

be independent

theorem)

Let the random variables ~ and

and q denoted as

where U is borelian mapping X l - ~ - X 2 . The r e g r e s s i o n

operator

of ~ with respect to

~

i d e n t i -

f i e s the b o r e l i a n mapping U iff

Proof. N e c e s s a r i t y. Let us consider the random variable = q -U~ we have

o Taking

its expectation

and using the assumption U~=E(q[~)

73 S u f i c c i e n c y. C o n s i d e r i n g

the conditional

expectation

of

with respect to ~ we have

By the virtue of the property 1.3

holds. Hence

E(

)=uS

holds with probability i# which was to be proved. 1.5.

Interpretation

of identification t h e o r e m

in domain of controll

theory.

I

U Fig. 1

Let us consider the system described by the borelian We may

observe the stochastic

operator U.

processes ~ and ~ in the input and out-

put of this system. A s s u m i n g that the trajectories realized

of stochastic

in the system are elements

mentional

can be transfered

and

and the p r o b l e m of

into the domain of the infinite

di-

probability theory.

The identification

t h e o r e m says that in the case shown in Fig. 1

if ~ and ~ are independent

operator ~

~ , q

of a separable Banach space we

may regard them as Banach valued random variables identification

processes

and E ~ = O we can search the regression

with respect to ~ to find the borelian

T h e n due to the identification

operator U.

theorem the problem of identifi-

cation can be reduced to the problem of finding the regression tor on the basis

of the observed realizations

of stochastic

opera-

processes

in the input and output of system. The next chapter deals with the searching of the regression in the case when the space X is a H i l b e r t

space.

operator

74

2. P r o p e r t i e s tor

in

a

o f

t h e

Hilber

t

R e g r e s s s i o n

O p e r a-

Space

2.1. Basic remarks. Let H

(i=i,2) denote separable H i l b e r t Spaces. We assume that l all the r a n d o m variables appearing in this chapter satisfy the condition

It is easy to verify that for the above considered

r a n d o m variab-

les following Lemma holds. LEMMA 2.1. If q ~ c J

and a H -valued r a n d o m variable

~

by condition

!llE(~Ir~) (~)~2p(d~) < o
2.2. The r e l a t i o n s h i p b e t w e e n the r e g r e s s i o n space and t h e l e a s t Let ~ be a ~ - v a l u e d

~ i the

(i=i,2). C o n s i d e r

operator

in H i l b e r t

square m e t h o d . r a n d o m variable

riable. We denote the scalar product Denote by

i~-measurable

is

2

and ~ a H 2 - v a l u e d r a n d o m va-

in the space H 2 as

~-field of b o r e l i a n

the class ~ o f

operators

<. , .>

subsets

of the spaces H. l acting from H 1 into H 2

w i t h the following properties: i) U are b o r e l i a n

operators

u ii)

-i

i.e. measurable

(B)~931

in the sense

for B~T~ 2

[

For the r e g r e s s i o n

operator

in H i l b e r t

space the following theo-

rem is true THEOREM

2 .io The functional E { I ~ - U ~ 2 attains its m i n i m u m in the class

by the condition

]m~IM(x)II2p~(dx) < ~

iff

holds with the p r o b a b i l i t y 1 i.e. iff U is the regression

operator M.

75 P r o o f. By the properties the assumed condition

of scalar product

]JE ( ~I~ ) (~) II2P(d~) < ~

in H i l b e r t

space and by

we obtain

II<~.II~-~(~;~)~2+EIJ~(~I~-~I2+2E I< ~ ~t~), ~(~I~)-~>]

~ II~ ~

Considering

the last term we have

By the lemma 2 .i we obtain

It follows

from the properties

1.1 and 1.2 that

Thus we have

The last t e r m in estimated sum equals zero and the remaining ones are non negative,

thus the expression to be estimated attains m i n i m u m

iff the equality

holds with the p r o b a b i l i t y i, which was to be proved. 2.3. A p p r o x i m a t i o n

theorem.

Let a H 2 - v a l u e d r a n d o m variable ~ be independent r a n d o m variable

~ . Let us assume that the expectation

from H l - V a l u e d of ~ is equal

zero. Let a r a n d o m variable ~ be expressed as

where A is a linear operator. A c c o r d i n g the operator A is the regression dora variable

~

theorem

operator of q with respect to the ran_

~ . By the virtue of the theorem 2.i the operator A, as

the regression class

to the identification

operator minimizes

the functional E I I ~ - A ~ 2 in the

. The m i n i m u m of this functional we denote as 6 2 . We have

~2~ll ~ -A~ 2

76

Let f{e(q ]a)~i=is2 j=l,2 denote orthonormal bases in spaces H i ''~'~ 3 (j=l,2) . Then we may show r a n d o m variables ~ and ~ uniquely in the form

~=2- t i e*' i

and

Let us determine N

the r a n d o m variables

~(N)~e(1). ~>i i By the above

~

and

~

~(N)

iei(2) and N

q(N)

as

~(N)=<~, ~ i e(2)i

notations the following t h e o r e m holds.

T H E O R E M 2.2 (Main theorem) ded and if operators A

If the linear regression

operator A is boun-

exist and satisfy conditions N

i~(~)% ~(~)II<~in where the m i n i m u m A

w i t h matrices

for

is taken over all the finite dimensional

~=1.2

....

operators

of order N, then

N N-~o~

N

where

P r o o f. Let us denote

By the t h e o r e m 2 oi we have 2

2

5.<5N We may show the expression A A~ and a n a l o g o u s l y

= ~

uniquely

in the form

~ e (2) aji ~ i j

N AN~

~-~ (N). (2) : ,•i ~ j~ a ji ~ie j

Thus we may express the errors

~2 and

6 2 in the following way N

and

l

<%

(N) = 0 for i=N+l, N+2, T a k i n g into account the fact that a 31

s2--~

c

.-,

a..

ti)

j

.~°# we can

77

The operator A N minimizes

the functional

~2N' since it is easy to see

that

,<

•~

( j-

%i

)

J

F r o m the last expression we can obtain 2 where

~ ~N,{ aji ~i) 2

It is easy to see that

( ~J-/aji--: ]~i)2--~0

because the respective

series are convergent.

Let us consider the first term of the expression have

but

N

/'

~

of the M

N

. We

m

C~+4 3x

,],~

]

F r o m the above considerations

and from the fact that the operator A

is bounded follows that

~N~

0

summarizing the obtained results we have

~2

2

where --7--O Thus

~2 - ~ - ~ 2

N

which was to be proved.

We may call the above proved theorem the approximation because

it gives the way of approximation

of the linear bounded reg-

ression operator A by means of the finite dimensional when all the conditions

The approximation bounded regression

operators AN0

are satisfied. This approximation

stood in the sense of the m i n i m i z a t i o n t h e o r e m enables

is under-

of square mean error. in a general case of a linear

operator the derivation

rithm. This makes the statistical

theorem

of the approximative

estimation

algo-

of the operators A

and

N the use of computer for c a l c u l a t i o n

possible.

78 2.3. Approxima_tive alqorithm. Let us assumpt that E [ =0 and E ~ =0. On the contrary we may consider the random variables ~ ' = { - E ~ Let us denote by ~ ( N )

and

and ~'= ~-E~

A ~(N) ~ (N) the covariance matrices

defined in the following way

and

/}~)~<~):[cov(%, h)]i,j~N THEOREM 2.3. If deter(N)% 0 then the operators A N determined by the theorem 2.2 can be expressed in the following form

~(N) r o o f. ~et ~s oonsider the e~pre~sion ~ l;~ (N)_~ ~ I~)11~" ~rom the properties of scalar product and by the above notations we may obtain N

N

N

E ]I~(N)-AN ~(N) II2=E]I ~ ( N ) I 2 + ~ c o v

= ,~-~a(N)

( ~i ~ ~k' ~

(N)

n N --24~ ~ V ~ a (N) . COV ( We have the following equation after the differentiation of the above expre s s ion

~

~a(N)

Jo k

coy< {k• h o ):°°v( ~jo ~io I

for io, jo--i~2, ..., N Using the notation of ~ ( N )

and

A

~(N) q (N)

we can write

~ ° A~(N) = J~ ~(N) ~ (~) Because the matrix

~(N)

by the assumption is non singular we have

[(N) %(N)

"

~CN)

This ends the proof. From the theorem 2.3 the approximative algorithm follows. Determination of the elements of the covariance matrices on the basis of the observed realizations needs the statistical approach• which will considered in the next chapter.

79

3. S t a t i s t i e a 1 ximation sion

of

a p p r o a c h linear

t o

bounded

t h e

a p p r o-

regres-

operator

3.1. Statistical approximation theorem. Let us denote bY(l~

(i)

, analogously by Yk

- k-th realization of the 1-th component of

- of the random variable ~ . We can determine

the estimator of covariance cov( ~i" [j) by the formula _

cov(

1

~

j)n -1

(i) (j)

The above estimator converges stochastically to the covariance coy( # ~j). the matrices If we ~i denote

A (n) ~(N) and ~ ~(n) {(N)

~(n)

[

1

~-~

~ (N) by the formulas

(i) (j)]

and j[{(N) q (N) =

-_--f

t,~ Xk

i,j& N

all the elements of those matrices will converge stochastically to the elements of the matrices

~{(N) and

~(N)

q(N)"

Let us denote ~(n)= ~(n) (n) N ~(N) " ~ ~(N) q(N) We raise the problem of convergence of the sequence of the funcionals -, (n) ,,~E II q-A N ~ to the value

$2 t, when N-~ o~ and n - ~

By this notions the following theorem is true. THEOREM 3 .I. If the Hl-Valued random variable ~ and the H24valued random variable ~ are independent and if det A (n)} O~ and E ~ < ~o, then ~(n) .2 NO2

limEH -

where

P r o o f. By the theorem 2.1, by the H~lder inequality and using the properties of scalar product we have 2 ~A 4 EII~ ¢(n) ~2

2

% _ %~(n).J 2+

2.E LJ(%-J%ll2

80

Let us consider

2 II . By the H~lder' s inequality

now the term E ~nN'-)A"N ( ~

we have

j}

tl

Using again the HSlder's

inequality

and by the theorem

6.31

([2]) we

have

r (N)

limE ]a: .'- (al n) ) i ']4--0 Since

Summarizing

the obtained results we have 2 ^ (n) , 2< 2

where ~-~ 0 n Thus by the above results

if

n --~

and by the theorem

2.2 we obtain

lira ~ II~- A N(n) jj2 ; A2 which was to be proved. 3.2. Example__=. Let us consider approximation described

an example

theorem

to the

by the differential

of application

identification

of the statistical

of the dynamical

system

equation

--- + a ~Z + a2Y= x d2y dt 2 ldt Let us assume that the inputx(t) process

~(~t)

the interval

and that the function

[O,T]

same assumptions

concerning

identification

functions

processes

integrable

~ (t)=

~ and q we can

theorem to the identification

any orthonormal

way

(1) ak

functions

~k(t) choosed for

on

space L 2 . By the

must be shown in the following

(l) z (~k a r e

is square

of Hi!bert

all the others

of a stochastic

system.

All the observed

where

x(t)

. Then x is an element

apply the statistical the m e n t i o n e d

is a trajectory

the

considered

of

81

case respectively. The observed signal must be sampled and the coefficients given realization

a(1)for k

1 are determined by the condition ~ [ ~ 4 z(1) (ti)-~a(1)~ k ~k(ti)I2=min

The coefficients the above square mean

are determined error

Having all the coefficients make computations

in such a way that they minimize

of the observed realizations

of the matrices

i ,N ~ ,

~ (N)

ning of the number N giving a good approximation is not solved in this paper. After ^ A(n)we

(N) and A(n)Determiis the problem,

the computation

may estimate the output signal

which

of the estimator

as ~A N(n.} Obvmously,

depends on the choosing of the orthonormal

we can

the matrix 1c.) N

system.

Conclusions The main purpose of this paper was the transfer regression

of the notion of

into the theory of Banaeh valued random variables. This

approach enables the formulation

of identification

main of the infinite dimensional

probability

theory.

In the special case, when the random variables we state that the regression

operator minimizes

This makes the proof of the approximation of the linear bounded cal approximation

regression operator.

theorem enables

problem in the do-

are Hilbert valued

the square mean error.

theorem possible in the case Given further the statisti-

practically

the identification

of

linear systems. The problem unsolved

in this paper is as well the approximation

of the nonlinear regression

operator.

It seems to the author that the general cation problem presented these problems.

formulation

of identifi-

in his paper should help with solving also

82 R e f e r e n c e s [ii Dunford N ~

Schwartz

[2]Goldberger A.S. N e w York

J. - Linear Operators,

- Econometric

- London

Theory,

and Semi-groups,

[41 Mourier E. - Les element aleatoires de L ' I n s t i t u t Henri Poincare,

151 R i e f f e l N. - The R a d o n - N i k o d y m Transactions

John w i l e y and Sons. Inc.,

- Sydney 1964

[3] Hille E. -Functional A n a l y s i s

Annals

Part i, London 1958

N e w York 1948

dans un espace de Banach, 13(1953),

pp. 161-244

t h e o r e m for Bochner

of the A m e r i c a n M a t h e m a t i c a l

Society,

Integral, 131

(1968),

pp. 4 6 6 - 4 8 7 [6] R e n ~ i A.

-Probability

theory, A c a d e m i a Kiado,

Budapest 1972

AN APPROACH TO IDENTIFICATION AND OPTIMIZATION IN Q~ALITY CONTROL

G. Romanin J a c u r Laboratorio di Elettronic~ Industriale del CNR Pa~ova (Italy)

W. Runggaldier Facolt~ di Statistica Universit~ Padsva (Italy)

I. INTRODUCTION I n what f o l l o w s a p r o c e d u r e of s i m u l t a n e o u s i d e n t i f i c a t i o n and c o n t r o l i s p r e s e n t ed, applied to the problem of quality control of a production line where items are classified as either good or defective. There two different subprcblems may be considered : I ) Checking the process performance, which is actually an identification problem. 2) Designing a sampling procedure t where the defective items sampled may be replaced by non-defective ones. The sampling procedure is to meet some given requirements, generally keeping the average outgoing ratio of defectives below a desired level t at the minimum sampling cost. This is actually a problem of optimal control.

1.1 The model. Taking ideas from dual control, we suggest a unified procedure for the si,mltaneous identification and control, as shown in fig. I.

p(_l

p,N plant

~I

-

f)

sampler

Pd and .... control Fig. I The output stream of a production plant, supposed to contain a proportion p of defective i%ems, is formally grouped into lots of N items each, with N a suitable number. A sampler, operating at frequency f, inspects n = Nf items from each lot, and replaces the d defectives found by nondefectives. The average outgoing ratio of

84

defectives thus becomes p(1 - i). In fact the average number of defectives still in the lot after sampling is given by Ep(dto t- d) = p . N - p.f.N = pN(i - f)

(I)

(dte t : total number of defectives in the lot). The numbers n ~ud d are used both to get better and better estimates of p, (identification), and to update the sampling frequency f (control). The cycle repeats for each lot. 1,2 the ing all

The objective® Designing the procedure in such a wa~v that it not only identifies value of p,but also makes us aware of any change of p, at the same time guarantee that the average outgoing ratio of defectives does not exceed a desired level Pd' this at minimum total cost. We assume p changes in only one direction~ namely it increases, i.e. the production process can only deteriorate.

2. ID~qTIFICATION AND CONTROL IN CASE OF CONSTANT PARAMETER For expositor2 purposes the procedure is first presented assuming p does not change

with time.

2.1 The identificati0nn Following the ideas of Bayesian statistics,we consider p distributed according to an a-priori beta distribution with parameters a and ~ . Having observed d defective items among the n sampled from the lot, we obtain an aposteriori distribution~ again a beta distribution, whose parameters are ai+ I =

a

+ d.

~i+I ~

~i +

(~C di)

(The index i refers to the lot). Initially we take aO = ~0 = I (uniform distribution of p, equivalent to absence of information), As the variance of the beta distribution is ~iven by

2

)2

(3)

we see that the larger is the number of items sampled, the smaller is ing an incrasingly accurate identification.

2 . 2 The c o n t r o l .

Let Pl be t h e r a t i o

Sampling with frequency f,

pl(1 - f ) . Bein~ p ~ o ~ t ~ t ,

of defectives

the average outgoing ratio

s(h)

o

, impl Z

i n t h e n e x t l o t t o be sampled. of defectives

becomes ( s e e ( l ) )

= p, t o =e~t *he ~ 4 e ~ t i v e ( 1 . 2 ) , t h ~ sempting

frequency f m~st satisfy the condition

p(1 - f) & Pd

(4)

85

cost

along with minimizing the cost, which here is the sampling supposed to be a linear function of f, i.e. A * f (A: cost of sampling N items). The only information we have about the value of p is its beta distribution updated so far. Denoting by m and ~ its mean value and standard deviation respectively, by the Chebyshev inequality the probability of p exceeding ~ = m + Ko is at most I/K2. We therefore may suppose p not greater than ~ and solve for f by substituting in (4) ~ for p, thereby satisfying our objective except for a probability with known upper bound. For what follows a good value for K was found to be K = 3. Then f becomes

f. ~ - p./~ (~

: -+

3~)

Initially f is set equal to I - Pd equivalent to ~ = I. Fig. 2 shows a flow chart of the procedure described so far (The dotted cases are not to be considered for the present case).

/ ~Pd /

,

\

II I nt=dt=nt= 0

if:

i

\ . . . . . . l. . . . J

I -Pd

nt

F

= nt+n = i %+d

In=d=O

I

l

c+

I

% :

I (nt,dt)

i+

m=m

item

yes

.

,,/

~

~o = o (nt,dt) 'I I

C

O

rn

1,

m-a

o~

f: ~- Pdl~ I

0~

Id = d+1

I \

f ....L _~ (,

X

xl \

'~ /

%

p

"- - ----F - _ . x

In -- n+1 E yes

)a~le n e x t / ~ \fre~7' /

Fig. 2

86

3. THE GE~[ERAL CASE OF THE PARAMETER VARYING WITH TIME Relaxing the condition introduced in the previous section 2., the ratio of defectives p in the production is now allowed to vary with time. Following cur objective we then have to decide from lot to lot whether p has changed. For this purpose a testing procedure is derived.

3.1 Definin~a t e s t i n ~ u r e . Being at the beginning of a new lot we denote again by p its ratio of defectives. For a p that does not change, the beta distribution u p ' t e d so far m a y b e considered to hold. Then,(see Sec. 2.2),except for a small probability~we have p < ~. Also E(Pl)=p • Therefore~if pl < p we decide for no change in the lot. In this case for the identification and control we proceed as for p constant. If however pl ~ ~t we decide a change has occurred. In this latter case a new identification is started and the control procedure has to be changed. To reach our decision, as the v~lue of Pl is not known, the testing procedure has to be combined with the sampling. This amounts to determining a value c such that if the number d of defective items sampled in the lot is less than c,we decide for the first alternative (no change). If however while sampling, d reaches the value o, the second alternative is chosen (a change has occurred). At this point two remarks have to be made. Since,deciding for the first alternative,we chose the sampling frequency according to (5),which guarantees our objective (the average outgoing ratio cf defectives does not exceed p=) only if E(p~)< p, we have to make sure that on the mean the testxng procedure recognlzes the sltuatlon pl ~ p. Since when ( p l ) ~ ~ and the sampling frequency is f, on the mean the number of defectives found E(d) exceeds the value Nf~9 to make sure our objective, c has to satisfy the restriction •

A

o ~ Nf~

(6)

The second remark refers to the control procedure signaled. If~ sampling a lot with frequency f, up to tives d was fo%md, the total number of defectives in is expected to be E(dl5 = d/f. Therefore, as soon as we ~ s t expect a number of defectives already passed given by (see (~5,(655

~(dl - d ) ~

~d - ~ : ~ d ( 1 _ f ) ~ n- ( ~?

to be used when the change is some moment a number of defecthe lot d I up to that point a change zs signaled (d=~ ~ Nf~), through the inspection process

- f) : ~ P d '

so that to guarantee in any case for the average outgoing ratio of defectives not to exceed pd ~ from the moment a change has been signaled, we cannot tolerate other defe~ tives to pass through the inspection process~which implies a sampling frequency f =! to the end of the lot.

3.2 Implementin~ the te_st~_~ urocedure. Our approach to determine the value of c follows the lines of Bayesian testing. We substitute the two composite hypotheses of 3.1 (see 7 below) by two corresponding simple ones (7'5 (75

!Pi <

~)

<7')

~Pl = m

[Pl

87

(m is as before the mean value of the beta distribution updated so far). The probability for a type - I error (to decide for the second alternative while the first one holds) is then g~ven by (8). Similarly for a type-II error (to decide for the first alternative while the second one holds) the probability is expressed

by (9). n

PI : ~

[d'n)md(1-m)n-d = I - ~ d=c

e~t

o-1

[d'n)md(1-m)n-d

(8)

d=O

d=O where n is the value of Nf rounded to the nearest integer, i.e. the number of items sampled from the whole lot. Introducing cost factors C I and C_ l for the type-I respec tlvely type-II errors and assumin~ equal a~-priori probability for the two alternative hypotheses (7'), the cost related to the testing procedure is given by

c:~(o IPz I

+

c~I PIi)

0o)

We want the value of c to minimize this cost. Supposing for a moment such a value of c exists and is unique, we find it by examining how C varies for two successive values of o (c and e + I). This amounts to ~lculating the d~fference

F

.,o. -.n-o

0(o+I )-C(c) = ~c1(n),LOnp [ l-p)

o

- CIm (l-m)

n-o] J

and examining the sign of the expression within brackets~ i.e.

cn~c(I-~) n-c- oimC(l-m)n-c

(12)

Increasing the value of c by one, the first term of (12) ~s maltiplied by ~/(I-~)

and the second one by m/(1-m), where obviously p/(1-~) > %/(I-m). As a result, (12) and consequently (11) is a strictly increasing function of c, implying the minimum of C, if it exists, to be unique. The corresponding minimizing value of c is the one at which (12) changes sign (from negative to positive). For reasonable choices of C v and Cvv this changing of sign, if any, occurs in the range c > O, as (12) is negative formic = O. Otherwise by the choice of C I and CII a sampl ing one by one would have been more convenient. Since (see(6)) the value of c is not allowed to exceed Nf~, if (12) does not change its sign in the range from 0 to Nf~, we set c = Nfp. Instead of determining at the beginning of each cycle the corresponding value of c and use it for the test, we may proceed in a recu~sive manner thereby greatly simplifying the testing procedure. We start by calculating (12) for c -- O. Then, ~o~ each defective item found during th~ s~mpl~g, we calculate (12) rec~arsively for the next value of o. As soon as either (12) changes sign or the number d of defectives sampled reaches the value Nf~, we decide for the second alternative (change) con, inn_ ing the sampling with f=1 to the end of the lot and starting a new identification. If however by the end of the lot none of the two situations has occurred, we decide for the first alternative (no change) and proceed with identification and control as for p constant. The whole procedure with the details can be seen from the flow chart of Fig. ? inserting in the dotted oases X and XI the flow segments of Fig. 3 (the dotted oases Y~ YI, Z are not to be considered here). Sampling the first lot t the flow segment X is left out.

88

O

~ The testing procedure

'J PI no

" -

Pn =

(1-~)

Nf

(I-~) ~f

¢ initializing values for testing procedure in next cycle

\

z ~

>

7=- J--nt=dt=O J

Fig. 3

~-3 Cost factors and double testing. If we make the wrong decision that a chs~ge has occurred, the n t items sampled since the present identification has started and from which we derived the relative information, are no longer considered/as a new identification begins. Thsrefore, t~ reach the same stage of information as before, n t mere items have to be ~mpled. This imulies the choice of CI= A/N -nt (remember from 2.2,A is the cost of sampling N items). CII is chosen as ~he cost of having a lot (N items) whose outgoing ratio of defectives on the mean exceeds the value Pd" At the moment a type-I error occurs~ n t might be large, implying a large value of C I and consequently of the cost related to the test. In order to Ooesibly reduce this cost and also to improve the reliability of the test, the procedure described in the previous section is extended to a double testing urocedure including a second test to be performed in the lot that immediately follows the one where a change has been si~aled, with the aim of making a check on the decision taken in the previous lot. The modified procedure ~s as follows. If, during the sampling of a lot~ the test leads us to decide that a change has occurred~ we go on sampling with f=1 to the end of the lot and, while a new identification is started t the old one is continued. In the following lot, sampled with frequency f corresponding to the new identification, along with the normal test the second test, based upon the old identification, is performed. If this second test confirms the previous decision, we definitely abandon the old identification and go on with the new one. If however the second test rejects the previous decision, only the old identification is continued and the sampling frequen cy for the following let chosen according to it. With the modifications just seen, the cost factor C I for the normal test can be reduced to the value A, as in case of a wrong decision taken in a lot, previous its rejection in the fellowing~ we have to sample about N items. The second test follows the lines of the normal test determining c' such that for

89

d _~c' by the end of the lot, the change previously signaled is confirmed, otherwise rejected. The value of c' however is not obtained this %~me by minimizing a cost, but by setting it equal to Nf~ ~ where ~'= m' + 3~' (m',~' referring to the old beta distribution). This choice of c' is motivated by the following two considerations. First, the lot is sampled at higher frequency, implying more accurate decisions. Se condly~ the cost factors related to the second test are difficult tc determine being partly due to chance. The whole procedure as described so far can be seen in more details from the previous flow charts (Fig. 2, Fig. 3) inserting the flow segments Y and YI of Fig. 4 in the corresponding dotted cases. (The dotted case Z is not yet to be considered).

: double test

n~

=

nt/

d t' = d t

1 saving of the previous identification before starting a new one

n t = n~ d t = d't

+ 3

I

:° l Fig. 4

In Fig. 5 a diagram is shown, where on the horizontal axis each segment represents a let and where on the vertical axis we plot the corresponding sampling frequency. The * indicates the point where a change signal occurs. In the let following the one in which the double test is performed, the sampling frequency is represented by the upper dotted line in case the change has been confirm ed, otherwise by the lower one. The values of f used for this diagram are derived from the results of an actual simulation run with N = 400, p = 0.1, Pd = 0.06.

90

1.O

.5

/t

|| /J

I

-~

I

J

3.~..

Minimizin~ total cost. The minimum value of the cost C related to the test depends upon f, becoming larger as f becomes smaller. If we want to take into account not only the sampling cost (A-f for a lot)~ but also the cost C~ minimizing their sum (our procedure in f&ct contemplates both control and identification), we note that a reduction in the value of f might not compensate the corresponding increase in the value of C. As f~ given by (5), depends upon G , which (see (3) and considerations following it) reduces as the procedure goes on, from the moment the saving in the sampling effort (A. ~f) is less than the increase of C (~C), we should not decrease further the value of ~ keeping it e ~ a l to its last value that we denote by ~ . The ~ e a s just outlined are implemented as follows. First the cost C related to the test has to be calculated I in other words we need the values for PI and PI- that can be calculated recursively using the values A P I and ~PTT made a l r ~ l y available by the flow segment X of Fig. 3. For this purpose {he segme~{ Z of Fig. 6 is inserted in the oc~respendin~ ca~e of segment ~.

--

= - (1-m)~NNf I

PII

[.... t

PI = PI + B C - A P I

[~PI

]

1 PI =

I + z~ PI

PII=~PII,BC = I

Fig. 6

+

Fig. 7

~he s~,mbo! BC in the segment Z of Fig. 6 stands for the binomial coefficient (Nf). The values BC~ PI and PI- h~ve to be initialized, which can be done in segment XI of the flow modified as in iFig. 7. If in section X of the flow the test is fullfilled (d = o), we have already the values of PI and PI we need. If not, at the end of the lot we will go through the I steps of section X again simulating a fictitious sampling, until the test is full-

9i

filled. Entering section Y we then calculate the value of C.(This last part is not reported in the flow as it is not difficult to imagine how it performs). Next we proceed to the comparison of ~C (difference between the value of C just calculated and the one of the previous lot) with A- ~ f ( ~ f is the difference between the sampl ing frequeney used in the previous lot and the present one). The comparison is made except if a change has been signaled in the previous lot. If d C exceeds A'd f, ~ will be kept constant ( ~ = ~ min) for the lots that follow. The comparison is not performed~ if previously ~ had already been set equal to a min, the same if we are at the end of the first two lots. In case a change was signaled that has been confirmed, a new identification begins and the whole procedure starts anew. If the change has not been confirmed, we proceed as described above excluding from consideration the lot, where the double test is performed. Both costs considered above depend not only upon f, they depend also upon N, considered here an input chosen in a suitable way. A suitable N means it should neither be too large, to allow a convenient updating both of the identification and the control, nor too small, to prevent an excessive randomness in the ratio of defec tires from lot to lot. We found a reasonable choice for N the number 500.

4. CONCLUSION Comparing the procedure described here with the standard procedures used in ~ a l i ty control,we foundthat we are better off.The above procedure in fact is an adaptive one,while those are not.There is however a cost to be paid for adaptivity, namely the computing we need for each cycle. The way we approached the identification of the possibly time-varying parameter p is not the only possible one,as also a regression analysis could serve the scope.If however the variation of p shows a rather irregular pattern,a regression analysis could not be sufficiently adequate and to obtain better ad~utivity~a procedure as the one described would be preferable. Looking for possible generalizations of the above method,we remark that for each lotpthe quality control problem as described here may also be visualized as a particular decision process over a Narkov chain,where transition probabilities are only partially known and may also vary with time.We therefore think the method may be extended to such more general problems. As the restriction,that the expected ratio of defectives in the final production should not exceed the value Pd, involves the unknown parameter p,one could also think of approaching similarly stochastic programming problems with restrictions involving unknown and possibly time-varying parameters.

Acknowledgement:This work has been supported by the Consiglio Nazionale delle Ricerche,Rome (Italy).

IDENTIFICATION

DE

CEA

Jean

I,

1

:XEMPLES

INTROOOCTION,

Oens oerteins identifier m~me,

M.

OOMAINES

S.

P,

PARC

VALROSE

06034

-

NICE CEOEX

:

probl6mes,

ou 8 contr61er

d'identification

ou de contrBle,

l'@l@ment

est un ouvert delR n ou, ce qui revient

au

sa fronti~re~

Soit ~

uns f~mille d~ouverts

probl~me

not6 J[9)

les prob!~mes

de jRn ~ ~ l'oovert 0 ¢ ~

on associe

un

; on a donc une fonction

qui se posent sont alors

les suivants

:

PROBLEME 1 :

inf J(~)

PROBLEME 2

:

d6terminer

~

opt

ell

tel que

j[~opt ) _< jr~,

En g@n6ral,

ld d@finition

de J utilise une "fonction

d'~tat"

"observation".

o~ y~

ast "la" solution ECO,y)

= o

dens un espace ~

de l'6quation

et une

93

un espace de {onctlons ou de distributions

En g ~ n ~ r e l , ~ s e r a sur £ , et E(9,y)

d~{iniss

= o sera une ~quetion aux d~riv@es pertielles,

Si H d6signe un nouvel espace et C# un op~rateur de V darts H, C~

:~

~

H, on observera C~

y~ , Ze r~sultat de l'observation

sara d~sign~ par hg ~ on aura doric le fonction coot :

J e) = llce ye

-

hglt2 H

EXEMPLE 1 : Darts cat exemple,

l'6quation

d'6tat est une ~quatlon de transmission

: k dEsigne un nombre

positi¢ donn~,

et f u n

donn6 darts L2(D)

- k

:

AYl + Yl = ~lo

(~)

- AY2 + Y2 = |I ~

(£')

(1.i) k aYl an

D=~UFU£

y2 22] an~

It)

Y2 = o L ' o p @ r a t e u r A est l e l a p l a c i e n on a donc d @ f i n i restrictions

; ~ pertir

Si K C D, on peut a l o r s

sont Yl et Y2" consid@rer l e s f o n c t i o n s

coots suivantes

ou [i.3]

J(~)

= ~ItY~

(~)

de 9 l e t des donn@es f i x e s ,

l e s # o n c t i o n s Yl st Y2 ou mieux l a # o n c t i o n y£

~ ~ et ~ ~'

~16ment

- hgll2Hi(K~

k, #)

dont l e s

94

Si K C ~ , on peut consid@rer

{1.4J

une autre ¢onction

= ½1lye

coot

- hgil 2 L2(KI

un probl~me trouver

qui entre dens

le cadre de cat exemple

est le suivant

la forms optimale O

d'un di~lectrique de constants opt soit m i n i m u m dens un domains donn@ K,

qu,'un champ 61ectrique

:

K tells

EXEMPLE 2, Best

une b o u l e

de c e n t r e

Cet

de rayon donn6 r 0 =OU

2 UB

On peut identifier

la recherche

de

~ cells de B ou mieux ~ cells du centre C de la boule B : DanslR 2 les coordonn6es

de C seront

d@sign6es p e r a = ( S l , a 2) Ainsi,

dens oe cos la "vraie" variable

A partir de e on d69init Ya solution

sere a L

de

:

i

- by

CloS}

~Y _ ~ - o

I

L On peu~ maintenant

+ y

[~]

= 91

I (~)

y = o

{F)

d@finir des 9onctions

Oens cat example B pourreit

representer,

coots somme dens l'sxemple par example,

1 ;

la section droite

d'qn tuba par lequel on apporte ou on 6vaoue de la oheleur. De 9e~on

plus g@n~rale,

on pourrait

ram~ne clots ~ une minimlsstion ble delR n [dens l'exemple de l a b o u l e B ) ,

9a±re d6pendre 2 de

ae

IRn ; On ss

d'une 9onction J d~ginie sur un sous ensem-

2, c'est l'ensemble sO peut se trouver

le centre

95

Notons qu'au point de vue num@rique pour cheque @valuation de J il faudra discr@tiser (5) et r@soudre le syst6me approch@. Ce genre de probl@mes a 6t@ 6tudi@ par KOENIG et ZOLESIO [th6se de 3i@me cycle, Unlversit@ de NICE).

EXEMPLE

3

:

11 s'aglt de la famille des probl@mes ~ fronti@re variable ; ce genre de probi@mes act @tudi@, avec les m@thedes pr@sent@es dens cette communication, par A. DERVIEUX et B. PALMERIO. 11 s'agit de d@terminer y e t

~=T

F t, e < t < T de fagon qua les relations (8) et (7) aient lieu

~t=T ~t [I.6]

~t=o O=U~

4y = f

~_!Y ~n

= h

(Q) {F t ]

YC''t=°) = Yo [']

[~o ]

y = g

[r t)

o < • <ST (1.7] iss @l@ments 4, g, h, y sont donn@s. On peut transformer ce probl@me en un probl@me de minimlsation

: ~ tout Q

on assocls la solution yQ de (1.6) et on pose

JcQ) = i I T JlyQ - gl 12 dt o O@terminer y e t

Otels

L2[Ft )

qua (i.6] [i.7) solent v~rifi@es revient ~ trouver

Qopt par lequel J[Qopt ) = o

Dens cs qui suit nous nous int6ressone ~ des ouverts ~ C

D c~ D est

donn@, ~ @tent un compact delR n. Si ×~ d@signe le #onction caraot@ristique de ~ , on d@signe p a r ~

l'ensemble des u, u =

XQ~

~

96

TOPOLOGIE s u r ~ o u

sur~

:

On ddslgne par ~ A ~

la difference sym~trique entre 0 et ~

On dire qua ~n----~ 0

Iorsque mes[~ ~ ~n ) ~

o lorsque u

Cela revient ~ prendre sur ~ une topolog±e du type LP[o)~

2

EXISTENCE O'UNE SOLUTION

En g@n@rel,

!'existence d'une solution du probl6me 2 aura lieu si par exemple

est compact et J continu, bles d'ouverts qui soient En voici un example Solt

~ +~ 1 ~ p < +~

B

I1 est donc important de conna~tre des ensem-

co@pacts

pour, per example,

le topologle L2CO)

:

un ouvert non vide centenu dens D ~ soit

2[e,h,r)

le famille des

0

ouverts ~ de D qui contiennent de cane suivente

:

sommet x, d'ang!e 20

V ~ ~ •

N ° et qui v~rifient

la propri6t6 uni~orme

~[O,h,r), ~ x e 9~, il existe un cane C

de c8t6 h tel que V y e Q N B ( x , r ]

8[x,r) d~signe la boule de centre x et de rayon r

P

on eit y e

x C

de X

C

:

97

D, CHENAIS THEOREME

a d@montr@

:

le

E[8,h,r]

un compact

dans L2[D].

Notons qu'on peut aussi caract6riser de Lipsehitz

3

relative

DEVELOPPEMENT

Soient ~

caract6ristiques

associ@es du type J[

o~ Tl[U,6U] sup@rieur

uni~orme

NECESSAIRE D'OPTIMALIIE.

F~ d e A L , u et u + 6u les {onctions

deux 61~ments

un d@veloppement

par une condition

des ~ ( ~[8,h,r].

DE J ; CONDITION

et Q + ~

[3.1]

E(8,h,r]

~ la fronti@re

~ on s'int~resse

aux fonctlons

J qui admettent

:

u + ~u]

: J[u]

[resp, T2[u,~u]

+ Tl[U,

~u]

est un in#iniment

+ T2 [u,~u]

petit d'ordre

1 (resp d'ordre

~ I, par exemple d'crdre 2]

On suppose que [3,2]

Tl[U,~U]

o5 v - - - ~ ~LC

S[u,v) LP[o]

est

= S[u,

lin~aire

1 ~ p < +-

et

, sl

u + ~u]

continue p'

v~rifie

- S[u,u] ; s i p a r exemple ! + ~ p p'

= 1, e l o r s

S[u,v]

s'~crire

r

S[u,v] [3.3]

= ]

~

[ Gu

un cas i n t ~ r e s s a n t

D

v[x]

dx

~ LP'[D]

est

eelui

En t e r m e s d ' o u v e r t s ,

[3.1]'

G Ix] u

o~ p = l

~ G ( L~'[D]. u

on p e u t 6 c r i r e

ces r e l a t i o n s

J [ ~ + 8~] = J[~] + T I [ ~ , ~ ]

sous l a f o r m e

+ T2[~,~]

[3.2) ' o~

,,,

)

S[~,~]

est additive

et continue

98

f SC£,II] = I

[3,3)'

G~(x) dx

II ,, G~ e L p ' ( o ) Notons que si 6~ + C

£, 6£ - C~

I

,~ + 6£ = (~ U 60 +]

\

~8

OR

[3.4)

TI(~,6£) =

f

G~[x] dx -

6a +

S

G£[x) dx 6~"

EXEMPLE 1 : Darts le cadr~ de l~exemple i d u n ° i, en choisissant J d ~ i n par (i,2), introduisons

F

k~Pl + P2 = (y - Yd) i

i-A

p2 + p2 = (y- yd)IQ

I (3,5)

l'~tat "edjoint" p£ = (pl,P2) par :

[a')

Pl = P2

I

k ~Pl =

_~P2,

-~

(r)

P2 = o aiors,

~Q)

(~)

on o6montre que

(3.83

G~

= (k - i)

avec

Vv(x) = grad, v [ x ]

Vy~

Vp~

cf. J. CEA, A, GIOAN, O, ~ICHEL.

99

O ~ ~

V

~

est un point critique de J. s i i l

tel

que ~ + ~ & ~ ,

mes [16~1)

~ r

N o t o n s qua dons un c a d r e h i l b e r t i e n , condition

analogue

est ~rad J(u)

En t e n a n t

compte de ( 3 . 1 ) ' ,

[3.2)',

existe 8 > o tel qua

;

16~ I = [~ + ~ )

lorsqu'il

n'y

apas

TI(~,~] 2

o

A Q de c o n t r a i n t e ,

la

= o. (3.3)'

et

[3.4)

on d ~ m o n t r e

la

PROPOSITION 3.1 : Q est un point critique d e ~ s i

[3.7] Naturellement

et seulement si

G£{x)

< o

V x68

Go[x]

> o

V x~

on a l a

Si £ est optimal alors £ est critique. Ainsi les relations

(3.7] constituent une

une condition n6cessaire pour qua l'ouvert ~ soit une solution du probl~me 2,

4

APPROXIMATION D'UN POINT CRITIQUE.

Ella est bas~e sur les conditions

[3.7] d'optimallt6

: soit £ , de~inissons

T(~) per

x e T[~)< si

l'ensemble

o6 G~

s'annulle

(3.7]'

£

On a donc l ' a l g o r i t h m e

f

> G~(x]

< o

e s t de mesure n u l l e ,

= T[~)

suivant :

~O donne £n+ 1 = T [ £ n]

n

=

oji~...

[3.7)

s'~crit

lO0

Naturel!ement,

11 ~auOra modifier cat elgorithme si 0 ~

n'entreine pas

T[~)~. Cat algorithme a donn@ d'excellents r@sultats num@riques, ayent en g@n@ral lieu apr@s une ou deux iterations,

la convergence

toutefois,

le convergence

th6orique de cat algorithme reste encore ~ @teblir.

CAS PARTICULIER

1

[Etudi6 par BENDALI -

DJAADANE - Th@se de 3i~me cycle - Universit@ d'Alger]

L~ouvert ~ sst d6fini comma l'image d'un ouvert fixe B delR n par une application t----# x = XCa,t) qui d6pend d'un param~tre a e K K est un ensemble compact et J e s t

CIR p.

; Ainsi la "vraie" variable n'est pas ~ mais a

done une fonction de a :

Pour d@terminer grad J(e], on commence par calculer Tl[a,ga)

c

=

j

G~[x] du -

I

g~[x) du, o~ en r@alit~

~+~ G est indic@

TI[e,~a)

=

par a ~ en ~ait cette relation s'~crit

~ Ga[X[e+~e,~)]

J(a+~a,t)

dt -

j

Ge(X(a,t))

B cO J repr@sente l e Jacobien du changement de v a r i a b l e .

J[a,t]

dt

B

A partir de l&, en

d@veloppant G

et J, l~in#iniment petit 6tent ~a, on obtient la dife@rentielle e de J par rapport & 6a et donc le gradient de J en a. On psut alors utiliser n'importe quelle m6thode du type gradient.

CAS PARTICULIER

2 :

-[a-$amiii~-~-d]ouverts

[@tudi@ par KOENIG et ZOLESIO] de~R n est d@finie comma @tent l'image d'une partie (1-

d'un espace de Banach A par une application u v@rifiant 1as propri~t6s s~ivantes

:

u est injec~ive

La "vraie" variable a ~

~

peut alors Etre un @16ment delR n [example 2) ou

un @l~ment d~un espaee eonctionnel

[probl@me "& fronti@re libra")

On salt eussi dens ce cas expliciter le gradient.

101

CAS PARTICULIER 3 : Dens ~ 2 pour simplifier,

~ est rep6r6 ~ l'aide d'une fonctlon

x;

La "vraie" variable est ici ~ .

1

Oervieux

et Palmerlo obtiennent,

en passant par les relations

m~diaires

[3.1],

[3.2),

int~r-

[3.3),

une

relation du type : 0

J(~+

1

h ,

) = J(~]

+ h

Ge 0

(t).

~(t]

dt ....

/

Catte lois h est l'infiniment

petit

; on a n6glig~ les termes d'ordre

sup6rieur ~ 1 ; Cette relation permet de savoir dens quelle "direction" pour d~terminer J ; de plus si on avait t --4 on pourrait

obtenir ~aeilement

CAS PARTICULIER

4

= ~

du gradient,

iu~z~

le gradient de J par rapport ~

et en les modiflant

approximations

CIR p,

a.

les principes de la m@thode classique

pour les app3iquer au probl~me actuel, un algorithme

sont boris mais inf~rieurs toutefois successives.

: ouverts

~ sera rep@r@ par un ensemble d'indices

~i ; En reprenant

J.CEA, A, GIOAN et J~ MICHEL proposent tats num@rlques

a~K

~i ; en fait les w i sont des "@l@ments ~inis"

de D, deux ~ deux disjoints.

In

il faut sa d@placer

teEO,l],

:

On suppose que ~ = i ~ l

=

~(a,t)

convergent,

Les r~sul-

& eeux de la m6thode des

I02

BI

BL

IOGRAPHIE

Nous donnons une bibliographie cas partiouliers BERGMAN S., SCHIFFER

@tudi@s

tr@s sommaire,

limit@e aux

;

M.

Kernel functions in mathematical

and elliptic differential physics,

Academic Press,

equations

New-YorK,

1953

CEA J., GIOAN Aoj MICHEL J. @uelques r@sultats

sur l'identification

A para~tre dens "CALCOLO"

CHENAIS O.,

Un r@sultat d'existence de domaine,

de domeines.

1973.

dens un probl~me d'identification

note eux C.R. Acad, Sci. PARIS,

t 276,

[12 F6v.

1973].

OERVIEUX.

PALMERIO.

A. B.

Identification Oirichlet,

(20 Novembre

HADAMARD.

J.

de fronti@re

dens l e c e s

d'un probl~me de

note aux C.R. Aced Sci. PARIS,

t.275

1972).

M6moire sur le probl~me d'analyse relati# & l'@quilibre des plaques ~lastlques dadamard,

C.N.R.S.,

encastr@es,

PARIS,

oeuvres de Jacques

iB68.

KOENIG. M.

Sur la location d'un domaine de forme donn@e,

ZOLESIO,

cycle,

J.P°

I,M,S.P.,

Universit~

de NICE,

th~se 3i~me

13 Mars 1973.

THE MODELLING OF EDIBLE OIL FAT MIXTURES J.O. G r a y - J.A. Ainsley Dept. of Electrical Engineering U.M.I.S.T. Sackville Street, Manchester U.K.

Introduction

This paper is concerned with a practical application of straightforward modelling and optimisation methods to an industrial process which promises to yield very attractive economic returns.

The process consists of the large scale batch pro-

duction of edible oil fat mixtures such as margarine and shortening products.

The batch production of edible oil fat mixtures requires the mixing of quantities of raw material to meet both economic and taste criteria.

The latter is the more

difficult to quantify but it is found that the variation with temperature of the ratio of the liquid to solid phase of the mixture has an important bearing on its taste and tactile properties and hence on the acceptability of the product.

A

measure of this ratio is obtained by dilatometerymeasurements at specific temperafumes.

The dilatometerytemperature profile of a compound oil fat mixture is~ how-

ever, a highly non-linear function of the behaviour of the individual constituents and much skill and experience is required by the analyst in the determination of constituent fractional proportions to ensure that the final compound meets the required dilatation temperature specification.

Accurate computer models of this

non-linear behaviour for a range of compounds would be a valuable aid to the designer of edible oil fat mixtures both in speeding the design process and reducing wastage due to error.

This paper deals initially with the derivation from experimental data of regression models to describe the dilatation temperature profiles of binary and tertiary interactive oil fat mixtures.

These models are subsequently incorporated in a computer

design program to calculate mixture constituent ratios which will yield a best fit to any specified mixture dilatation temperature profile.

Economic restraints are

incorporated by setting an upper bound to the fractional proportion of any of the constituent components.

Such bounds will vary with the seasonal changes in raw

material costs and are at the discretion of the designer.

Dilatation and the oil correction factor

If a quantity of solid fat is heated, the volume of fat during heating can be represented by the curve ABCD in Figure i.

Between point A and point B the fat is

104

completely solid and between C and D it is in a purely liquid condition.

The

dilatation of a fat at any temperature is defined as the expansion during isothermal melting and is represented in Fi___gure 1 by the difference in respective volumes between the partially melted fat at point P and the subcooled liquid fat at point R. The dilatation is usually denoted by the symbol D T where the subscript refers to the temperature at which the dilatation is measured. Dilatation figures are usually 3 related to 25 grams of fat with the volume being expressed in mm .

The measurement of dilatation depends on a knowledge of the subcooled liquid line CQ which is normally difficult to determine experimentally due to the inherent instability of the subcooling process.

CQ must therefore be obtained by extrapola-

tion of the easily measured liquid line CD.

The coefficient of the oil thermal

expansion has been shown (1) to be of the form E = k + yT The constants I and 7 have been given a range of values by various experimenters (1) and, depending on the value used, the resulting dilatation figure can lie within a spread of 36 dilatation units at 20°C.

This spread narrows as the degree of extra-

polation decreases and, as it is impractical to determine coefficients k and 7 for every possible combination of fat and oils, some compromise figure must be determined and generally adopted in the measurements.

A degree of uncertainty is thus

present in any quoted dilatation figure and this will be reflected in the expected accuracy of any analytical model derived.

The modelling of binary mixtures

The data from six binary oil mixtures were available for modelling.

The mixtures

were as follows: !.

PALM OIL + LARD

2.

HARDENED FISH OIL + LARD

3.

HARDENED FISH 0IL + PALM OIL

4.

COCO OIL ÷ PALM OIL

5.

HARDENED FISH OIL + COCO OIL

6.

LARD + COCO OIL

The dilatations for each of these mixtures was measured at various compositions and the following temperatures: 20°C, 30°C and 37°C.

A typical set of experimental results is shown in F igur e 2 which demonstrates the general non-linear form of the mixing process.

To improve the symmetry of the raw

data a dimensionless dilatation d was adopted where

105

f(X ) d =

Here X ,

AX~ + BX8

X~ are the constituent concentrations,

A,B

and

- g(X )

are the respective pure component dilatations

f(X A)

is the raw data

g(X)

is a least squares regression model obtain from the raw data.

Polynomial forms of up to fourth order were determin~using a regression analysis computer program to obtain the optimum g(X e) function and, for each polynomial form, both the total and residual variances were calculated with the significance of the model being examined by means of the F test method (2).

A typical regression

result is shown in TABLE i where the F test percentage figure indicates the significance of the model.

The standard error is given in terms of dilatation units when

the model is applied to the original data.

The 95% confidence limits of the poly-

nomial coefficients are also listed.

TABLE 1 A typical regression result given by the F test method.

Binary mixture.

Hardened fish oil lard mixture d20 = 1 . 0 - 0.65XF + 0.57X~, 0.i 3 X ~ - 0.06X~ F-test 99%

Standard error ± 7 units

Coefficient

Limit

1.0

± 0.0O4

-0.65

± 0.01

0.57

± O.01

O.13

± 0.01

-0.06

± O. 01

The predicted error for the best d20 models of mixtures 2, 3, 4 and 5 lay within a band of approximately 40 dilatation units at an F test significance level of 99%. The most ill fitting model was that of mixture 1 where the predicted error band was 150 dilatation units.

At 30°C the error band was small for all mixtures due to a

more accurate oil correction figure and the smaller absolute values of the dilatations.

106

Modelling of tertiary mixtures

Four mixtures of oils consisting of three interactive components were modelled. These were: Io

HARDENED FISH OIL + PALM OIL • COCO OIL

2.

HARDENED FISH 0IL ÷ LARD ÷ PALM OIL

3.

HARDENED FISH OIL • LARD + COCO OIL

4.

LARD ÷ PALM 0IL ÷ COCO OIL

The measured dilatations of these mixtures were available at various concentrations and the following temperatures: 20°C, 30°C and 37°C.

In this case, as shown in

Figure 3,there are two possible ways of proceeding to formulate the model and a range of polynomial regression models were postulated which had the two general forms DT = K O t Kla + K2b • K3c + K4aa ÷ K5b~ ÷ K6c~

DT = K 0 ÷ Kla + K2b + K3c + K46~ac ÷ K56abc • K66aab where a, b, c = AX~ BX, CX ~ac ~ 6bc ~ 6ab = dac }A + el, % c

IB + CI' dab IA • B I

= i, 1.5, 2.0, 2.5 ..o The predicted D20 figures for for best models of mixtures i, 2 and 3 were within a band of 50 dilatation units at an F test significance level of 99%.

The correspond-

ing D30 and D37 figures for these mixtures were 40 and 20 dilatation units respectively.

The best model for mixture # produced corresponding error bands of 80, 90

and 94 dilatation units.

A typical regression result is shown in TABLE 2.

Here a

variable is declared redundant if the chance that it is redundant is in excess of

50%. TABLE 2 Typical regression result given by F test Tertiary Mixture Hardened fish oil/palm oil/coco oil mixture D20 = 1178 - 3.3FX F - 0.96PXp - 1.5CX C 0.19dFp FX F + PXp * 1.65dFc FX F ~ CX C ÷ 0.32Dcp CX C ÷ PXp + O°OO1 FX F 0.002 PXp F-test 9g.g

2

- 0.0004 CX C

2

2

Standard error ± 22 dilatation units

107

Coefficient 1178

Limit

Limit

Coefficient

± 4

0.O02

± 0.00017

-3.3

± 0.035

0.0004

± 0.00018

-0.96

± 0.049

0.19

± 0,039

-1.5

± 0.048

1.65

± O.037

± 0.00003

0.32

± O.O59

0.00!

The term - 0.0004 is redundant

Nuclear magnet± ¢ resonance spectroscopy

Nuclear magnetic resonance spectroscopy methods can be employed to give a direct reading of liquid in a solid/liquid sample and thus, in this case, it is possible to determine the solid fat index at any temperature and hence the dilatation figure. Due to errors in the oil correction calculation, more accurate models will only result if a more absolute measurement criterion such as the solid fat index is adopted.

Some preliminary measurements have been taken using this method and a

comparison with similar dilatometerymeasurements

is given in Figure 4.

Future

experimental data will be based entirely on NMRS measurements.

Synthesising o?timum three component oil mixtures

A computer program based on the simplex hill climbing routine and incorporating the best regression models of tertiary oil mixtures was devised to calculate the composition of a three component oil fat mixture which meets a required dilatation temperature specification. of computer store. (i)

The program which is written in Fortran IV requires 8K

There are eight steps in its execution.

The desired mixture dilatation values are entered.

(2)

The appropriate oil mixture is chosen.

(3)

Pure component dilatation values are entered.

(4)

The maximum desired value of each constituent in the mixture is chosen. This value will be determined by economic or other criteria at the disposal of the analyst.

The stoichiometric mixture relationship is assumed to be

satisfied. (5)

Starting point of search is chosen.

Data entry is now complete and the following steps are automatically undertaken by the program: (6)

By altering each constituent by ±0.5% six points circling the chosen starting point Po are obZained as shown in Figure 5.

(7)

The objective function e is obtained at each of the seven points

108

e = (E20 -o $20)2 t (E30 - S30 )2 + (E37 - $37 )2 where E20 ~ E30 ~ E37 are the mixture dilatations at 20°C~ 30°C and 37°C calculated from the regression models and $20, $30, $37 are the specified mixture dilatations.

(8)

The smallest objective function is determined and this point used as an initial point in a new search.

The search is terminated if the initial point has the

smallest value of objective function.

If the search meets a boundary then the

optimum solution is that point which crosses the boundary.

As shown in

Figure 5 the search near the apex of the triangle is essentially constrained to two directions.

No facility exists in the simple hill climbing program used to determine whether the minimum obtained is local or global.

Instead a map of the objective function is

printed out which can he visually scanned by the operator to obtain the global minimum search area~

Typical computer results are shown in TABLE 3 for four oil mixtures where a constant arbitrary dilatation profile is specified.

This profile was appropriate

for the first two oil mixtures and a good design resulted.

The D37 figure specified for mixture 3 was, however, unrealistically low as LARD PALM COCO mixtumes are generally characterised by D37 values in the range 50 ÷ 120 dilatation units.

The attempt by the hill climbing routine to meet this figure had

a subsequent deleterious effect on the D30 figure obtained.

A similar situation

occurs with the HSO,LARD.PALM.D30 mixture.

Discussion of results

The regression methods used in the derivation of the mixture models were found to be generally successful in dealing with the inherent non-linearities of component interaction and, once dilatation estimate errors are eliminated, better models should be obtained. Possible improvements in the optimisation program include the incorporation of a sensitivity routine to inform the operator as to which component variation has the greatest effect on the dilatation figures and the insertion of a command interceptor.

The latter routine would allow the operator to re-enter the design program

at any point, change a parameter value and exit directly to ascertain the effect of his action on the dilatation contour.

This would enable a fast iterative design to

be completed with a minimum of data entry.

109

As the entire program only occupies some 8k words of computer store it is suitable for use on a small, inexpensive, dedicated computer which could also be used for online control of the mixing system from the tank farm.

Alternatively, the program is

suitable for use on a time shared terminal connected via a land link to a remote central processor.

The economics of the latter approach are now being investigated.

A computer program of this type will completely remove the tedium of design calculation and ensure a specified dilatation temperature profile for the final product batch.

The associated reduction in product waste due to design error will, of

course, result in significantly lower production costing over a wide range of product mixtures.

Conclusions

The oil correction calculation gives rise to a large error in dilatation readings, particularly at low temperatures, and this error is reflected in the derived regression models.

A more absolute measurement technique than normal dilatometery

methods is thus necessary if accurate models are required.

The optimisation

program developed for synthesising oil mixtures is simple in concept~ easy to use and gives good results provided a realistic dilatation contour is specified.

References

(1)

Boekenoogen, H.A. 1966 Vol. i.

(2)

Perry, J~H. Book Co.).

Analysis and Characterisation of Oil Fats and Fat Products,

(Chichester: John Wiley - Interscience Publications).

Chemical En$ineers Handbook, 1963, 4th Edn. (New York: McGraw-Hill

~H

~ t

3

<~ ~

%

\ ~lcJ

<:

D

D

D

,7

0 ),

o

3

e---~

3

°

°

°

c,--]

X

3

~

B ©

73 Q

~

0 0

i......

Magnitude of Non Linearity of Three Component Model

FIGURE 3

Magnitude of Non Linearity of Three Component Model

113

Comparison of dimensionless dilatations obtained from dilatometry and NMR e -- dilatatometry x NMR

Palm o i l / L a r d x

® Ill

1.0

®

mixture @ X

0

X

x

X

0

0

X~

x

X

dzo

0

O

~opalm oli

~OOyo

,~

Hardened fish o i l / L a r d mixture I-0

~xgx x

0

0

o

~o hardened fish oil --.

too% FIGURE 4

114

~oo%

\ \

/ 8

~oo%

'C

\

C

too%

FIGURE

5

TABLE3

D37

Dilatations

0.2

Composition B 3 =

-4

-i

CDilatation~D30

( Unifs

3D37

2

9

196

D30

D37

552

C

P

O.1211 0.0485

H

O. 8303

D20

D20

[~rors

bilatation

Calculated

Mixtures

Optimum

C

P

Fractional

0.5

B1 =

Desired

B2 :

129

H

8

0.97

Maximum

D30

Component 251

535

685

D20

Assumed 203

PALM

HFO

MIXTURE

10

24

748

COCO

ARBITARY SPECIFIED MIXTURE DILATATION

2

7

15

12

207

565

0.2

C

L

H

81

172

600

LARD

0.O688

O. 1852

O. 7~70

i.O

0.97

8

203

658

HFO

D20 = 550;

C

L

H

IO

24

748

COCO

0.1330

O.4147

166

567

66

-34

17

76

C

P

L

91

225

625

L

C

P

PALM

D37 : iO

0.4523

0.2

I.O

i.O

169

306

790

LARD

D30 = 200;

iO

24

........ !

748

COCO

O.1806

0.0090

151

586

22

-4 9

36

32

P

L

H

141

266

695

LARD

O.8104

I.O

I.O

0.95

O

193

680

HFO

P

L

H

91

225

525

PALM

Lm

FREE BOUNDARY PROBLEMS AND IMPULSE CONTROL .......

~,

. . . . . . . . . . . . . . . . . . . . . .

J.L. LIONS University of PARIS Vl and

I .R. I .A. ,Rocquencourt

INTRODUCTION

Free boundary problems

classicaly arise in Physics,

one of the simplest

examples being the melting of ice. We do not intend here, by no means, to review this family of problems,

which are very numerous,

been studied by a large number of different methods. give brief indications inequalities

(V.I)

on the possible

in these problems,

~e only want to

use of the so-called variational the interest being there to

connect the free boundar E problems and the optimization the subject of Section

theory ; this is

I below.

We consider next impulse control problems, and the Author

and have

as introduced in

A.Be~souss~u

[3] 1)~

One can show that the optimal cost function is characterized solution of a new free bpundar2 problem~ Mathematical problems

solved in Section

(somewhat)

I lead us to introduce

extends the V.I : the &uasi-variational

as introduced

in [2] by A. Bensoussan,

a new tool, which inequalities

(Q.V.I),

N. Gomrsat and the Author for the

stationary case and in [3~)by A. Bensoussan tion case.

by the

analogies with

and the Author for the evolu-

~nis is the object of Section 2.

Complete reports ~n the many problems which arise along these lines will be given in [3]3),

4), 5) [9]. Other problems directly related to this

note are studied in [11]o The results briefly presented

in Section 2 extend the results obtained by

various authors in the so-called s - S policy.

Let us quote here the works

of Scarf [12], Veinott

[14]. (Cf. also the bibliographies

The methods

in this short note show the strong connections

introduced

which exist between free boundary problems management

and in techniques

in Physics,

of optimization.

of these works).

~n problems

of

117

I.

FREE BOUNDARY PROBT.~S AND V.I.

I. I An example. I~t

O be a bounded open set of ~n(1), with smooth boundary F. Let

be a partial differential operator given in n (1.1)

~

~ by

~

A~ = - i,Dj=1 x~i(aij(x)

n .i)+ i=I ~ ai(x)

i + a°(x)~

where aij, a i, a o E L~(O). We suppose that A is elliptic in the following sense : if ~ , ~ 6 HI(o) (2) and if we define (1.2)

a(~,@) =i,j=IJO~ [ ai ~ (x)

i x~i 8~ dx + i~I

ai ~ l

@ dx +

a ° ~@ dx

then

a(~,~) ~ ~ II~lli(o) ,

(1.3)

~>0.

The problem we want to consider is the following : we look for a function u

in

O

such that

(1.4)

Au - f ~

(1.5) (1.6)

0

in

O,

u - $ g 0

in

O

(Au-f)(u-%) = 0 in

~u (1.7)

where

~

f, @

are given in

tire associated with

O

(5) =

o

O, and where

~-~ denotes the conormal deriva-

A .

(I) In the physical applications, n =1,2,3. In the problems of impulse control that we study in Section 2, the dimension n cau be very large.

(2) HI(o) = {~I ~, ~x . . . . '~x E T'2(O)}, ~ being real I -

li~N~1(e)- i--D1

valued ;

n

i

dx

+

~

dx.

(3) Actually this~boundary condition should be dropped at boundary points (if any) where u = %.

t18

Of co~rse

O is divided in two regions where r e s p e c t i v e l y Au = f and

u = # , with

(1.4) and (1.5) being satisfied everywhere,

gions being not given ; the interface

and these re-

regions is a

between these two

free surface. ~e now transform the problem that it is equivalent

(1.8)

a(u,v-u) u <~ ~.

The problem wing

!. 2

u

> (f,v-u)

into a V.i.

One can check

satisfying

VV~<~

,

(1.8) is what is called a variational inequality

[13] and

sis (1.3),

(1.4)...(1.7)

to finding

[10]; it follows from these works

the problem

(1.8) admits a unique

(V.I) follo-

that under the hypothe-

solut~gn.

Various remarks.

Remark

1.1.

A large number of problems of this type,

in Physics and in Mechanics

in stationary

or non stationary

one has to use the tool of V.I. of evolution, ples and variants etc.. Remark

arising in elasto-plasticity,

are studied

lead to problems

cases ; in the latter case, introduced

in [ 10]. Exam-

in non-newtonian fluids

in the book [6].

I. 2.

In the problem alluded

to in Remark

I. I , one obtains a V.I on the

. A very important

~'physical" tmknown function

u

theory is due to C. Baiocchi

[I] who showed that classical free boundary

problems

arising in infiltration

ph3Tsical" unknown function, used, permits

step forward to the

theory can be reduced - on a new " non

say u -

to a V.I. on u . This idea, properly

the solution of a number of ether free boundary problems

of. [ < [ 5 ] [ 7 ] . Remark

I .3.

For the problems implemented

of V.I. numerical methods

are known and numerically

; a report is given in the book of Glowinski,

and the Author [8].

~r4moli~res

;

119

Ii.

IN~ULSE CONTROL PROBLEMS.

2.1.

Formulation of the problem.

We give here a very formal presentation

; for precise

statements we

refer to [3],

I) and 3). We explain the problem on an example.

that we have,

at time

t, an amount

x = [xl .... ,Xnl

We suppose that we know the cumulative s > t ; D(t,s)

can be deterministic

double sequence

Vxt = I t ~ Oxt <

and where ~xlt

,

2 Oxt,

We denote by here,

;

T

02xt

3 <'" " < °xt

t

and

Vxt

is a

are the instants where we place orders, are the amount of goods we order at time

this double sequence

the horizon,

to fix ideas,

that

is deterministic

or stochastic.

it can be finite or infinite

; we suppose

T < ~ .

The cost function i~ defined as follows (2.2)

A policy

t ' ~xt ; Oxt " ~xl ; "'"

2 ~xt ' --- E ~ n

I ~xt

-.-

demand D(t,s) between

or stochastic.

(I) .

:

(2.1) where

of goods

We assume

Jxt(Vxt) = E

:

f(Yxt(S),s)ds

+ Nxt

t (where one should drop the Expectation if the problem is deterministic), where

i)

: Yxt(S) = state at time s = amount of goods at our disposal at time s (this is uniquely determined D(t,s)

ii)

is given and if Vxt is chosen)

iii)

;

f(x,s) = holding cost by unit of time if = outage cost Nxt

if

by unit of time if

x > 0 , x < 0 ;

= number of orders we place during the period

It, T] ; we can assume here that we have to pay I for placing an order.

(I)

~herefore the dimension in this problem is equal to the number of items and consequently n can be very large.

~20 This is an impulse control problem. We want to minimize

(2.2) with res-

pect to "all" possible policies (2.1). Let us define (2.3)

u(x,t)

=

inf Jxt(Vxt)

V~at we w~ut to show (for precise statements and proofs, we refer to the works of A. Bensoussan and the Author referred to in the Bibliography), is that

u(x,t) can be characterized by the solution of a new free

boundary problem and that this free boundary problem can be studied by using the technique of u ~ s i y a r i a t i o n a l

inequalitie ~ (Q.v.I.).

2.2. Free boundary problem.

One can show that

u(x,t)

satisfies the following set of inequalities

and equalities (I) : ~u (2.4)

-~

(2.~)

(2) +Au-

u

-

f~
~(u)

u(x~)

(2.7)

= 0

t E [t o ,~]

•

g 0 ,

~_ Cu+ Au-f)(u-~(u))

(2.S)

x~n,

= 0

,

where in (2.5) (2.8)

~(u)(x) = I + inf.u(x+~),

~ = {~i }' ~i >/ O.

Of course (2.2) obviously follows from the definition.

The proof of

(2.5) is immediate : if we consider the ~articular policy consists in ordering

~ (~i>~ O) at time

~xt which

t, then :

u(x,t) = inf Jxt(Vxt) ~< inf Jxt(Vxt) = I + inf Jx+~,t(Vx+~,t) hence (2.5) follows. The proofs of (2.4)(2.6) are much more complicated. (I) Of course one has to make here precise hypothesis on the nature of the Demand D(t,s) ; cf. A. Bensoussan and the A., loc.cit. (2) The operator A is determined by D . It is a second order (resp. first order) operator in the stechastic (resp. deterministic) case.

121

let us observe now that ( 2 . 4 ) . . . ( 2 . 7 )

is indeed a free boundary problem.

I~virtue of (2.6) there are two regions in say

S, one has

~

x [to, T ] ; in one region,

:

(2.9)

u = M(u)

and in the complementary region

au + A u - B-~

(2.1o)

The interface between

S

C

one has

:

f = 0 •

and

Some remarks are now in order

C

is a free boundary.

:

Remark 2. I. u ~ M(u) defined by ( 2 . 8 )

The non linear operator since to compute neighborhood

M(u)

of

is of non local type,

x it is not sufficient

at point

to know

u

in a

x.

Remark 2.2. Many other examples

or operators of the type of

arise in applications.

M

as given by

Cf. A.Bensoussan and the Author,

loc.

(2.8)

cir.

Remark 2.3. From the solution

u

one can derive the best policy (which exists)

this is an extension of the "s - S policy", also the Bibliographies

introduced

;

in [12][14](see

of these works).

Remark 2.4. In Section 2.3. below, we briefly consider the stationary case, over in a bounded domain blem is then to find

u

O of

~n

with smooth boundary

Au - f ~< 0 , in O

(2.12)

u - M(u)

(2.13)

We assume that ~)

~< 0 ,

(Au-f) (u-M(u))

~ A

= 0 ,

(2)

au

(2.14)

F (I); the pro-

satisfying

(2.11)

and

and more

=

0

on

I~

is given by (1.1) sm_d satisfies

(1.3).

CZ. [311) 3),for the interpretation of the problem. The introduction of a bounded domain O eliminates some technical difficulties. (2) Actually this boundary condition should be dropped at boundary points where u = M(u).

122

2.3.

Quasi v a r i a t i o n a l inequalities

The a n a l o g y between ( 2 . i l ) difference is that

.o.

(Q.V.I.).

(2.14)

and ( t . 4 )

...

(1.7)

is

clear

; the

~ , w h i c h is g i v e n in (1.5), is now "replaced" b y

M(u) which is not given. One can check that (2.11)

o.. (2.14) is equivalent to finding

u

such

that

(2.15)

I

a(u,v-u) ~ ( f , v - u )

V v ~ ~(u)

,

It u ( m u ) . The p r o b l e m (2.15) is called a quasi v a r i a t i o n a l inequalities One can show the existence

(Q.V.I.).

and uniqueness of a m a x i m a l solution

w h i c h can be obtained as the limit of solutions of V.I. Let us introduce

(2.!6)

u°

s o l u t i o n of the N e u m a n n p r o b l e m :

a(u°,v) = (f,v)

V v E HI(o)

u n , n >i 1 , as the solution of the

and let us define

a(u~,v-u ~) ~ ~,v-u n)

(2.17)

V.I.

:

7 v ~< m u d - l ) ,

One can show that (2.18) and obtain

u u

o

~ u

I

~

..° ~ u

n

~

..°

~ s o l u t i o n of (2.15), as the limit (in LP(~)

and HI(o) weakly)

of

u n as

V P

finite

n~

Remark 2.5. Algorithm (2.17) used jointly w i t h n u m e r i c a l technique for solving V.!. gives n u m e r i c a l methods for solving Q.V.I.

Cf. [2][9].

Remark 2.6. For n u m e r i c a l purposes,

one is of course faced w i t h the problem of dimen-

sionality. We shall report elsewhere on the use of d e c o m p o s i t i o n methods and sub-optimal policies.

123

BiBZIOGRAPHY

[I] C.Baiocchi,

C.R.Acad. Sc., 273 (1971), 1215-1218.

[2] A.Bensoussau, M.Goursat and J.~.Lions,

C.R.Acad.Sc.Ps~is, s@ance du

2 avril 1973. E3] A.Bensoussan and J.L.Lions, I) C.R. Acad. Sc. SEance du 2 avril 1973, 2) C.R. Acad. Sc. S@auce du 2 avril 1973, 3) Book in preparation, 4) Work to the memory of I.Petrowsky, Ousp.Nat. Nauk 1974 5) Int. Journal of Appl. mat. and Optimization. E4] H.Br~zis and G.Duvaut, C.R. Acad. Sc. Paris, t.276 (1973), 875-878. E5] H.Br~zis and G.Stampacchia, C.~,Acad. Sc. Paris, 276, (1973), 129-132. [6] G.Davaut and J.L.Iions,

In@quations en M@canique et en PhEsique,

Dunod ~ 1972. [7] G.Duvaut,

C.R. Acad. Sc. Paris, June 1973.

[8] R.Glowinski, J.L.Lions and R. Tr@moli~res, Book to appear, Dunod, 1974. [9] N.Goursat,

Report LABORIA,

1973.

[10]J.L.Lions and G.Stampacchia, Comm. Pure and Appl. Math XX (1967), 493-519. 11]J.P.qaudrat and ~.Viot, Report LABORIA, 12]H.Scarf

1973.

~ath.Neth. in the social sciences. Stanford Univ.Press, 1960.

13] G. Stampacchia

C.R.Acad. Sc. Paris. 258 (1964) 4413-44~6.

14] A.E. Veinott

J. SIAM Appl. Nath, 14,5, September 1966.

A CONVEX P R O G R A ~ I N G

~ETHOD IN HiLBERT SPACE A~D ITS APPLICATIONS

TO 0PTI~3~L C0h~ROL OF SYST~I~ DESCRIBED BY PARABOLIC EQUATIONS Kazimierz Nalanowski Polish Academy of Sciences Institute of Applied Cybernetics Warsaw, Poland 1. INTRODUCTION Some optimal control problems for systems described by partial differential equations can be reduced tion of a convex functional J(y)

[2, 6] to the

problem of minimiza-

on a closed convex and bounded set D

in a Hilbert space. However, this set (the so called attainability set) usually is not given in an explicit form. Therefore the direct minimization of the functional is very difficult. On the other hand it is comparatively easy to find the points of support of D by given supporting hyperplanes° It was Gilbert who first proposed[4] to use these points of support for construction of a sequence minimizing J(y). He considered the case of a quadratic functional defined on n-dimensional Euclidean space. It turne$ out that the speed of convergence of Gilbert's algorithm is comparatively law. Several attempts were made to improve the speed of convergence of Gilbert's procedure these methods delt with quadratic functionals defined

on

~, 8, 9]. All n-dimension-

al Euclidean space. In this paper all modifications of Gilbert's method are reduced to one general scheme, which is applied to the problem of minimization of a convex functional on closed, space. The use of this scheme

convex

and bounded set

in a

Hilbert

is proposed for solving an optimal

control

problem for parabolic equations. Some computations were performed and ~he three modifications of the method were compared on the basis of obtained numerical results. 2.C0kr~

PROGRA~L~IING PROBLEM AND AN !TERATiVE PROCEDURE FOR SOLVING IT

In a Hilbert space H there is given a closed, convex

and

bounded,

i.e. weakly compact [2,11] set D. ~oreover, a non-negative real convex functional J(Yl is defined on the space H. It is ass mmed that J(y) is two times continuously differentiable and its Hessian J"(yl satisfies the following conditions:

125

n~zll2~ (J"(ylz, zl ~< Nllzll2) Y y ~ D ,

Vz~H

(I)

where 0 < n ~ N ~ . The condition (I) implies that J(y) is strictly convex. Our purpose is to find an element Yopt g D called an optimal element satisfying the relation J(Yopt) = inf J(y). (2) y 6D Since J(y) is lower semicontinuous functional as a convex ~1] and the set D is weakly compact the point Yopt exists and it is unique due to the strict convexity of J(y). At the point Yopt the following necessary and sufficient condition of optimality has to be satisfied (-J' (Yopt) ,Y) ~

(-J' (Yopt) , Yopt ) ~/~& O.

(3)

To determine Yopt an iterative procedure is used based on minimization of some quadratic approximations of the functional J(y) on closed and convex subsets of D, which are constructed successively. To this end we define functionals Ji (y) ~ J(Y) + (J' (Yi)'y - Yi ) + ~ N(Y - Yi' y - Yi ) ,

(4a)

2i(Y) ~

(4b)

Ji(y)

and

J(Y) + (J~ (Yi),y - Yi) + ~ n(y - Yi' Y - Yi ) "

$i(y)

are quadratic functionals and Ji(Yi ) = ~(Yi ) = J(Yi ) •

(5a)

It follows from (q) that $i(Y) ~ J(Y) ~ J i ( Y ) We denote by

Yi 6 D i

the unique element satisfying the relation

Ji-1 (yi) =

Let

(Sb)

i~ Ji-1 (y)" Y£D i

(6)

Yi+l £ D by any arbitrary element such that (-J' (yi), y) ~< (-J'(yi), ~i+I ),

i. e. Yi+l is any point at which -J' (yi) supports D.

V y £D,

the hyperplane ~Ii

(7)

orthogonal

to

Theore m . If the family of closed, convex and bounded sets Di~ D is constructed in such a way that

[yil u

c Di+1

(8)

then the sequence {Yi~ is strongly convergent to Yopt"

126

The proof of Theorem is given in Appendix q. As it is shown in Appendix 2 the optimal value of the functional can be estimated as follows

max

O, J(y~, ) +

+

#(yopt~ ~

(J' (Y~) ' ~+~I (91

#(yi ) ,

where

In the particular case where

J(y) is a quadratic functional of the

form J(yl

= (z - y, z - yl

(IoI

where z ~ H is a given element, which does not belong to D, we have n= = N = 2,and the condition (71 takes on the form (z-yi,yl~(z-yi,Yi+11. In this case the estimation (9~ reduces to the following one 1 ~ max~ i

,

~

(z-~ ,z-y~l

(z-Y°Pt'z-Y°Pt) ~(z-Yi,z-Yi~(11)

3° SOi~ METHODS OF CONSTRUCTING THE SETS D i Note that

according to

Theorem the

iterative procedure

is con-

vergent to the optimal solution if the condition (81 is satisfied matter what are the forms of the sets

D i. However the speed

no

of con-

vergence depends very much on the shape of D i. We present three methods of constructing the family D i known from literature.In all these methods the sets D i are characterized by

fi-

nite number of parameters. a. Gilbert's method [4] This method is the simplest one. We start from any arbitrary point YO ~ D and at (i+q~-th

iteration

we

choose

as

Di+ I

the

segment

In this case the element Yi+1 can be easily found from the formula Yi+1 = Yi + ~C i+1(yi+q - yi I , where

n i+I = ~ n Hence the points

(-J' (yi),

i+n - yil

' ~(Yi+1 - Yi' ~i+I - Yi ~ J

(121

Yi+1 are determined in a very simple way.However

127

the speed of convergence in terms of the number of iterations is very [4]. This slow convergence is due to the fact that at each step we use only the information obtained in the last iteration (namely yi ) and we do not take into consideration the previous ones. To overcome this disadvantage the following modification was proposed: low

b. Barr's method [3] In this method as in the previous one we select an arbitrary tial point Y0 and we put D 1 = Fy0, Yl]' but as

o

coo

co v

F 1 : conv ~ i '

,oo

2 Yi'

...

-..

Di

ini-

we choose

:

o

i+1] ' Yi J '

(131

where y.3 l (j = 1, ..., i+1) denotes the vectors spanning the set D i . Every element y £ D i can be represented in the form a convex combination of y~, i.e. i+1

j=l i+1

where ~Cj ~ 0 ,

~ ~ j=l

= I.

(1~)

Hence~ in order to find the element Yi we have to determine the values ~ of the coefficients < J at which the function Ji_l(~) assumes its minimum subject to the constraints (14). This functional can be rewritten in the form i+1 A

%-I(y) = J(Yi-1) + (J' (Yi-1), ~ ,

~JY~) +

j=l -

,

: °o - [ < c i , ~- > + ~ ~ ,

d~

Yi

Yi-

=

Bi ~ > ]

where Co = j(yi_l) + 1 N(Yi_l, Yi_l)

are (i+l)-dimensional vectors Bi = { _ I

N(y~, yk)~i+l/i+l is (i+1)× (i+l}-dimensional

matrix)

('15)

128

~

denotes the inner product in (i+l)-dimensional Euclidean space. ~inimization of (15) subject to (14) is a typical finite dimensional quadratic programming problem and it can be solved using any of well knowuprocedures [5]. Note that in this method ter each iteration, what in To avoid this difficulty D i were proposed ~, 7, ~ , sion of the vectors

the dimension of B i usually increases afturn increases the time of computations. some modifications of the construction of that makes it possible to limit the dimen-

~ i to a given number.

c. Nahi-V~hee!er~s method [8] Both methods presented up to now were general in the sense that they could be applied to any set D. The third method takes advantage of a specific form of the set D. In linear optimal control problems as a set D we choose the so called attainability set.We consider the case of scalar control function with constrained magnitude and performance index depending on termin~I S ~ a ~ : We put

D=

y

t

, y=

A(t)

u(t) at : lu(t)l

,

(165

0

where: u(t) ~ R 1 is a scalar control functionl A(t) is a continuous linear operator from R fl into H, T is a fixed time of control. The set D defined by (16) satisfies [~ all conditions of Theorem. The appropriate elements y will be found by means of determining control functions u('l corresponding to y according to (16). It is easy to check ~, ~ that the control ~i+l(t) corresponding to the element

Yi+l satisfying (7) is given by 5i+l(t) = sgn (-J' (yi5 , A(t))

(175

As Uo(-) we choose any arbitrary piece-wise constant function satisfying the condition

~uo(t)1% I,

rn

V t £LO,~ ]

(18)

Let

I toj)

(,95

be the set of all points of discontinuity of Uo('). The function Ul (') is derived from (17). By 1 we denote the set of all points of discontinuity of Uq('). Let us assume that this set has a finite number of elements and let us put

129

We introduce the set U I of all piece-wise constant functions isfying (q81 with the points of discontinuity at t~. We put

[

T

D1 = ~ Y

L ~ ~ o~v~o~ ~

sat-

: Y =S 0

A(t)u(tl

dr; u(.) £UII ]

•

(215

{~o~ ~ { ~ ~ o~ ~ ~ ~o~eove~ ~ e ~e~ ~ ~

closed and convex [8], hence it satisfies all the conditions of Theorem. By ul(. ) E U I we denote the control functions which corresponds to the element Yl satisfying (61. In the same way as before we denote by Ui+ 1 the set of all piece-wise constant functions satisfying (18) with the points of discontinuity belonging to the set = ~k where tiJ and ~ki+I denote the points of discontinuity of ui( "I and ui+1 (") respectively. It follows from this construction that usually the number of the elements tJi increases at each iteration. Let the number of these elements be L i. In the same way as D I we define Di =

y • y =

A(t) u(tl at; u(.) ~ u

•

(23)

0 Now we shall show how to find the element Yi satisfying (65. Let us denote j+l ti li~ =

/

A(t)

at,

j = o, I,

..., L i

(2g)

t~ Li+1 where t~ = 0, t i = T, and i~i are given elements of the space H. It follows from the definition of U i and D i that

j=1

where ~c j are the values of the control function u(.) in the intervals (t j-l, tJ)Thus to find it is enough to determ/ne the coefficients ~_~ ui(.) J.

130

satisfying the condition

(26) that minimize the functional Li+1

~i_I(~) : j(yi_1) + (J (Yi_l) ,~

•Li+l

~Jl~) +

Li+l j=l

\j=1 =

~

j=l [ + <~t, Bio~>]

c o-

(27)

Hence we obtain a quadratic form similar to (15) and to find oL~ any procedure of quadratic programming in a finite dimensional space can be used as it was proposed in Barf's method. ~. APPLICATION TO 0PTIMAL COETROL OF S Y S T E ~ DESCRIBED BY PARABOLIC EQUATIONS The method of programming presented above can be used to determine optimal control of systems described by linear parabolic equations. Let V and H be two Hilbert spaces such that V ~ H C,V', and let A be a linear, continuous and coercive operator from V to V'. Let U be another Hilbert space and G e ~ (U; HI • For t 6 [0,T] we consider the following equation dy(t)/dt + A y(t) = G u(tl

(28)

with the initial condition y(01 = Y0 a H. For each ugL2(O,TIU)

y(u)

e

equation

(28a)

(28) has

[6] a unique

solution

c(o,~; R).

A function

u

~

L2(0,T! U) is called an admissible control if

u(t) 6 %~

for almost all

t a [0,T] ,

(29)

where l I c U is a closed, convex and bounded set. The optimization problem is to find such an admissible control u0 which minimizes the functional ~(y(T; u)) . (30) Let us denote D = {y(T)

a ~

~

d~a__V_( tI +

A y(t)

= G u(t))

y(O)

=

Yo

: u ( t ) c~

ll~j . (:31)

It can be shown [6] that D is closed,convex and bounded, therefore

131

to find minimum of J(y(T;u)) we can apply the procedure described in Section 2. To this end we shall show how to find Y-i+1 satisfying (7). Let us introduce an adjoint equation -(dp(t)/dt)

+ A ~ p(t) = 0 ;

(52)

p(T) = -J~(y(T| u i)) .

(52a)

It is easy to show [6] that the condition (7) is equivalent to the following one (G~p(t),

Ui+l(t)) ~ (G~ p(t), u)

Vu~and all

for almost t

e[O,T].

(33)

Thus to find Yi+l = Y(T;Ui+l) we have to integrate two equations: first (321 in order to obtain Ui+l from (331, and then (28). Having ~i+l(t) and Yi+1 we can apply the procedure of minimalization described in Section 2. As a numerical example we consider a system described in an oblong (0,1) x (O,T) by heat equation ~y(x, t)

~2y(x, t) x2

~t

o

(54)

with initial condition

y(x, O)

:

o

(34a)

and boundary conditions

~.x(o, t~

:

o;

9x(1, t)

t)]o

(54b)

~x

The cost functional is a quadratic one J(ul = (z - y(T), z - y ( T I ) = ~ [z(x) - y(x,T)~ 2 dx 0

(35)

where z g L2(O,T) is desired final temperature distribution.Two types of constraints imposed on control function are considered (a) m %u(t) ~< M for almost all ta[O,T]j (361 (b) functions (u(t) are given [I, qO] as solutions of the eq. d~t(t) = _ _~I u(t) + ~

6

u(O)

=

v(t),

Oj

(37) (37a)

where p ~< v(tl ~ P. To perform computation eq. (34) was discretized both in space variables. The following numerical data were taken case (a) ~ = I; m = -1; M = 1; T = 1; z(x) ~- 0.45; 0.5 A t = 2.5 10-2; A x = 10-1~

(37bi time and

132

case (b)

~ = 10; ~ = 0,04"; p = O; P = 1; T = 0.2; 0.4; z(x) ~ 0.2; A t = 4"× 10-3; 8 × 1 0 - 3 ; ~ x = 5 X lOW 2.

The computations were performed for all three methods using a computer 0DRA 1204 and obtained results are listed in Table 1. The considered control functions are

piece-wise constant

and the

last column of the Table I gives the number of discontinuities obtained in each method. This number characterizes the simplicity of obtained control. TABLE 1 No. of iterations

(a)

35 5

9oo J4.4"80 y" 10 - 3 37@ I 1-576 ~" 10 - 3

6

3911 I 6 . 1 6 3 x 10-4.

z(x) - o.45

Time I

J(y(T; u~

Type of constraint s

Estimation of error

NO. Of diseont.

4.320 ,,, lO - 3

10

sec. i

1 . 5 7 6 ~ 10 - 3 1 . 7 6 9 x 10 -zt"

5 5 ii ,ii,

(a) ~(x) ~

o.5

4" 5

(b) T=0.2 sec

(b) T = 0.4 sec •

14 6 6

800 11. 599 ~ 10 - 2 185 I 1 , 5 3 0 ~" 10 - 2 615 I 1.4"89 × 10 - 2

323 4:5

6

8.372 × 10 -4 6.659 x 10 -1(

3

8,10 -2 9000

576 14.552~ 10 -2 4151

1.360 × 10 -3

I # . 5 4 6 ~ 1 0 -2

10 -5

5

1.296× 10 - 4 3-20@× 10 - 4

4 4 19 7 6

7

540 13"414 ~10 - 4

3.O90× 10 - 3 3 . ~ 1 4 × 10 - 4

6

3647 1 4 " 9 8 9 ~ 1 0 -6

# . 9 8 9 × 10 - 6

26

1 -- Gilbert's method; 2 -

659 1 3 . O 9 0 ~ 1 0 -3

2

Barf's method; 3 - Nahi-Wheeler s method, i

An example of the plots of values of functional vs. the number

of

iterations for different method is given in Fig. 1 and the values

of

the functional vs. computation time in Fig. 2. The forms of optimal control and the final temperature

distribu -

tions obtained using Nahi-Wheeler method are presented in Fig.3. It follows from the obtained results that the speed of convergence expressed in terms of the number of iterations is the lowest for Gilbert's method. On the other hand the time of each iteration for method is comparatively low. Hence the total computation time the same order as in Barf's method and several times Ns~i-Wheeler's . As far as the accuracy and simplicity

lower of

this is

of

then

in

obtained

control is concerned the Nahi-Wheeler's method is the best, Barf's is not very much inferior and Gilbert's is much worse than two others.

133

I0-~

fO-3

o

)':1'~'k'/o'/2'/e'/e.

'~

50

no. o f itepado~

I Itl

J

I

I

I

II1~

ioo

t

I

Iooo

t

t[soY7

Fig.2. Value of functional J(u) vs. computation time. Case (a) z(x)---- 0.45

Fig.1. Value of functional J(u) vs. the number of iterations. Case (a) z(xl ~ 0.45

y(x,r)' 7"~

0,45

0,2

O.'J

O..'/45 I

O.5

I

-1.ofCase(a), zfx)=-O,45

I

L

~.Ox-

I

v(t)~

L

L

0.5

1.Ox-

0.!

o.2t/sec.7

Jk _ 0

Case(b), 7=0.2sec

Fig. 3. Form of control and final temperature distribution Summing up it seems that the best is Barr's method.

In

the

case

where the high accuracy is necessary Nahi-Wheeler's can be applied. REFERENCES 1. Arienti G., Colonelli A., Kenneth P. Cahiers de IRIA 2,81-106(1970) 2. Balakrishnan A.V., Introduction to optimization theor~ in a Hilbert space, Springer Verlag, 1971. 3. Barr R.0. SIAM J. on Control 7, 415-429 (1969) 4. Gilbert E.G. SIA~i J. on Control 4, 61-80 (1966)

134

5. Hadley G. ~ Nonlinear and d,Tnamic programming,Addison-Wesley,1964 6. Lions J.L., Optimal control of systems 5overned by partial differential equations, Springer Verlag, 1971 7. btalanowski K., Arch. Autom. i Telemech. 18, 3-18 (1973) 8. Nahi N.E., V~aeeler L.A., IEEE Trans. AC-12, 515-521 (1967} 9. Pascavaradi T., Narendra K.S., BIA]{ J. on Control 8,396-402(1970) 10. Sakawa Y., IEEE Trans. AC-11, 420-426 (19661 11. Veinberg M.M., Variational method and method of monotone operators, Gostiechizdat~1972 (in Russian} APPENDIX 1. PROOF OF THEOP~J~I First it will be shown that the sequence{J(y~)] is non-increasing. Indeed,substituting in (#a) y = Yi+l,and taking-lnto consideration (5), (6) and (8) we get (1.1)

J(Yi+l ) ~< Ji(~i+l ) ~< Ji(Yi ) = J(yi ) •

On the other hand sequence ~J(y~)] is bounded from below J(Yopt ) . Therefore it is convergent.-Weare going to show that

by

(1.2)

iim J(yi ) > J(Yopt ). i ---~o~ To this end first we shall prove that

(1.3)

lira (-J~ (Yi)' Yi+l - Yi ) = O. i --~o~ Let us assume that (1.3) is not satisfied.Then taking into account (7) ~e conclude that there exists a constant 6 > 0 such that for every integer m > O there exists a subscript ~ > m such that

(-J'(Y~), ~ +1 - Y9 )~ ~.

(1.4)

6

Let us denote

Y = Yr_ + ~ ( ~ + I - y ~

)' ~

(o,I).

u

It follows from (6) and (8) that

5~ (~) ~

~ (~ +1),

Vz~(O,ll.

Taking into consideration (4) and (1.4) we get 1 2

(-J'(y~l,y2 +I-Y~)>- ~IIY~+I-Y~

+~(-~'(y~l,~+1-y~

i+

where

(1.5)

~ =

sup

Ily - z l l 2 < ~ .

y,z e D

l>~tting

where

05= rain{l, £/N~]1 we get

{ ~N ' I

> 0 does not depend on ~ .

Taking into consideration (5b) and (1.6) we obtain

~(~) + - (J~11(~)'4 J(~) ~+1 - ~) J(~+l)\< ~?(Y~+I)+ ~ J~+l -

+

~ ]

135

or which contradicts the convergence of the sequence ~J(yi) ~ . This tradiction proves (1.3). Now let us prove 41.2). Denoting Y~ = Yi + ~ (Yopt -

Yi ),

con-

~g[0,1]

we get from (~) and (5)

(1.7)

~i (y~) : J(Yi ) + ~ (J' (Yi) 'Yopt - Yi ) +

nj~ 2 IIYopt - Yiil 2~<

~< J(ya ) ~< (1 - ~ )J(yi ) + ~ J(Yopt)' From (7) and (1.7) we have

(1.8) (-g' (yi) 'Yi+l - Yi ) >~ ('J' (Yi)Yopt - Yi ) ) J(Yi ) - J(Yopt ) ~ O. 41.3) together with (1.8) prove (1.2). For J(y) is weakly lower semicontinuous,the set D is weakly compact and Yopt is the unique element satisfying (2) then (1.2) implies (1.9)

Yi

~

Yopt"

To prove the strong convergence we note that it follows from (1) that

(1.1o)

nlly- zll ~< (J'(y), y - z) - (J'(z), y - z),

Yy,z

eD.

Hence from (7) and (1.10) we get

(1.11) nAYopt - Yi II~< (J' (Yopt) 'Yopt - Yi ) + G-J' (yi) ,Yopt - Yi )~< 4

(J' (Yopt) 'Yopt - Yi ) + (-J' (Yi) '~i+1 - Yi ).

For i --~ ~ the first term on the right hand side of 41.10) tends to zero by (1.9) and the second by (1.3). Hence (1.12)

lira l~Yopt - yill = O i--~ APPENDIX 2. ESTI[vL%TION OF THE ERROR

q.e.d.

It is obvious that the following estimation takes place (2.1)

J(Yopt) ~< J(yi).

To estimate J(Yopt) from below we take advantage of (5b) and find the estimation of Ji(y ) on D. Consider the situation where the hyperplane M i separates the set D and the point at which $i(y) assumes its global minimum.lt takes place iff o. Using (4) we rewrite this condition in the form (2.2)

(J' (yi) , J' (yi) + n(Yi+q - yi )) > O.

In this case the minimal value of

$.(y) on

D is not greater then

},36

the minimal value of this functional on M i. This minimal value is assumed at the point

y, where

~i(y~ : ~J (Yi)" Taking into consideration that Yi+1 ~ ~i we get from (#b)

~

( (J' (Yi)' ~i+I - Yi) %

+ I) (J (yi), Yi+1 -Yi )" From (5b) ~ (2.1)~

the error of

(2.2) and (2,3) we obtain i-th iteration.

the estimation (9) of

ABOUT SOME FREE BOUNDARY PROBLEMS CONNECTED WITH HYDRAULICS CLAUDIO BAIOCCH_!I, Laboratorio di Analisi Numerica del C. N. R. PAVIA,

ITALY

N.I. DESCRIPTION OF THE PROBLEM Two water reservoirs,

of different levels, are separated by an

earth dam, which is assumed homogeneous,

isotropic and with impervious

basis. We ask for [he steady flow of the water between the reservoirs, and in particular for the flow r e g i o n ~ unknown part y= ~(x)

(or, in other words,

for the

of its boundary).

Denoting by D the dam, and by Yl and Y2 the water levels

(with Yl

greater than Y2; see picture I) the mathematical problem can be stated as follows

(see for instance L8J for the general treatment of this ty-

pe of problems):

]

Find a subset ~ function u=u(x,y)

~tu=Yl ( ~n

of D, bounded from above by a continuous decre~

sing function y= ~(x),

such that there exists in ~

an harmonic

which satisfies the boundaryGconditions:

on AF; u=y 2 on BC; u=y on C C ~

and FC~ ;~nU-0 on AB and FC

.

denoting the normal derivative; we point out that on the "free

boundary" we have both the Dirichlet and the Neumann condition).

x

PICTURE I

I38

In a paper of gle,

the origin

W(X,y):] y

LU(X,t)-

the new unknown W ~0;

g=-

St

for

I proved I

D is

a rectan-

AF and BE are vertical,

(x,y)6 ~

w(1-~w)

if

AB is

, setting:

function w satisfies

is defined

that,

;

w(x,y)=

0 for

(x,y)6D--~

the relations:

= 0

on D;

w=g

on

9D

on f~ D by the formula:

(Y-Yl) 2 (y_y2) 2 2 on AF; g= 2 on BC; g linear on AB; g=0 elsewhere.

The system precisely,

(I)r (2) can be studied

as a varionational

inequality;

d e n o t i n g by H I (D) the space of square summable

on D, whose K :

picture

is in A)

AW~I;

where g(x,y) (2)

(see _;I]' /2j)

30,a6X]0,yIC(in

D =

horizontal;

(I)

1971

first derivatives

~ v6HI

are also square summable,

(D) ; U ~_.0 on D; v : g

functions

and setting:

on ~ D~

the function w satisfies: (3)

w E K;

~v~K:

Vice-versa tion;

~_[gradw,grad(v-w)]dxdy~_

it is well known

(see ~9 ~ ) that

and we can proof that w solution of = ! (x,y) ~ D; w(x,y) ~

(4) ~

(and obviously boundary

problems

inequalities

theory;

(see [3], L4],ES~);

pe of the dam is a t r a p e z i u m

(see picture

w ~0;

A

w _~ I;

{ DyW=y-y I on AF Denoting

by

ge" of the dam

-q

w(1-~w)

treatment.

to a wide class of free

here I will

corresponding

theo-

and we will see later

allows us also a new numerical

cribe one of these problems,

(5)

on ~

) we get a solution of the free

At present we can apply similar methods boundary

setting:

In this way we obtain an existence-uniqueness

rem, via the v a r i o n a t i o n a l that this approach

(3) has a unique solu-

(3) is such that,

0 ~ ; u=y-D ~ w

graph of ~ = ~ - g D

problem.

~(w-v)dxdy.

limit myself to des-

to the case where the sha2). System

(I) becomes now:

= 0; w=g on FEB; w linear on AB;

(we reobtain w=g on ~ D if AF is vertical) . the value of DxW on AB, q represents

(which is now unknown;

in the previous

the "dischar

case i~.-was

2 2 2 -I). Now we can define, for real q: q= (YI-Y2) a ) 6 H I (D) ; v ~ 0 on D; v=g on FEB ; Vx=- q on AB] Kq = (~v and we must modify a bilinear

(3) into the formula

form associed with

"natural boundary

condition"

- A

(like

on AF is

9 ~

(6) following

(where a(u,v)

.~gradu,gradv dxdy) ):

whose

is

139

(6)

W q 6 Kq; ~ V 6 K q

:

Actually we can proof que value

q%

a(Wq,V-Wq) ~

(Wq-V)dxdy + 2

~AF(Y1_y )

(V-Wq)dy

(see (4 3 and ~7J ) that there exists a uni-

of q, such that

(4) with w = Wq~

gives the solution

of the free boundary problem; and this unique value of q can be carac~ terized as the unique value of q such that the corresponding w

solu-

tion of (6) is a "more regular" function, for instance w q 6 CI (D~.

PICTURE 2 N.3. NUMERICAL APPROACH In the cases where the shape of the dam is simple gle, or a trapezium) we discretized the problem

(like a rectan-

( (3) or (6) respec-

tively) by a finite difference scheme. In the first case we must solve the set of inequalities: I Wh~0;

AhWh

~ I;

Wh(1-AhWh ) = 0

in the interior gridpoints~

(7) wh = g

in the gridpoints of the boundary

where h is the mesh amplitude and ~ h sation of the Laplace operator;

is the usual 5-points discreti-

system (7) has a unique solution and

can be solved by means of an iterative scheme, with S.O.R. and proje~ tions

(for the details see L~J,[~J) ;

A h will be defined as the union

of the mesh with center in the gridpoints where w h > 0 ; Uh = Y-~2Wh result (8)

setting also

( ~2 discretizing Dy) we have the following convergence

(see L-SJ) :

u h converges to u in H 1 discrete; ~

= interior of lim ~ h . h-~0

140

In the second case we must m o d i f y i W h ~- 0; ~ h W h (9)

w h = g on FEB;

an a p p r o x i m a t i o n q(h)

in the interior g r i d p o i n t s

on A~7, ~ 2 Wh = Y-Yl on AF

J 1 W h = -q

and we m u s t add to the s o l u t i o n of of q ~

of the e q u a t i o n

fh(q)

that

is a convex,

q,--~fh(q)

(7) into:

Z I; W h ( 1 - ~ h W h) = 0

(9) an a l g o r i t h m for the choice of

In order to do it we look for the tooth

= 0 where fh (q) = ~ 2 W h (F); it can be proved s t r i c t l y d e c r e a s i n g function, w h i c h as-

sumes o p p o s i t e signe in two known points q0 and ql; the u n i q u e tooth of fh(q) q=q(h), we can proof notations

setting q(h)

and w h as the u n i q u e solution of

[seeO~J)

that q(h)

similars to the ones used in

c o n v e r g e s to q e

as

(9) w i t h

and

(with

(8) ) the c o n v e r g e n c e result

(8) is still valid. The n u m e r i c a l results o b t e i n e d by a p p l y i n g our m e t h o d show a good a g r e e m e n t w i t h the ones r e p o r t e d in the literature; for some comparisons.

we refer to ~ 5 #

We want to p o i n t out that the main advantages

of our m e t h o d from the n u m e r i c a l p o i n t of p r o g r a m m i n g and the speed of e x e c u t i o n

v i e w are the s e m p l i c i t y of

(in c o m p a r i s o n w i t h a classi-

cal d i f f e r e n c e m e t h o d we see that both e x e c u t i o n time and F o r t r a n sta tements n e c e s s a r y to p r o g r a m the a l g o r i t h m are reduced to about one third) ; m o r e o v e r our m e t h o d is r i g o r o u s from the m a t h e m a t i c a l point of view. REFERENCES I C. BAIOCCH[. Sur un p r o b l ~ m e ~ f r o n t i ~ r e libre t r a d u i s a n t le filtr~ ge des liquides ~ travers des m i l i e u x poreux. C. R. Acad. Sc. Paris

273 ( ~ 2 ~ )

1215-1217.

2 C. BAIOCCH!. Su un p r o b l e m a di frontiera libera connesso a q u e s t i o ni di idraulica. Ann. di Mat. 97 (1972) 13 7-127 3 C. BAIOCCH!, V. COMINCIOLI, L. GUERRI, G. VOLPI. Free b o u n d a r y problems in the theory of fluid flow through porous media: n u m e r i c a l approach. To appear in C a l c o l o 4 C. BAIOCCHI~ V. COMINCIOLI, E. MAGENES, G. POZZI. Free b o u n d a r y pr~ blems in the theory of fluid flow t h r o u g h porous media: e x i s t e n c e and u n i q u e n e s s theorems. To appear in Ann. di Mat. 5 C. BAIOCCHI~ E. M~GENES. P r o c e e d i n g s of the S y m p o s i u m "Metodi valutativi nella F i s i c a - M a t e m a t i c a " , Acc. Naz. Lincei, Rome dec 1972o 6 V. COMINCIOLI, L. GUERRI, G. VOLPI. Analisi n u m e r i c a di un p r o b l e m a di f r o n t i e r a libera c o n n e s s o col moto di un fluido a t t r a v e r s o un m e z z o poroso. Publ° N. 17 of Lab. Anal. Num. Pavia ( I ~ I ) 7 V. COMINCIOLI. Paper in preparation. 8 M. E~ HARR. G r o u n d w a t e r and seepage. Mc Graw Hill, N.Y.I 19 62 9 G. STAMPACCHIA. Formes b i l i n ~ a i r e s coercives sur les ensembles convexes. C. R. Acad. Sc. P a r i s , 2 5 8 (1964) 4413-4416.

METHODE DE DECOMPOSITION APPLIQUEE AU CONTROLE OPTIMAL DE SYSTEMES DISTRIBUES A. BENSOUSSAN, R. GLOWINSKI, J.L. LIONS (IRIA 78 - ROCQ~NCOURT

- FRMNCE)

INTRODUCTION Le but de ce travail est d'Etudier l'applieation des techniques de d6compositioncoordination ~ la resolution numErique des probl~mes de contr$1e optimal gouvernEs par des Equations aux dErivEes partielles.

L'approche retenue est celle procEdant par

d6composition du domaine o~ on ramgne, par un proegd~ it6ratif, la resolution du problgme de contr$1e optimal initial, ~ celle d'une suite de sous-problgmes de m~me nature relatifs ~ des sous-domaines rEalisant une partition du domaine initial. On dEgagera les principes de la mEthode sur un probl~me de contrSle oO l'Equation d'Etat est lingaire elliptique et le domaine rectangulaire, mais les techniques dgcrites cidessous s'appliquent ~ des gEomgtries plus compliquges et ~ des gquations d'Etat non linEaires,stationnaires

ou d'Evolution.

I - ETUDE DU PROBLEME GLOBAL I.I - NOTATIONS - HYPOTHESES. On eonsidgre un domaine

~$ ~

de la forme F2

Figure 1

et on note

~

U ~ [J % U ~

= r]

la

F4

fronti~re de ~ . L'Etat du syst~me

est donne par "Az = 0

z]

(1.1)

%Ur4

=0

F I UF2 oE

~z v = (Vl,V 2) est le contrSle ; v i E n2(Fi ) , i=l,2. Dans (I.I), ~n

repr6sente la

dgrivEe normale orient~e vers l'extSrieur de f~ . Le critgre est dEfini par (en remarquant que

(1.2)

J(v) =

/

J~

z = z(v) = z(x;v))

Iz(x) - Yd(X) 12dx + ~ I

lr

v2(x)dl~ 1 UF2

;

~ > 0 .

142 On consid~re !'ensemble

(reprfisentant les contraintes) !

(1.3)

2

l~ad = lad x Ua d

i

,

convexe fermg de L2(Fi ).

Uad

Le probl~me global est d~fini par (1.4)

Minimiser

J(v),

v E Nad .

1.2 - CONDITIONS NECESSAIRES ET SUFFISANTES D'OPTIMALITE. II est standard (cf. J.L. LIONS caract~ri~e

(I1 ) que (1.4) possgde une solution unique u,

par la condition d'Euler suivante

J/~

(1.5)

(Y-Yd)(z-y)dx + ~) !F

u(v-u)dF

>~

O

1 UF2 g

vEl~ad

, z=z(v),

On introduit l'gtat adjoint d~fini par £

t -AP = Y-Yd

(1.6)

y=z(u),

u ~ ~ad "

dans

plr3Ur4 = 0 I~ rlUr2 =o ;

alors (cf. J.L. LIONS

(I)),

(1.5) fiquivaut

¢

(1.7)

/ d

(p+~u)(v-u)dF ~ O,

Ii sera commode de r ~ c r i r e variationnelle.

'

u ~ Nad "

!es conditions d'optimalit~

sous forme d'une in~quation

On introduit !'espace de Hilbert

(1.8) o~

V v C ~ad

rl U~

={z] zE Nl(~), zlr3Ur4 =o}

v

HI(~) est l'espace de Sobolev d'ordre I.

On notera que (y,p) E g x g

. On a alors

t.1 Le.. triplet (y,p,u) (o_~ u est le contr61e optimal et

Thgorgme

y

l'~tat optimal) est

solution unique de !'in~quation variationnelle

(1.9)

f~ grad +

b

y grad z dx - / Fl U F2

[

(~)a+p) (v-u)dF

J F1 o3

a

et

b

(I) Le rSle de

uz dF + a(

> 0

grad p grad q dx -

lqz,q EIr ,v ~ l~ad , Y , P E ¥

Ur2

sont deux constantes positives arbitraires a

et

b

appara~tra au n ° 3.2

(1)

f

(y-yd)q dx)

, uE

l~ad ,

143

II - DECOMPOSITION 2.1 - HYPOTHESES - NOTATIONS. Pour ~conomiser les notations, on consid~re une d~composition de sous-domaines

~l,

~

(la g~n~ralisation F 31

an deux

sous domaines s'en d~duit aussitSt) r ~2

n

n

F2 ~2

F41

F42

Fisure 2

07, avec des notations ~videntes F3 = F3! U F32 F4 = F4; UF42 =

~! ~ ~

, suppos~e r~guli~re.

Pour i=|,2, on introduit les espaces de Hilbert (2.])

Vi = { z E

On pose

H](~i) I ziF3iU

F4 i

=0}

.

Yid = Yd I ~. i

On introduit le probl~me suivant : trouver (yi,Pi,Ui) fiant (2.2)

SY] = Y2

sur

LP]

sur

P2

dans

%ri x ¥ i

i x t;ad v~ri-

et

grad Yi grad z.dx - /

u.z. d F.

+

=~,2 1

(2.3)

i

/

a( i=1,2

~. 1

bC

1

(~u i + pi)(vi-ui)d ri) >

i=! ,2

• .

0

If z i, qi' vie %'i x gi

l

et

(2.4)

Zl = z2 [I

' ql = q21~

On a alors Th~or~me 2.1 Ii existe une solution unique de (2.2),(2.3)

donn~e par :

XUad

144

(2.5)

i Yi = restriction de y

~

~i

t

Pi = r e s t r i c t i o n

de

p

~

~i

l ui = restriction

de

u

~

ri

< D~monstration

D6finissons (yi,Pi,ui)

par (2.5). Alors d~apr~s (].I) et (1.6), on a

~AYi = 0 (2.6)

dans

Yil F3i U F4i

~i =

0

~Yi = ~-IFi ui ~f

i -A Pi = Yi - Yid 1

(2.7)

dans

%

=0

I pilr3i U r4i ~I

= O

Fi

f<

(2.8)

i (Pi + 'J ui)(vi-ui)d ri >~ O. J ri

Les relations (2.2) sont v~rifi~es (propri~t6s des traces dans HI(~)). Par ailleurs, d'aprgs LIONS-MAGENES (I) , on peut d6finir ~Yi ~ H- ~ SPi -~ (2.9) et de plus, on a (2.10)

~Yl

~Y2

SPl

~P2

7nll

8oient alors (2.8)

zi' qi

Fi v6rifiant (2.4),

Ay i z. dx + a [

= ~nl

--I

v i E Nad " On a, d~apr~s (2.6), (2.7),

(- hpi-(yiYid))qidx + b

f

1

1

(~ui+Pi)(vi-ui)dri] ~0. 1

D'aprgs !a ~ormule de Green, il en m~sulte ~Yi zid(~i) ) + a !( ]~ grad Pi grad qidx ! (]~| grad Yi grad zidx _ ~I. ~--n i i i -

(yi-Yid)qidx '~i

~--~qid($~i )) + b ~i

~ i

(pi ÷ Vui)(vi-ui)dri ~ 0 ri

et on vgrifie ais6ment, grace g (2.6), (2.7), (2.4), (2.11) que (2.11) entra~ne (2.3).

145

Soit r~ciproquement une solution de (2.3). On pose (2.12)

y(x) =

~ Yi(X) Xi(X) i=1,2

p(x) =

~ Pi(X) Xi(X) i=1,2

o~ Xi(X) est la fonction caract~ristique de ~i" II est classique que (2.2) entra~ne Y,P E Hl(g). Comme ZIF3 i = YilF3i

'

PIF3 i = PiIF3 i

YIF4 i = YiIF4 i

,

Pl F4 i = Pi IF4i

on a bien y,pE1f

. On v~rifie alors ais~ment que (2.3) implique (1.9). •

2.2 - LAGRANGEN GENERALISE. Bien que (2.3) soit une in~quation variationnelle non symm~trique (et donc ne provient pas d'un problgme d'optimisation), on peut lui associer un lagrangien d~fini par I i= ,2

/

+ a (i=~+

i

grad Yi grad zidx - /F ui zi dFi + i /~i grad , Pi grad2 qidx - /~i (yi-yid)qidx)

(2.13)

+ b [i=I,2

Fi

+ /I %(Zl-Z2)dl

(~ u i + pi)(vi-ui)dri ) + o~

+ a /~ ~(ql-q2)d~

, ~ ~-~

(~)

Th~or~me 2.2 i

1

II existe (yi,Pi,ui) E F i x ~i x N ad et %, ~ E H-2(~ ), v~rifiant les contraintes (2.2). e t ~ > 0 , pour tout zi,q i EIi, viEN~d " Une telle solution est de plus unique. D~monstration Comme (yi,Pi,Ui) est solution de (2.2), (2.3), elle est d~finie de mani~re unique par (2.5). Soient alors (yi,Pi,ui,%,~)

et

(yi,Pi,Ui,%',D')

deux solutions ~ventuelles. On a (2.14)

/ (~-%')(z]-z2)d~ +/

(~-~')(ql-q2)d~ = 0

Mais l'application .

z 1

+

zil ~

V zi,qi ~ ~i .

146 1

de

¥i

->

H2 (I)

est g image dense et (2.14) implique alors

d'oN l'unicit~. Par ailleurs, on v~rifie que SY ! I

~Y2

X°--Tnl SPi

=

I

= --~n I~P2 = - ~n[l

III - METHODE DE COORDINATION 3. i - PRINCIPE DE LA METHODE. La m~thode propos~e est it~rative. Supposons que nous soyons ~ l'~tape ~= ,~k, "~= k

. On cherche alors

yk , pk , u k

r~alisant ~ >i O, V

dans le Th~or~me 2.2 et pour les valeurs pr~cgdentes de %

k, avec

zi,qi,vi comme

et N . Le probl~me se d~-

compose aussitSt en Yi = 0

dans

~i

=0

r3iUr4i (3.1)

k

N~n

r. i

-~n ~

= ui

= (-I)i- k k

~pk = yk

Yid

dans f~i

1 i

?

pk

i (3.2)

I

= 0

i r3i U r4i

?pi[k "~n

k --~n ~ F

F. = O

= (-l)i

k k k (~) ui+Pi)(vi-ui)d r i >. o,

(3.3)

Ilk

vviE

i Na d •

l

Les relations (3.1)~ (3°2), (3.3) s'interpr~tent comme les C.N.S. d'optimalit~ du problgme suivant : l'~tat du syst~me est donn~ par (3.1) et le eritgre est donng par (3.4)

jk(uik) =

. ( y k - Yid)2dx + x) i i 7~ i Fi

i

;~ Yi dE "

147

ik+l , ~k+! par les formules suivantes (o~ S est l'application de

On calcule ensuite

dualit~ de H½([ ) +

H~ (I))

(3.5) k+l o~

k p

= ~

k

k k k + 0 S( Pl - P2 )

est une suite de nombres r~els, born~e.

3.2 - LE PROBLEME DE LA CONVERGENCE. On v~rifie tout d'abord la relation suivante Igrad(Yi-y~)12dx i=1,2

.

~ i=1,2

1

-

a

l

/g

I i=1,2

(u.-u~)(y~-y.)dF. z l l l z

.

Igrad(Pi-pk) 12dx .

a

/~ (

~ i=1,2

k

k

Yi-Yi ) (Pi-Pi)dx -

.

i

1

(3.6) - b~

I F (ui-uk)2d Fi + b . i=l ,2

i=! ,2

/F

i

Fi + (Pi-Pi)(ui-ui)d k k

. i

k k k (~/-~)(P] - P2 - P] + P2 )d~

>~ O,

d'o~ encore (avec des notations ~videntes)

i/

1

1

~ lui-u~T~ + ~

k

i

(3.7)

b ~ I%-.~12 + T~b /r

+

.k - Pi 12dFi + IPi

. a g

)2dx +

i 1

a 1

~ i s d'apr~s l'infigalitfi de Poincarg, et la continuitfi de l'application trace, il existe une constante C telle que (3.8)

~.

,grad z,2dx ~

C (

~ z z2dx 2 d F+ i ~ ) . .

i

i

i

de sorte que (3.7) et (3.8) impliquent ;

(3.9)

-

-

z Z~

i 'ui-u~'2(b~ - ~

) + ~

+ a ~

~

0 "

.

~ (yi-Yi) dF i

148

Supposons donc que 0 4y C- > ! °

(3.10) On choisit

¥-~ < 2C, b = r} et e n f i n

b

~] <2C, <

a<

2C-a

a 2C

de f a ~ o n que

2~>~+ bce q u i e s t p o s s i b l e ditions,

il

existe

(3.11)

- C(

s i ~) > ~ rl une constante

, ce qui est C > 0

vfrifi6,

telle

grace

~ (3.10).

Darts c e s c o n -

que

~ t l y i - y k [ t 2 + l[pi-pktt2 + [ u i - u i l z

) +

+a

>~0 .

Posons ~k = ik _ l m

k

=]~

k

-~

On a ~k+l

k -

ok ~, k k+ , ~
=

~?+] _ m k

=

0

k

. k

k+

,

S(pl-Pl-P 2 P2 )

d'o~

(3. ]

+ 2

0

Jl

( -X)(YI-Yl - Y2

Y2 )d~ "!<

+(0k)2cl

~ l[yk-yi I12 + i

Okf~ ( xk-x)(Ylk - Y] - Y2k + Y2 )all

+ 2

k est une constante. De (3.12) et de relations analogues ~crites pour m , on

o{i C 1 d6duit

+]l12 f{ +

a211m k+] I!-

~< ll%kll2 + a2 IImkll2 + (pk)2c2( ~ Ilyk - yfi2 + llpk _ pill2)_

(3.13) - 2C @k( ~ l[yk - yill2 + !I k 12 k 2 P i - Pi + lui-uil )" i II est clair que l~on peut choisir la suite Ok de mani~re que dgcroissante et convergente. k Yi De plus

xk, ]k

IIKkll2 + llrnkll2 soit

II en r6sulte que +

Yi '

k Pi ÷

Pi '

k ui ÷

ui "

demeurent dans des born6s. Un passage g la limite dans ~>~ 0 montre la convergence faible de % k ~k respectivement vers k,u . On a ainsi dgmontr6 le

149

Th~orgme

3.1

Sous l'hypothgse (3.10), on a (3.14)

k Yi

(3.15)

u~

(3.16)

%k , k

÷

k Pi

Yi'

÷

Pi

~u i

dans

L2(q)

÷

~ ,D

dan§

dans

Yi

H- ~ (~) faible.

IV - APPLICATION NUMERIQUE 4.1 - POSITION DU PROBLEME. On a pris

~ = )0,4 (x) 0,I( et d~compos8 ~ en quatre sous-domaines ~i,i=I,2,3,

4, comme indiqu~ sur la figure 3, les notations ~tant des variantes ~videntes de celles du N°2.1. Le critgre ~tant celui du N°I.I, relation (1.2), l'~tat du syst~me est donng (avec f#O F car cela fac~lite la construction de sol~tions exactes) par : ,,3b F32 ,,, -33 -3/: '

rI

aI

~

~

12 ~2

F41

~3

F42

I

F2

34 ~4

F43

- $z = f

(4.1)

~

23

Figur e 3

F44

dans

zlr3u F4 = 0

~]rl u rE

v

avec (4.2)

v Eq~a d = ~ =

L2(FI ) x L2(F2 ).

4.2 - CHOIX DU PROBLEME TEST. Si dans (4.1) et dans le crit~re on prend

f

et

Yd dgfinis, respectivement,

par : (4.3)

2 2 f(xl,x 2) = 2 (-x I - x 2 + 4x I + x2)

(4.4)

2 2 Yd(Xl,X2) = -(x2-x2)(Xl-4X |) - 8 v

On vgrifie faeilement que le problgme de contrSle optimal Min tion unique :

v6~ Vl(X2) = -4(x2-x2)

(4.5) v2(x ) l'gtat y e t (4.6)

2 -4(x2-x 2)

l'~tat adjoint p correspondant ~tant donn~s par : 2 2 y(x|,x2) = -(x2-x2)(xl-4Xl)

J(v) admet comme solu-

150

2 P(Xl,X 2) = 4 ~(x2-x2).

(4.7)

4~3 - MISE EN OEUVRE DE L'ALGORITHME DE DECOMPOSITION - COORDINATION. L'algorithme du N=3.1, relatif g une dgcomposition de ~ en deux sous-domaines se g~ngralise ais~ment g la decomposition du N°4.1. Au vu des relations (3.1), (3.2) il faut rgsoudre ~ chaque itgration un probl~me de contrSle optimal, elliptique, dans chaque

~i' i=1'2'3'4 ; l'approximation des Equations (3.1), (3.2) a Et~ effeetuEe R

partir d'une discr~tisation par diffgrences finies de pas h=~o , soit environ 400 points de discrgtisation par sous-domaine. Les sous-probl~mes de contr$1e ci-dessus ont ~t~ r~solus par la m~thode du gradient dont la mise en oeuvre est facilitge en remarquant que les matrices de discr~tisation, assoei~es aux ~quations (3.1), (3.2) sont identiques, sur chaque sous-domaine, et indgpendantes de i et k; ces matrices gtant sym~triques et dEfinies positives on peut alors utiliser une factorisation de CHOLESKY faite une lois pour routes au debut du programme. 4.4 - RESULTATS NUMERIQUES. Dans les deux exemples qui ont EtE traitEs, on a initialisE l'algorithme de dEcomposition - coordination en prenant %1 = O, ~I=o et on a pris

S=Identit~ dans (3.5).

Exemple ! Ii correspond ~ v=5 ; pour O k = O =2 on a max Iyl-Y21 l l = 0.653,

max Iyl-Y2 4 4 I= 1.5 x 10-2 , max ly 10 I - Y2IO I= 1.8 x 10-4

On a am~lior~ la vitesse de convergence en utilisant une strategic de p type steepest descent permettant d'obtenir k important par rapport ~ la m~thode avec P

k

max IY~- - Y~I- = 1.8 x lO-4 fixe.

variables de d'o~ un gain

B

E x~mp le 2 Ii correspond ~ p

k

~=O.|25

; le problgme a ~t~ rEsolu en utilisant la stratfigie de

vari@bles ci-dessus ; ~ ~tant plus petit que dans le probl~me pr~cgdent la conver-

gence est plus lente soit

I

max ly I - y~l = O.506,

maxly ~ - y~I = 2.10 -3,

Y l= |o4 °

Remarqu..e. 3 . ! Dans l e s deux e x e m p l e s p r f i c ~ d e n t s , l a r ~ s o h t i o n

~ chaque i t g r a t i o n ,

des s o u s -

p r o b l g m e s de e o n t r S l e o p t i m a l p a r l a mEthode du g r a d i e n t n ' a j a m a i s demand~ p l u s de trois

it~rations

internes

(pour un t e s t

d'arr~t

assez sfivgre). ~

Remarque 3 . 2 Pour ~ de l ' o r d r e

de l0

-2

on n ' a pu o h t e n i r

l a c o n v e r g e n c e de l ' a l g o r i t h m e

de

dEcomposition-coordination la condition (3.10) n'Etant visiblement pas satisfaite. B Temps de calcul Dans les deux exemples precedents, de l'ordre de 1 minute pour 5 iterations de coordination.

151

[I)

J.L. Lions

ContrSle optimal des syst~mes gouvern~s par des ~quatlons aux d~riv~es partielles, Dunod-Gauthier-Villars (1968).

(I)

J.L. Lions - E. Magenes

Probl~mes aux limites non homog~nes (T.l), Dunod (1968).

A P P R O X I M A T I O N OF O P T I M A L CONTROL PROBLEMS OF SYSTEMS D E S C R I B E D BY B O U N D A R Y - V A L U E M I X E D P R O B I ~ M S OF D I R I C H L E T - N E U M A N N TYPE P. COLLI FRA_NZONE L a b o r a t o r i o di Anaiisi N u m e r i c a del C.N.R.- Pavia,

Italia

INTRO DUCT ION We report here some results on optimal control of systems d e s c r i b e d by b o u n d a r y - v a l u e m i x e d p r o b l e m s of D i r i c h l e t - N e u m a n n type for second order linear e l l i p t i c p a r t i a l d i f f e r e n t i a l operators.

These results are

a c h i e v e d in the framework of the d e t e r m i n i s t i c theory

as d e v e l o p e d in

Lions'

book

[I] on optimal control.

We consider here the case of a boun

dary control on the D i r i c h l e t condition.

The state of the s y s t e m is the

solution of a m i x e d n o n - v a r i a t i o n a l problem. The initial control p r o b l e m I) is a p p r o x i m a t e d by a family I ) of £

control p r o b l e m s of systems d e s c r i b e d by v a r i a t i o n a l p r o b l e m s of Neumann type and o b t a i n e d by a p p l y i n g the p e n a l i z a t i o n method. sults are reported. tion problems,

C o n v e r g e n c e re-

M o r e o v e r a family Iv) of finite dimension optimiza-

w h i c h converges to the initial p r o b l e m I), is considered.

Results of some n u m e r i c a l e x p e r i e n c e s Statin9 O f the_problem;

notations

are reported.

(~)

and definitions.

Let 9 be a b o u n d e d open set of sional m a n i f o l d and let ~ be a(n-2)

Rn whose b o u n d a r y F is a(n-1 ) dimend i m e n s i o n a l m a n i f o l d which separates

F into two open n o n - e m p t y disjoint sets F ° and FlSUCh that:

F=FoU r i U X

(t

For u , v E H 1 (a) (I)

,

FoNr1:¢

,

ton r l = X

) , set:

a(u~v)= ~ ~ ~ai~(x) ! ~ ~x. ~u ~v dx+lao~x. (x) uvdx i,j=l 3 l

Hereafterf we assume the f o l l o w i n g regularity h y p o t h e s e s on ~ and on the

(~) C o m p l e t e l y d e t a i l e d proves will (Sc. Mat.

appear on "Rend. Istituto Lombardo

and Nat.).

~V -~2 (~])~ i=I, .... n}; (i) HI ( ~ ) = { V : V ~ L 2 (~) ,8-~--~

for the definitions of real

1

Sobolev spaces Hs(~) chap.

I.

and HS(F)

with s ~ R, see L i o n s - M a g e n e s

[IJ

153

the c o e f f i c i e n t s i)

in the

form a(u,v):

3~=£ and ~ are C 2 - m a n i f o l d s ,

ii)

locally

(2

aij (x) E C 3 (~) , ao(X) ( C I (~)

on one side of F,

)

n ~ aij(x)~i~j>~I~l 2 ,~=(~I,~2 1,]=l

iii)

~ is t o t a l l y

..... ~n ) 6 R n, V x 6~,

~>0

% (x) >.0, V x ~ The

form

(I) is c o n t i n u o u s

order elliptic (2)

A=

The adjoint

-

on H I (~) and defines

the second-

operator:

. . 3x. ~ (aij (x)3~--~.)+ao(X) 1,3=i l 3

form a ~(u,v)

the o p e r a t o r

and b i l i n e a r

differential

is d e f i n e d

A ~ associated

by:

a'(u,v)=a(v,u)

, V u , v 6 H I (~) ;

is:

n

(3)

_/X-'[

A*=

~

i,j=13xi (aji

( X ) ~ , ) + a o (x) 3

S t a t e of the s [ s t e m Set T = { t : t 6 H 1 (~),roYot=O}(3) and ~{H -T (~) , ~ ( [O,1[, unique

solution

of the v a r i a t i o n a l

a~(w,t) = Which

is e q u i v a l e n t

<~,t> -TIQ T,~

let w £ T be the

equation:

,V t 6T

to the h o m o g e n e o u s

(4) m i x e d pr.oblem of D i r i c h l e t - N e u -

mann type :

(5) A~w=~ in ~, roYo%r=-O, r l Y A ~ W = O U n d e r the a s s u m p t i o n s i),ii) it follows from the r e g u l a r i t y Shamir

[I]

W ~ The space 2

(3 (4 (s

ck(~), mes

H 3/2

defined

-~

(~)

,

by

V q>O

by

k positive

continuously

integer,

is the space

differentiable

The b r a c k e t You=trace

< ,> indicates

of functions

which

the d u a l i t y

operators between

from~to

at£ e x t e r i o r

~. For trace

direction theorems,

cosine

F ° and F I.

H s and its dual space.

n ~u of u on F, Y A U = ~ aij ~tx ;~~--~cos(n,x i) , c o n o r m a l i,3=I 3

ve to A on F, c o s ( n , x i ) = i - t h

are k-ti-

in ~.

We denote by r o and r ! the r e s t r i c t i o n

mal

results

:

derivati-

of n, n b e i n g th nor-

see L i o n s - M a g e n e s

[I].

154

XA~,~ (~)={v:v ~ H ~/2-D(~) ~ A~V ~ H -T (~) t rOYoV=O,

rlTA~V=O}

e n d o w e d w i t h the n o r m ~2

~,_

=IIvll 2

+ITA~vJI

is a Hilbert space for ~ E ] 0 , 1 ] and i & [O,1[. The o p e r a t o r A ~ is thus an a l g e b r a i c and t o p o l o g i c a l i s o m o r p h i s m of XA~,~ (~) on

(~) for each ~ ~ ]O,

and

7 6 [O,

.

Chosen

[o½[ by t r a n s p o s i t i o n , (4) T~

there exists

a unique y 6 H ~ (~), V T 6 [O,}[ s o l u t i o n of:

= + (6) , -~ ~ - p ~ p,~ ~,F O -~rFo . 3/2~o

gWfXA,

r (a)

By d e f i n i t i o n y is the ~'weak ~' solution of the m i x e d problem: (5)

Ay=f in ~

The r e g u l a r i t y

, roYoy=go,

rlYAy=gl

result y ~ H ilz+~ (e)

follows from V i s h i k - E s k i n ' s w o r k

[I] and therefore the b o u n d a r y condi-

tions may be i n t e r p r e t e d a c c o r d i n g to trace theorems. We point out t h a t the state y of the s y s t e m is solution of a boundary-value p r o b l e m of n o n - v a r i a t i o n a l type;

this means that the state y

does not b e l o n g to H 1 (~). C o n t r o l space In many applications,

it is very f r e q u e n t l y e n c o u n t e r e d the case in

which the control is e x e r c i s e d through the b o u n d a r y ry conditions). U s u a l l y controls however piece-wise

(i.e. on the bounda-

are d e s c r i b e d by m e a s u r a b l e

constant controls

functions;

are often used.

We shall c o n s i d e r the case of a b o u n d a r y control on the Dirichlet condition on F . If we w a n t to develop our theory in a Hilbert space fra o and still allow p i e c e - w i s e constant controls (or controls with

mework

even s t r o n g e r singularities)

then we are led to choose as control space

(6) The m a p p i n g w ÷ r l Y o W is a linear continuous map of T ° = { v ; v E H 3/~-° (~), roYoV=O},

o< I , in HIo-~(FI ) , the b r a c k e t

lity of H-I+~ (FI) and HIo-°(FI ) .

<,> then indicates the dua-

155

q~the

following

Sobolev space:

~ = H ~ (Fo) where ~ 6 I0,½[ For each v ~ ~ (6)

, let y(v) 6 H ~ + ~ (~) be the unique solution of:

P) { Ay(v)=f in ~, roYoY(V)=go+V,

Observations

space

We confine ourselves

to the case in which the trace of y is obser-

ved on F I. The mapping v+rlYoY(V) H

(rl);

rlYAY(V)=g 1

is an affine continuous

map of ~ i n

in order to avoid as much possible the use of spaces H s, s not

an integer, w e may consider in particular se as observations

rlyoY(V) E L 2 (FI) and then choo-

space: Y=L 2 (FI)

Control problem Given ~ad

a closed convex and bounded subset of q]~

Yd ~ Y With every control v £ ~ ,

we associate the "cost function":

J(v)= I IrlToY(V)-ydI2d~+II v2d~ F1 F0 We shall consider the optimization I)

Find u ~ % d

,

I~0

problem: minimizing

J(v) over

~ad

We obtain the following result: Proposition

-I-

(7

Problem I) ad~.its a unique

)solution u, termed optimal

control which is characterized by the inequality: (7)

I[rl~oY(U)-Yd]'rlYoY(V-u)d~+l I u(v-u)d~>10, Vv ~ ~ad F F 1 o Let us now trasform (7) by introducing the adjoint p(u), unique solution in H 1 (~) of the problem: (8)

A~p(u)=O

By a regularity

in ~, roToP(U)=O,

r17A~P(U)=-(rlYoY(U)-Yd)

result of Shamir the solution p(u) belongs to H s/~-n (~)

V n>O. This allows us to apply Green's quality (7) into the equivalent form: (7) For i>O uniqueness

formula and thus transform ine-

follows from J(v) being strictly

it follows from the uniqueness operator A.

convex;

for i=0

of the Cauchy problem for the elliptic

!56

"p(u)'v-u> U

+'i o

Therefore

the system

u is the optimal

(6) , (8) , (9) admits

a unique

solution

{y,p,u}

where

control.

A p_pproximation of I) The main task in the numerical a good approximation variational

scheme

solution

of problem

for the state equation

I) is to produce

(6) which

is of non-

type.

Set

_~ [

as (w,~)=a(w,?)+s

we approximate

problem

(9)

) ro7o w'roYo'~da F o

P)

, w ;9 6

Hl

(~)

(6), by means of a family P g ) of variational

problems : for each ¢>0,

(I o)

i variational

let Ys (v) be the unique

equation:

a s (y¢ (v) ,~)=

L.

in H 1 (f2) of the

i +s-I

P'~

-P'~

solution

(go +v)" roYo ~da+

r

V2'rl

-{%'rl o

where we assume

g! E H -I/2 (F I) while

The variational

equation

value problem

of Neumann

pc)~Aya (v)=f~

go and v belong

(I0) is equivalent

to ~

.

to the following

boundary

type:

in

(II)

[ ~YAY~ (vJ +Xro '

(i0) ¥ o y a (v) =ga (V)

(8) We note that:

~

(9) We note that:

for v ~ H i

on

F

~ gs (v) =

go+V,

on F °

¢gl '

on F 1

(u)=roYA~P(U) +}~u.

+[ IroYoVl i2 ] , since off o

(~)~

IIvl i2 -<M(~) [ ~ I [ ~ - I I 2 + 1~ ci~ - 1- i ' = oF~

F O is a non-empty

open set of F, it follows

that a (v~v)>~811vl i g ¢ (small enough);

~ V v 6 H- (~) ~ where S>O is independent i~ a (u,v) is therefore coercive on H 1 (~).

(I0) We note that boundary

condition

on F is of "natural

type".

of v and

157

where ×r ° is the characteristic function of F0 on r. Let us consider the family I¢) of control problems hereafter described: Xs)

Find us ~ d

minimizin~ J s (v) over %

d

where Je (v)= I IrlYoY¢ (v)-y~'2d~+k I v2d~" r F 1 0 We then obtain: Proposition-2For e a c h a>O, p r o b l e m I E ) a d m i t s a u n i q u e s o l u t i o n u ; moreover a c necessary and sufficient condition for u s to be an optimal control for I¢)

is

Pa)

that

the

following

as (ye,~) =

equations

and inequalities

be

satisfied:

+a-I I (go+Us) .roYo~d~+ , -p,~ p,a r -~'rl ~/z'r I o V~ 6 H l (~)

P~)

aa~(Ps,*)=- I [rlYoY¢-Yd]rlYo*d~, F1 I (-~roYoPs+lu¢) (v-us)d~>.O , ro

~ 9 ( HI ([~)

Vv

£ ~d.

We have the following convergence result: Theorem

-IAs

s÷0 we have:

u e converges to u weakly in ~ (strongly in L 2 (FO) ) Furthemore : ya÷y in H ~ (~), V T <1, ps÷p in H I+6 (~), V 8< 1 and Js (us)÷ J(u). Numerical approximation n Let ~ b e t h e s e t P+

ordered

as

follows:

(hl ..... hn)'< (hi' .... hn)~--->hi~
there

exists

a linear

continuous

operator

~h o f

H1 (~)

158

in V h such that: lim i [ y - , ~ h y t l =0, h+o 1, Q

V y 6 H 1 (~)

! ',Y-~hYi Ik, ~.~ClhTS-kt tyl I s,Q , V y ~

S

(~),

~k ~

[0,I[

ts

[k,2]

~

C being a constant independent of h and y. We approximate problem P~)

(11), with the family of problem Pc,h ) :

*For each a>O and h6 ~

let yc,h(V)6 V h the unique solution of:

Pc,h) IIFa~ (y¢,~,w~)=_pi~~p,f~+¢-l.Fo[(go +v)rlYOwh d°+d/2,Fl
t ,

Set ~ = R +n

~ for each k6 ~

let ~ k

Vw h

~ V h .

be a finite dimension subspace o f ~

endowed with the induced norm, and let qk be a linear continuous operator of ~ in ~ k

such that:

v6~

, liml Iv-qk vl k÷o

I%:0

There exist approximations {Vh,~ h} and { ~ k , q k } which satisfy the staded h~:potheses; see, for example Aubin [I], Bramble-Schatz [1]. Finally,

for each k 6 ~

let:

~ k d a closed convex set in ~ k such that

Set c=M.lhl s, M positive constant and s>O, yc,h(V)=Yh(V)

and ~=(h,k) ,

let us consider the family Iv) of optimization problems: I )

Find u 9 e ~ k ad minimizing J (Vk) over ~ k ad

where

a,~ (vk): ] P1

/rlToYh(Vk)_ydl2d~+X

f

Vkd~2.

Fo

Proposition -3For any ~j problem Iv) admits at least one solution; solutions u

S FO Furthemore u

the set of the

of I V ) is characterized by the inequality:

u

V

Po is solution of I ) if and only if it satisfies the system:

159

Ph )

as (Yh'Wh)=

+s-I I (go+Uv)'roYoWh do+ P -p,~ p,~ ~ -~/2,r i ~,r 1 o ,V wh 6 vh

Ph )

aZ(Ph,Wh)=- I [rlYoYh-Yd]'rlYoWhd~, F

~w h e Vh

(111

i

I[ F

o oPh+lU ] O, V

O

The above described gence

approximation

is supported

by the following

conver-

result:

Theorem -2For ~=Mlhl s, M positive h÷O, k÷O independently)

constant

and s 6 1 0 , 1 [

the sequence

{u }, u

being

tion of I ) r converges weakly in ~ (strongly the unique solution u of problem I). Finally niga

some results

(Laboratorio

reported;

on numerical

di Analisi Numerica

computations

were carried

Centro di Calcoli Numerici Let us consider

experiences

del C.N.R.-Pavia)

(i.e.

a solu-

in L 2 (F o)) to

performed

on an IBM 360/44

- University

the following

as v+O

by G.Gazza-

and myself

installed

are

at the

of Pavia.

case

F1

Fig.

I

F1

FI

F° chosen ~ , Fo,F I, ~as intervals

in figure

of equal length

._(Ii) We observe

I_ that ~Ji(u

the advantage of the adjoint

{ i 1~let ~ k

and ~ a d

x~ be the partition

of F ° into k

be the set of functions

I

)=-[roYoPh+lU

of calculating

Z={(O.I,O),(O.9,0) = Points of discontinuity in boundary conditions

; the described

the gradient

state Ph on F O.

values

on F

o

piece-

scheme presents

through

the trace

160

wise constant

on the partition

~k

and valued between 0 an 7. 2 of R+. We denote by J~h the regular

Let h=(hl,h 2) be an element

mesh of points M = ( m l h l , m 2 h 2 ) , m i6 Z. We consider a h(M)=

the approximation

V h of Hi(a)

hereafter

2 Z [(mi-1)hi, (mi+1)hi] i=I 2

xi

n (1-1

i=l Vh={Zh,Zh=

Zhgh,Z h 6R}

experiences

-Ay(v)=O

hT

-rail

) , x ~ ~h (M)

8h(X)=

,

M 6 ~ h (~) Numerical

described:

, x ~ ah(M)

o

are performed

in ~%, yiFo=V,

on the model-problem:

a~ ?nlrl=gl

and

J(v)=~

ly(v)iFl-Ydl2d~

FI and we choose yd=zlF I

a)

and g l = ~ i F ]

x2 z=arct g x i_~12

whe re :

and then Yott=z,

Uott =

. ×=0.5 , 0.5 ~ X'$03 .¢'Tr

X2 b)

z=z - '--~arCtgx -O 65 1 "

The solution gradient

and then Yott=z,

of problem

the projection

ve constraint~ constraint

I ) has been

active.

namely,

of the gradient

and a coniugate

becomes

found

l

. ~ Uott = "z

, ×=0.~,5,x=0.65

LO ,O.&WX.~0.9

method with projection;

of descent

control

arctg~)

, o4,{×< 0.3~

direction

Figure

achieved

as direction

on the linear manifold

by act~

at the next step if no other

2 represent

in the specific numeric

by using the direct

the method using

cases

the approximated

considered.

optimal

1"0

m N

i~. 0

n"

H

II i~ ~t

II

II

II

H

0

II 0

v

ct

©

t

I

I

1

cn

II 0

~-

0

v

Z

II

~

II

Ix3

I1

0

0

0

0

0

0

0

°

........

I...........

I-

j

1

-1

o ~,

162

REFERENCES AUBIN,J.P.

[13-"Approximations

of elliptic boundary-value

Wiley - Interscience, BRAMBLE,

J.H.-SCHATZ,

A.H.

vol.XXVI,

[1~-"Least squares methods

elliptic boundary-value N.113,

pp.!-31,

BENSOUSSAN-BOSSAVIT-NEDELEC

COLLI FRANZONE,

problems"

de I'I.R.I.A.,

[13-"Approssimazione

P.

for 2mth order

Math.

Comp. voi.25,

(1971).

[I3 -"Approximation

le" Cahiers

problems"°

(1972).

zione di problemi

des probl~mes

n.2.

mediante

il metodo di penalizza-

misti di Dirichlet-Neumann

ri lineari ellittici

de contr~-

del secondo ordine",

per operato-

to appear on

BolI.U.M.I.. COLLI FRANZONE,

P.-GAZZANIGA,

G. [IJ-"Sull'analisi

ma misto di Dirichlet-Neumann littiche

del secondo ordine"

del C.N.R. LATTES,

R.-LIONS,

J.L.

tions" LIONS,

J.L.

per equazioni Pubblicazione

lineari elN.34 del L.A.N.

di Pavia.

E1~-"M~thode Dunod, Paris,

<13-"Optimal

numerica del proble-

de quasi r~versibilit~

et app!ica-

(1967).

control of systems governed by partial diffe-

rential equations"

Grundlchren

B.170,

Springer-Berlino,

(I 971 ) . E2~-"Some aspects

of the optimal control of distributed

rameter systems" vol.6, LIONS,

J.L.-MAGENES,

E.

E.

[1~-"Non-homogeneous Grundlchren,

~1~-"Regularization

I.M.-ESKIN,

I.G.

domains

B.181,

convolution

and their applications"

E13-"Application

Springer-Berlin,

6, pp.150-168,

~1~-"Elliptic

(133) , pp.15-76, YVON, J.P.

boundary value problems

and

(I 971) .

of mixed second order elliptic problems"

Israel J. of Math., VISH!K,

pa-

SIAM,

(1972) .

applications" SHAMIR,

Series in Applied Mathematics,

(1968). equations

in bounded

Uspehi Math.Nauk

22, I

(1967) .

de la p~nalization

bl~me de contrSle optimal"

a la r~solution

Cahier de I'IRIA n.2,

d'un pro(1970).

CONTROL OF PARABOLIC SYSTEMS WITH BOUNDARY CONDITIONS INVOLVING TIME-DELAYS P.K.C.Wang Department of System Science University of California Los Angeles,California U.S.A. In this paper, we consider various problems in the optimal control and stability of parabolic systems with boundary conditions involving time-delays.

These problems

arise physically in the control of diffusion processes in which time-delayed feedback signals are introduced at the boundary of a system's spatial domain.

To illustrate

the basic ideas, only the results for parabolic systems with the simplest forms of boundary conditions involving time-delays will be presented.

Results of a more

general nature are discussed in detail in reference [6]. I. PRELIMINARIES Let ~ be a bounded open set in R n with an infinitely differentiable boundary F and I denote a given finite time interval ]0,T[.

We consider a parabolic system of

the form: ~Y - by = f }t

in Q=~×I,

where A is the Laplacian operator.

(i)

The function f corresponds to either a distribut-

ed control or a specified function defined in Q. Let F 1 and F 2 be given disjoint subsets of F such that F = F I u F 2. ZI=FI×I,i=I,2.

Let Z=FxI;

We consider the following Neumann boundary condition involving a

time-delay: ~(x,t)

=A ~ i,j=l

cos(N,xi ) a ~ x (x,t) = q(x,t)

on ~,

(2)

J

where q(x,t) = ~(x){y(w(x),t-Y) + u(x,t)},

(3)

where ~ is a given C~ function defined on P with compact support in FI; cos(N,xi) is the i-th directional cosine of the outward normal N at a point x C F ;

u represents

either a boundary control or a given function defined in E; the time-delay T is a specified positive number; w is a continuously differentiable bijection of F onto F such that w(x)=x if x E F 2 and w ( x ) C F 2 if x ~ F I ; vanish on F. The initial data for (i) are given by

moreover, its Jacobian does not

164

y(x,0) = Yo(X),

x6~,

I

(4)

/

y(x,t') = @0(X,t~),

}

(x,t')@FX[-T,0[,

where y and 9~ are specified functions. Note that only ~^, the restriction of +0 2 o u to F X[-T,O[, is of importance here. In what follows, we shall first give sufficient conditions for the existence of a unique solution of the mixed initial-boundary value problem (1)-(4).

Then, various optimal control problems will be discussed.

For simplicity, let the final time T=KT, where K is a given positive integer. We introduce the following notations: Ij=](j-I)T,jT[, Qj=~×Ij, Z =FxI. and El=F~xI ] J 3 J for i=1,2 and j=0,1,...,K. Let Hr(~), r>~0, denote the Sobolev space of order r on ~.

For any pair of real numbers r,s>~0, the Sobolev space Hr'S(Q) is defined by Hr's(Q) = H0(I;Hr(~))~HS(I;H0(~)),

Q=~XI,

where Hs(I;X) denotes the Sobolev space of order s of functions defined on I and with values in X. For optimal control problems~ it is of importance to consider the cases where the control f or u belongs to L2(Q) or L2(E) respectively.

For these cases, we have

the following results: 1 ~ ~ ~ ~ 2 Theorem I: Let y ~@~,u and f be given with y ~H~(~) {resp. H2(~)}, ~n6H2'#(Zn) --'--2 o o o i i u u (resp. L2(S0 )}' ~E~½'~(S) {resp. L~(S)} and fEL~(Q) {resp. (HZ'T(Q)) ', dual of H~'~(Q))}.

Then, there exists a unique solution y~H2'I(Q)

{resp. He'#(Q)}

for

i

problem (1)-(4).

Moreover, y(.,jT)~HI(~)

{resp. H~(~)} for j=l ..... K.

The above theorem can be established by first solving the problem on QI using the basic results of Lions and Magenes [3;p.33,p.81] specialized to the case of (i)(4).

In a similar manner, the existence of a unique solution on Q2 can be establish-

ed by using the solution on QI to generate the initial data at t=T and boundary data on S 2.

This advancing process is repeated for Q3~Q4,... , until the final cylinder

set QK is reached°

Note that the solution y on Qj (denoted hereafter by yj), j=l,

..°,K, is required to satisfy the initial data at t=(j-l)T: yj(x,(j-l)T) = Yj_l(x,(j-l)T),

(5)

XE~,

and boundary condition (2) whose right-hand-side is given by

(6)

qj(x,t) = ~(x){Yj_I(W(X),t-T) + u(x,t)}. In order to apply the same results of Lions and Magenes to any Qj, we must verify that Yj-l and Yj_llZ. ,j=2,...,K, ql"

satisfy the same conditions as required for Yo and

This can be sh~w~ by making use of the trace theorem [3;p.9] and a result per-

taining to the continuity of Yj_l(.,t) on [(j-2)T,(j-I)T]

[2;p.19].

165 II. OPTIMAL CONTROL We shall consider optimal control problems for system (1)-(4) in which the control corresponds to either f or u belonging to a specified closed convex set UQEL2(Q) or U EEL2(E)

respectively.

Problem i: Let yo,@O and u be given functions satisfying the hypotheses of Theorem 1 so that for a given control fEUQ, a unique solution y(f)~H2"l(Q) y(.,T;f)EHI(~).

exists and

The problem is to find a f°EUQ such that j(fo)~j(f) for all f ~UQ,

where J is a cost functional given by

J(f)=ll~lY(x,t;f)-Ydi2dxdt d x d

+ t12!~Y(x,T;f)-YdTi2dx , Q + ~3 IQ(Nf) f

(7)

2 where 1i~0 and 11+~2+~3>0; Yd and Yd~ are given in L (Q) and L2(~) respectively;and 2 N is a positive linear operator on L (Q) into L (Q). For the above problem, it is known [i] that for 13>0 , a unique optimal control fo exists, moreover~ fo is characterized by j,(fo).(f_fo) = ~i In(Y(fo)_yd)(y(f)_y(fO))

+ X2

o

dxdt

o

(y(x,T;f)-YdT)(Y(x,T;f)-y(x,T;f ))dx

+ %3 [ (Nf°)(f-f°) dxdt ~ 0

JQ

for all fEUQ.

(8)

The above condition can be simplified by introducing the following adjoint equation.

For each f~UQ, we define p=p(f)=p(x,t;f) as the solution of

-

8p(f) St - Ap(f) = >~l(y(f)-yd)

in Q,

(9)

with terminal condition p(x,T;f) = %2(Y(x,T;f)-YdT) ,

(io)

xE~,

and boundary condition: ~P(f)(x,t) = 0 ~o

~ ~u (X,t)

for (x,t)e([r-w(supp(@))]×l)U(w(supp(@))×~)

= ~(w-l(x))I Jw(x)Ip(w-l(x),t+I;f)

(ii)

for (x,t) Ew(supp(~))x]0,T-y[, (12)

where J

denotes the Jacobian of w; w(supp~)) w under the mapping w and

is the image of the support of

166

n

~(x,t) DO

= ~ c°s(D'xi) i,j=$

~-a!i!(f)(x,t). DX,

(i3)

3

We observe that for given yd,YdT and f, problem (9)-(13) can be solved backward in time starting from t=T by first obtaining the solution P=PK on QK and terminal condition (19) and boundary condition DPK(f) DO (x,t) = 0

for (x,t) 6 Z K.

(14)

Having found pK ~ we may proceed to solve the problem on QK-I backward in time with terminal dats at t=(K-l) : PK_I(X,(K-I)T) = PK(X,(K-l)[),

(15)

xE~,

and with boundary conditions: SPK-I (f) DIJ (x,t) = 0,

(x,t)~[F-w(supp(~))]XlK_l,

(16)

DPK_I (f) -i t -i 21) ,(x,t) = @(w (x))iJw(x)IPK(W (x),t+T;f),

(x,t)~w(supp((~))XlK_ I, (17)

Note that the right-hand-side of (i7) is completely determined once PK is known. This backward process is repeated until the solution on the initial cylinder set QI is determined.

For fEL2(Q)~ the existence of a unique solution pK(f)~H2'I(QK)

with pK (.,(K-I)~)~HI(~) can be established by applying Theorem 1 to (9)-(13) with obvious change of variables and with reversed sense of time t'=T-t.

The result can

bel extendedl to Qj,I~<j~
Thus, we have the result:

Lemma i: Let the hypotheses of Theorem 1 be satisfied.

Then for given YdEL2(Q),

YdTEHl(~) and any fEL2(Q), there exists a unique solution p(f) EH2'I(Q) to problem (9)-(13). Now, in view of Lemma i~ we can proceed to simplify (8) using the adjoint equation,

It can be shown [6] by setting f=fo in (9)-(13), multiplying both sides of

(9) by (y(f)_y(fO)) and integrating over Q that (8) reduces to

Q(p(f°)+k3Nf°)(f-f°) dxdt >I 0 for all fEUQ.

(is)

This result can be summarized as: Theorem ~: For Problem i with cost functional (7) with Yd6L2(Q)'YdT ~ HI(Q) and %3>0, there exists a unique optimal control fo which is determined by the solution of (i),

167

(9) with boundary conditions (2),(11)-(12), initial condition (4) and terminal condition (i0) (all with f=fo). Moreover, fo satisfies (18). For the special case where UQ=L2(Q), (18) is satisfied when fo

=

-%~iN-ip(f°).

(19)

In particular, if N is the identity operator on L2(Q), then,in view of Lemma i, we have fOEH2,1(Q). To obtain the optimal control (19) in feedback form, we follow Lions' approach [i] by first considering the following set of equations with s E I: 3y3t- £y + % ~ - i p

= 0,

1 (x,t) ~ ~×]s,T[,

~p

3t

-

-

Ap

-

%1 y

=

-%lYd

(20)

,

with boundary conditions: =I~(x){y(w(x),t-y)+u(x,t)}

if t-T~s

I (x,t) 6~×]s,T[,

]

(21)

l~(x){~s(W(x),t-T)+u(x,t)} if t-T<s ~-~D(x' t) =I0'

^A (x' t) E ~s= (F-w(supp (~))x] s'T [(J (w(supp (~)) ×IK) '

1 (22)

~(w-l(x)) ]Jw(X) Ip (w-l(x), t+T),

(x, t) ew(supp (~))×Is, T-y [,

and with initial and terminal conditions: y(x,s) = Ys(X),

I

p(x,T) = %2(Y(x,T)-YdT),

I

x~,

(23)

where Ys is given in H 1 (~) and ~ is a given function defined in Fx[s-y,s[, whose 2 s I! 2 restriction ~s to F x[s-y,s[ is in H~'~(F x[s-T,s[). Note that (20)-(23) provide the solution to the optimal control problem associated with (i) for t E]s,T[, UQ= L2(Q) and with a cost functional given by T

T

Js(f)=%l Is I~lY(X't;f)-ydl2 dxdt + %21~ Iy(X'T;f)-ydTI2 dx + %3 Is I~ (Nf)f dxdt" (24) The problem with %3>0 has a unique optimal control in the form of (19). Consequently, (20)-(23) has a unique solution {y,p} also. In fact, for any given pair (Ys '~)s i I 2 EHI(~)~H~'~(F x[s-T,sD, the solution y,p~H2'l(~x]s,T[). Moreover, the following property can be readily established.

168 Proposition: Let {y,p} be the solution of (20)-(23) with s=0.

Define ~ , the system s

"state" at time s, by the pair (y(.,s),$s) , where

$,S(, t,) = i #P'O("'~)

f°r t' ~ {s=[<'O[ffi [s-~"s['

1 (25)

y(.,t*)[r2

for t'E[s-T,s[-~ s.

i

Then, for all pairs s~t in I, (26)

pC',t) = P(t,s)O s + rs<'-,t). where P(t,s) and rs(.,t) are defined as follows: (i) we solve the equations:

+ 3 -iy = o,

a-Z -Ay - ~t

o,

1

t (x,O6 ~x]s,'r[,

(27)

- kl~ =

with boundary conditions ~(x)~(w(x),t-T)

if t-T>~s,

) i

~(x,t) = I

+(X)~s(W(X),t-~)

if t-T<S,

ix,t) 6~×]s,T[,

(28)

i

^

~(~,t) = I o,

(~,t)e zs

(29)

@(w-l(x)) [Jw(X) [y(w-l(x), t+T),

(x,t)6 w(supp(¢))×]s,T-r[,

and initial and terminal conditions:

~(x,s) = y(x,~),

I xee,

V(x,T) = k28(x,T),

i

(30)

then (31)

P(t,s)o s = y(.,t); (ii) we solve the equations:

3t (x, t) ~ × ] s , T [, ~t

(32)

= -llYd'

with boundary conditions: (~(x){q(w(x)~t-T)+u(x,t)}

if t-T~s,

[ ~(x)u(x,t)

if t-T<s,

, (x,t) 6g~X]s,T[,

?q(x,t) =

i

(33)

169

^

~(x, t)

I0,

(~, t)e zs,

= l@(w-l(x))lJw(X)l~(w-l(x ),t+T), and with

initial

and terminal

(x,t) E w(supp(~))×]s,T-T[,

i(34)

conditions:

n(x,s) = 0,

1 x@~,

~(x,T) = 12(n(x,T)-YdT) ,

)

(35)

then (36)

rs(X,t) = ~(x,t).

Now, the optimal feedback control can be obtained by setting s=t in (26) and substituting the result into (19): f°(.,t) = -%31N-l(p(t,t)gt+rt(.,t)),

(37)

tel.

By making use of Schwartz's kernel theorem [4], it can be verified that the optimal feedback control (37) (with N being the identity operator on L2(Q)) can be represented in the form: t K0(x,x',t)y(x',t)dx' + "t-T-F 2Kl(x'x''t't' )#t(x',t')dF2dt'+rt(x,t)l

t'

(38) where {K0,K I} is the kernel of P(t,t). P ro~leg_m~: Let yo,~0 and f be given functions satisfying the hypothesis of Theorem i with uEL2(E). Let y(x,t;u) denote the solution of (1)-(4) at (x,t) corresponding to a given control u~ UE. cost functional is minimized over UZ:

We wish to find a uEU~ such that the following T

J(u)=ll jQ lY(x,t;u)-yd]2dxdt + %21 ]y(x,T;n)-YdT]2dx + 13 I

~s

(~u)u dFdt, upp (~) (39) where the %~s,y d and YdT are as in Problem i, and ~ is a positive linear operator on 0

L2(~) into L2(Z).! ~-- If yo,~O and f satisfyl the conditions of Theorem i, then for each u~U~, y(u) EH2'~(Q) and y(.,T;u)~HY(~). Hence J(u) is defined. Similar to Problem i, this problem has a unique optimal control u°EU~ if %3>0. Also, u ° can be characterized by %1 I(Y(U°)-Yd)(Y(u)-y(u°)) -Q + %3 I

dxdt + %2 I(Y(x'T;u°)-YdT)(Y(x'T;u)-y(x'T;u°))dx -Q T I (~u°)(u-u°) dFdt ) 0 0 "supp (~)

for all u~U~.

(40)

170

The foregoing inequality can be simplified by introducing an adjoint equation whose form is identical to (9)-(13). 3 ~_ a unique solution y(u)6H2'4(Q)

From Theorem i, for any u6L2(Z), there exists i If Y~dE L2 (Q) and YdTEH2(~),

! with y(.,T;u)~H2(~).

then the right-hand-side of (9) and (I0) are in L2(Q) and H2-(~) respectively. far

Simi3 3 to Problem i, we can establish the existence of a unique solution p(u)~H2'~(Q)

for (9)-(13).

Moreover, (40) can be simplified as

T I 0

(p(u°)~(x)+%3Nu°)(u-u°)

dFdt >~ 0

for all

(41)

supp (@)

in Problems 1 and 2, the cost functionals involve only deviations of the solutions from their desired values averaged over the interior of the spatial domain ~. Similar results can be obtained for optimal control problems in which the spatial averaging in the cost functional is taken over both ~ and its boundary F[6]. III. STABILITY In applications~ it is of interest to establish stability conditions for systems with specified forms of feedback controls.

Here, we consider system (i) with f=0

and with a Dirichlet boundary condition of the form: y(x, t)=~(x) {y(w(x) ,t-T)+u(x, t) }, where # and w are as in (3).

(x,t) ~ Fx[O,°°[,

(42)

In view of (38), we consider a feedback control which

is a linear function of the state O t (defined by (25)) and has a representation of the form: t u(x,t)= I Fo(X,X',t)y(x',t) -~

dx' + I I Fl(x,x',t,t')~t(x',t')dF2dt', -t_T~F 2 (x,t) 6Fx[O,~[.

(43)

We shall establish a sufficient condition for the boundedness of solutions of system (i) with boundary condition (43) and feedback control (43). Let Qt

denote the cylinder set ~x]O,tl[ for O
Let Yo6C°(~)

continuous ~unction in Qt I satisfying (i) in QtlU Stl where St=~x{t}.

and y be a Then the weak

maximum principle [5] asserts that the maximum of IYl in Qt I is attained on the part _

SOU(Fx]O,t!])

of the boundary of Qtl , i.e. max ly(x~t) i ~ max {ma_x lYo(X) I, sup ly(x,t)I}. Qtl ~ Fx]O,t 1 ]

(44)

we shall show by contradiction that if t q(x't)-I~(x)i Ii I1 + I •IFo(X'x''t)Idx' +

I IFl(X'x''t't')IdF2dt' II< 1 t-T JF 2

(45)

171

for all (x,t) EF×[0,~[, then the maximum of IYl is attained on S0 or max_ly(x,t) I ~ x6fi

m aXlYo(X) I

for all t~0.

(46)

First, assume that there exists a point (x*,t*)EFx]T,tl] such that [y(x*,t*)I= max ly(x,t)[.

(47)

Qt 1 Then, i n view of (42) and ( 4 3 ) , we have

Iy(x*,t*)I~
I 'Fl(x*'x't*'t)''Y(X't)'dxdtl t*-T Jr 2

~< n(x*'t*)'Y(x''t*)''

(48)

where N is defined in (45). It is evident that if (45) is satisfied, then (48) leads to a contradiction. Consequently, such a point (x*,t*) does not exist under condition (45). Now, assume that a point (x*,t*)EFx[0,T] exists such that (47) holds. from (42) and (43), we have

Then,

Iy(x*, t*)l~<1~(x*)I I I~0(w(x*), t*-T) 1 \ +

(I

IF0(x*,x,t*)Idx +

~2

fill

)

'

~Fl(X*,X,t*,t) Idxdt ly(x*,t*)l I •

-T

F

(49)

If we impose the condition that ~0 is continuous on E0 and satisfies sup [~0(x,t) l < max lYo(X)], E0

(50)

then I~0(W(X*),t*-T) I <

max lYo(X) I ~ ly(x*,t*) I

(51)

and ly(x*,t*)l < n(x*,t*)ly(x*,t*) I.

(52)

Again~ under condition (45),(52) leads to a contradiction. Thus, such (x*,t*) does not exist. Finally, we note that the foregoing results remain valid for arbitrarily large tI. Consequently, (46) holds. Thus, we have established: Theorem 3: Let y be a classical solution of (i) with f=0 satisfying boundary condition (42),(43) and initial data (4). Let Yo~C°(~) and ~0EC°(ZO ) such that (50) is satisfied. Then, under condition (45), ma~ly(x,t) I < ma_xly (x) I for all t~0. The conditions in the above theorem represent restrlctl~ns ~n the lnltxal data,

t72

the parameter in the boundary conditions and the kernels of the feedback control operators.

Physically speaking, the result simply states that if the feedback gain

and the maximum magnitude of the past boundary data are sufficiently small, then the solution will not grow with time. IV. CONCLUDING REMARKS In this paper, only parabolic systems with the simplest forms of boundary conditions involving time-delays have been considered.

One may consider optimal control

and stability problems for parabolic system (I) with a variety of more complex boundary conditions involving time-delays.

A few examples are given below:

~. Boundarr~ Condition Involvin$ Multi~l ~ Time-Delays : This a generalization of the Neumann boundary condition (2): M ~ ( x , t ) + Y(x,t)y(x~t) = ~(x) I ~ b m ( x , t ) y ( w ( x ) , t - T m) m=0 M +

where 0~T0<%I<°~<%M;

~

Cm(X,t)~(w(x),t-Tm)+U(X,t) I

(53)

on E,

~bm and cm are specified coefficients.

If Y,Cm~m=l~.~.~M are

identically zero on F~ the results of this paper can be extended to this case without difficulty. 2. Boundary Condition with Indirect Control: Here, the boundary condition is identical to (2) and (3) except that u is generated indirectly by u(x,t) = g(x)Tz(t),

(x,t)~,

(54)

where (o)T denotes transposition; g is a given mapping from F into Rr; z(t) E R r is the solution of the following system of linear ordinary differential-difference equations: M dz(t) dt

~'~

Fm(t)z(t-T m) + G(t)v(t)

(55)

m=0 with given initi~l da~a z(t ~) = ~(t ~ ) 6 R r,

t'6[-Tm,0],

(56)

where 0
The control v ( - ) ~ ,

a

One may consider the optimal control

problem involving the minimization of a convex cost functional of the solution y and the control v.

Also~ instead of (55), one may replace it by a functional differen-

tial equation.

Finally, one may consider similar problems for general parabolic and

hyperbolic systems with boundary conditions involving time-delays. will be discussed elsewhere.

These problems

173

ACKNOWLEDGEMENT: This work was supported by U.S.Air Force-Office of Scientific Research, Grant No. AFOSR-72-2303. REFERENCES : [i] J.L.Lions, Contr$1e Optimal de Syst~mes Gouvern~s par des Equations aux D~riv~es Partielles, Dunod,1968. [2] J.L.Lions and E.Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol.l, Springer-Verlag, N.Y. 1972 (Translated by P.Kenneth). [3] J.L.Lions and E.Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol.2, Springer-Verlag, N.Y. 1972 (Translated by P.Kenneth). [4] L.Schwartz, "Theorie des Noyaux", Proc. Int. Congress of Mathematicians, Vol.l, 1950, pp.220-230. [5] O.A.Lady~enskaja,V.A. Solonnikov and N.N.Ural'ceva,Linear and Quasilinear Equations of Parabolic Type,

Translations of Math. Monographs No.23, Am.Math.Soc.R.I.1968.

[6] P.K.C.Wang, "Optimal Control of Parabolic Systems with Boundary Conditions Involving Time-Delays", Univ. of Calif. School of Engineering and Applied Science Rpt. No. UCLA-ENG-7346, June,1973.

CHARACTERIZATION

OF C O N E S

I S O M O R P H I C TO C O N E S

OF C O N V E X

Jean-Pierre University

OF F U N C T I O N S FUNCTIONS

AUBIN

of P a r i s - 9

Dauphine

- Paris

INTRODUCTION The tions

a i m of this

A(U)

F(S)

of

subset

defined

lower

function

semi-continuous

exists

~

is the

characterization

on a t o p o l o g i c a l

S of a l o c a l l y

There

paper

convex

convex vector

a continuous

£ A(U)

space

of c o n e s

U isomorphic

functions

space

with

defined

F in the

of

the cone

on a c o n v e x

following

sense

map ~ from U o n t o S s u c h t h a t

c a n be w r i t t e n

in a u n i q u e

way

func-

in the

:

any

following

form ~(~ The ties

of

existence

lower

of

= g(~ u) of s u c h

we

This

shall

paper

i. E X A M P L E

stating describe

summarizes

Functions

:

g

~

an i s o m o r p h i s m

semi-continuous

A(U) o B e f o r e

functions,

where

convex the

F(S)

implies

functions

theorem

hold

that

all

for the

characterizing

the p r o p e r functions

such

cones

of

two e x a m p l e s . the r e p o r t

defined

[~ .

on a f a m i l y

of

convex

compact subsets.

i. 1 D @ f i n i t i o n s Let ~X a n d (i,

l

vector

spaces

supplied

with

their

weak

opologie s and

(I.

X ~ be two p a i r e d

2) ~ If

let us d e n o t e the

by

set of c o n v e x

L = {PI'

compact

.... , pn } r a n g e s

subsets over

the

of X family

of

finite

subsets

175

of X' and ~ L ( X ) the w e a k

= max l[ i=l,...n

topology

denotes the semi-norms of

of X, we shall

s u p p l y A w i t h the t o p o l o g y

d e f i n e d by the semi d i s t a n c e s

(i. 3)

6L(A'

B) = m a x { sup inf x6A yeB

~ L (~ - y) , sup inf yeB xeA

w h e n L r a n g e s o v e r the finite

Definition

We s h a l l

~.ay t h a t

and t h a t a

of X'

I. 1

AoCA i s '~onvex" i f

V A, B ~ Ao, q

(1. 4)

subsets

l~L(X - y)]

~

[o, 11 ,

~A÷(I-~)B~A0

function ~ from A° into ~ i s "convex"

if (I. 5)

V A, B ~Ao , ~ [ o , We s h a l l

denote

"convex"

functions

Remark . If

i] , ~ A

by A(Ao)

Ae

on A

o

i. 1

q

: X.

,~-

A~

.~

In p a r t i c u l a r , (i. 7)

t h e cone of l o w e r s e m i - c o n t i n u o u s

defined

i

(i. 6)

+(i-)B).< X~(A)+ (1-~)~(B).

A ~ A

~

is a c o n v e x

(A) = inf x&A

the s u p p o r t

|

)

q(A,

function,

q(x)

then

is "convex': on

A.

functions

p) = sup x~A

< p, x >

w h e r e p ~ X'

are '~ffine". We c a n p r o v e the f o l l o w i n g Proposition

Any f u n c t i o n

i.

result.

1

~

A(A,)

can be w r i t t e n

i n a u n i q u e way

i76

(I, 8)

~(A)

= g

(~(A))

0-(A)

,~ { p ~-........--)cr(A~ p) ] (

where

i)

(i. 9)

l[

ii)

g is a lower

semi-continuous

~

X'

function

on

~ X'

!o 2 Som_~e a p p l i c a t i o n s Let us i n t r o d u c e i) a f u n c t i o n ii} (i, i0)

a lower

where

S

space

Y

iii)

i)

is a c o n v e x

l

convex

subset

linear

"conjugate

~p)

A(A0)

semi-continuous

a continuous

and the

(i. 11)

9e

of a l o c a l l y

map L

functions"

= sup

function

e

g £ F(S)

convex

vector

L(X, Y)

defined

by

[~(A, p) - ~ ( A ) ]

where

p e X'

A~o g

%ii)

(q) = sup yeS

Pr0positio__~n If

{ ao

then

}

there

inf A ~ Ao

[< q, y >

- g(y) ]

y,

where q

io 2 6-

Ao i s

exists

such

that

g ,is

y e Y'

such

that

I V (A) + inf xeA

continuous

g(- L x)~

at

L ao

,

=

(I" 12) = - rain

[~'(L'

q6Y

We can d e d u c e result"

(q)]

from this p r o p o s i t i o n

for m i n i m i z a t i o n

Corollary Let P ~

q) + g ~

~ the f o l l o w i n g

problems

1. 1

Y be a c l o s e d

convex

cone

wi~h

o

P

(for the M a c k e y

If u ~ Y,

"duality

topology

let us i n t r o d u c e

T(Y,

Y'))

the subset

non-empty

interior

177 (i. 13)

Au =

{A e

If there

Ao s u c h t h a t there e x i s t s

exists

(I. 14)

(i. 15)

{ao}

L ao - u

then there

exists

inf

= inf

Ae =

~(A) Au

-

Ae

~((L'

q)

e Ao s u c h t h a t ~

~ ~

P~ p+ s u c h t h a t

[~(A)

- o(A, L' q) + ]

Ao - = - sin

[~L'

qeP

Brief

economic

Let X = ~ n price

systems,

a commodity

X' = ~ n

the space of prices,

u

~ n

set of c o n v e x c o m p a c t

subsets

o. e

Ao as a ~irm w h o s e p r o d u c t i o n

and w h o s e

~(A,

space,

vector.

We i n t e r p r e t A A - ~n+

+

P+ =f~.n+ the c o n e of p o s i t i v e

We d e n o t e by Ao a "convex" containing

q) - ~

interpretation

denote

a "demand"

x ~ A w i t h L x - u E P}

"maximum profit

p) = sup x e A

defined

function"

is

on P+ =

[~n

set is

+

We t a k e Y = X and L to be the i d e n t i t y m a p p i n g . T h e n the set A u is the set of firms w h i c h "demand" If

~

vector : Ao~

of firms,

A ,

the

u. )~

is a c o s t

t h e n the m i n i m i z a t i o n

the m i n i m i z a t i o n

can s a t i s f y

on

) ~ (A) -

function defined of

~

Ao of the p e r t u r b e d ~(A, ~) +

<4, u>

on

~u

on the set amounts

cost f u n c t i o n

to

t78

: Functions

2. EXAMPLE

defined

on ~ a l g e b r a s

2. io D~finitions Let ~i) (i. I)

A

~ii)

be a

o_algebra

m : Ae____~i3-~m

defined on a set

be an atomless

bounded v e c t o r - v a l u e d

measure. We shall setA ~ B if and only if re(A) = re(B) and the factor (2. 2)

~(A,

set

A/~

B) =

Ao

, supplied with the distance

~Im(A) - m ( B ) ~

By the Lyapounov

theorem,

re(A) is convex.

Therefore,

(e

3)

lw
A, B) = m-l(lra(A)

Definition

We s h a l l

(2. 4)

If

g

by

defined

q ( ~ ( l t Af B)) Remarks

on A~ s a t i s f y i n g

i

I ~(A)

+ (i -I )9(B)

2. 1 :

~m

(2. 5)

_____>~

~(A) belongs to

is a lower semi-continuous

= g(m(A)) any function

in a unique way in the form

convex

then the function %0 defined by

~(Ao) . Conversely,

In particular, AF____>-

toAo.

A(Ao) t h e cone of l o w e r s e m i - c o n t i n u o u s

function defined on m(A),

written

(i -I )re(B)) belongs

2. 1

denote

functions

+

~ 6 ~(Ao)

can be

(2. 5) where g is convex.

the functions A| ..... _) and (where p 6 ~ m ) belong to

A(A=).

179

. Let ~

= { 9 ~ L~(~,

and

L I(~, A,

~°6

A, [ml) such that 9 (~) &

Iml).

Then the function

~

: Ao ~-----9~[~ defined by

(A) = inf { ~%o[~) 8 (e)d]ml(~) I @ 6 ~

(2. 6)

belongs to

[O, 13 }

and ~

(~)dlm I (~) = m(A)}

A (Ao) .

2. 2. Some Applications Let

us

introduce

i) a function

{~eA(Ao)

ii) a lower semi-continuous (2. 7)

convex function g E F(S) where S

is a convex subset of a locally convex vector space Y ii) a continuous

linear map

L ~ L (~n

y)

and the "conjugate functions" defined by

l

i)

t~(p)

= sup

[ - ~(A)]

where p & ~ m

[ - g(y) ]

where q

AeAo

ii)

g ~ (q) = sup y£S

Proposition If there

2. 1

exists

then there

~ Y'

Ao • Ao s u c h

exists

q

that

g is

continuouS

a t L m[A0),

~ Y'such that

l inf [~(A) A ~ Ao + g( - L m(A))]

||= - min < qGY'

[~(L'

q) + g~(q)]

We can deduce from this proposition the following result"

for minimization

problems.

"duality

180

C o r o l l a r Z 2.

Let

(2.

P C Y be a c l o s e d

P

(for

the

If

u

Y,

6

~ackey let

Ii)

there

L m

then

that

the

L re(A)

a non-empty

interior

Y~))

subset

-

Ao & Ao s u c h

exists

q~(A)

-q

: inf

Au (2.

• (Y,

with

u

6

P}

that

- u ~ P

there

linf

cone

us i n t r o d u c e

exists

(Ao)

convex

topology

A u = {A ~ Ao s u c h

10)

If (2.

1

~

P+ s u c h

.that

[~(A)

-
L m(A)>

=-

rain [ ~ ( L

+
u> ]

A~Ao

12

=-

~(L

~ ~) +

~ q) -

]

qeP +

Brief For

instance¢

on the map

economic

set

interpretation we

~ of

associating

resulting u ~ m

can

"elementary with

from

regard

A as a set of decisions"

any d e c i s i o n

the d e c i s i o n

as an o b j e c t i v e .

A.

Then

A ~

and

"act"

Y =~m

set A

defined

the m e a s u r e

Ao an

We take the

"decisions"

m(A)& q~ m

and we

is the

m as the

interpret

set of d e c i -

U

sions~ If

~

whose

acts

: A o ~

of d e c i s i o n s ,

are

greater

is a c o s t then

the m i n i m i z a t i o n

than

function

the m i n i m i z a t i o n

on

A~-------~ ~ (A) -
~he o b j e c t i v e defined of

Ao of the p e r t u r b e d re(A)> + <~,

u>

~

u.

on the

on cost

set

Au amounts function

to

181

3. STATEMENT OF THE THEOREM OF ISOMORPHISM The two above cones of functions are examples of cones of " ~ -vex" functions 3. i. The cone of

y-vex

functions

Let l i) U be a set (3. I) ~ i i ) ~ ' I ( U )

be the set of probability discrete measures

I

~ = Z finite (where

ei 6(ui) on U

~(u i) is the Dirac measure at ui, o/ 0, 7 ~i = i) finite

Let us introduce (3. 2) ~

be a correspondance with non-empty values from

L

'I(U) into U

Definition

3. I

We s h a l l say t h a t a f u n c t i o n t~. u~ (3. 3)

(~) ~

~ ei ~ (Ui) for any ~ -

)~

is "~-vex"

~ ~i ~(ui) @

if

b°'l(u)

and " ~ -affine" (3. 4)

q(~)

= ~ ~i ~(ui) for any ~ : Z °i 6(ui) ~ ~'I(U)

Remark 3. 1 If U is a convex subset of vector space X, then usual convex functions are (3. 5)

~-vex functions where ~ is the map defined by

~ = Z ei ui whenever

~ = ~ ei 6(ui)

~

~'I(U) .

If A is the family of convex compact subsets of X, then the "convex" functions ~

£

~(A) are ~ - v e x

functions where

is the map defined by (3. 6)

~

= ~ ~i Ai whenever ~ = E

~i ~(Ai)

£

~'I(A)

182

°

If A is a o-algebra

vector-valued ~-vex (3. 7)

~

measure,

ei m(Ai))

~____~< m

is a bounded

then the functions

functions where

= m-l([

A

and m :

~

£

atomless

A(Ao)

are

~ is the map defined by ~ = Z ~i' @(A i)

whenever

3. 2. The cone of lower semi-continuous

~-vex

G ~' 1 (Ao) functions

Let us denote by (3. 8)

G the vector

space of

~ -affine

functions defined on U

We shall supply U with the topology defined by the semi-distances (3. 9)

@L(Uf v) = max i=l,...~n when L =

~ fi(u)

{fl . . . . .

fn~

- fi(v) I

ranges over the finite

subsets of G.

We denote by (3. i0)

~[(U)

the cone of lower semi-continuous

~-vex

functions

Theorem 3. 1

There

exist

(3. ll) !i) a locally convex vector lii)

a continuous

space F

map n from U into F

such that (3. 12) S = ~(U)

is a convex subset of F

and s u c h t h ~ t h e

map

(3. ].3) g ~ F ( S ) ~

is

~

an i s o m o r p h i s m

convex nuous

functions ~-vex

= g o ~ 6

from t h e

functions

~-vex

references,

cone F(S]

on S o n t o t h e

of l o w e r s e m i - c o n t i n u o u s

c o n e A (u)

of lower semi-conti-

on U.

The proof of this theorem, of the

fl~(U)

functions,

based on the the minimax properties

together with other results

can be found in

[i]

and the

183

[q

.

Jean-Pierre AUBIN E x i s t e n c e of saddle points for classes of convex and nonconvex functions. M a t h e m a t i c s R e s e a r c h Center Technical Summary Report ~ U n i v e r s i t y of W i s c o n s i n

(1972).

1289

NECESSARY

CONDITIONS

FOR PARETO O P T I ~ L I T Y JEAN-LOUIS

AND SUFFICIENT

IN A MULTICRITERION

GOFFIN

i.

Decision making

confronted

by a multiplicity

it into account directly, The concept control

theory

In addition maker

of goals,

a renewed

systems

and it seems necessary

has been extensively

theory and more recently

is

to take

studied

in the

in the field of optimal

([I] - E6]).

to the complexity

due to multiple

ce the consequences

of his actions.

trained perturbations

following

if the perturbations - or players

which influenif their pro-

can be modelled

the lines given in Refs

is quite fitting

of other decision makers

the consequence

the decision-

These disturbances,

cannot be assessed

This description

goals,

or uncertainties

bability distribution

Formally

attracted

and economic

rather than finding an a priori mix of goals.

is often faced with perturbations

the existence

- Montreal

have recently

in engineering

of Pareto optimality

realm of economic

Commerciales

INTRODUCTION

problems

attention.

SYSTEM*

ALAIN HAURIE

Ecole des Hautes Etudes

Vector valued optimization

CONDITIONS PERTURBED

as set-cons[7]

[9].

are caused by

- which can affect

of the decision of a coalition of players[10].

the system is described

by :

(i)

A decision

set X

(ii)

A perturbation

(iii)

A vector valued cost criterion

set Y

¢ : x × z ÷ R p , (x, y) ~ ,(x,

y) ~ (~j(x,

y)) j=l ..... p

In section

2~ the optimality

generalization In section

criterion

is precisely

defined

as a

of Pareto-optimality.

3~ the scalarization

process

is studied

for this class of

system. In section

4, a Lagrange multiplier

*This work was supported and $73-0268o

theorem

is proved.

by the Canada Council under Grants $72-0513

185 Notation

¢(.)

A (~j(.)) j=l,... ,p

~x

, ygR

Rp A: { z ~

p

x < y

if

Rp : O <

!

~x(X, y) will denote

xj .< yj

j = i,

....

p

z}

the gradient w.r.t,

x of the fonction

~(., y) at

p o i n t x. ~(x;

h) will

denote

the directional

the function ~j (x) 2. To every decision

g i v e n on

Given a preorder nimum

DEFINITION

y)

(X,-<)

in the direction

h of

@

OF OPTIMALITY

x in X is associated

~ ( x , Y) =a { , ( x , Any p r e o r d e r

derivative

: y~

a set of possible

outcomes

:

Y}

{~O(x, Y) : x ~ X} w i l l a minimal

induce a preorder

on X.

element will be called a Pareto-mi-

in X.

In this paper

the following

x ~ x' where

if

Sup

preorder

will be used

~(x, Y) _< Sup

Sup ~(x, Y) is the L.U.B.

Sup ~ ( x , Y) = (Sup ~ j ( x ,

of

~(x',

:

Y)

~(x, Y)

in

(Rp, _<). Note

that

Y)) j--I ..... p

Thus

the following

Definition

2.1 :

Sup {~j(x, implies

that

of Pareto

optimality

x* in X is Pareto-optimal y)

: y c Y} _< Sup{~j(x*

y)

is adopted.

if, for all x in x : : y~

Y}

~je.{l ..... p}

: y ~ Y}

V j ~ {i ..... p}

:

Sup {~j(x, We can define

definition

y) : y C Y} = Sup{~j(x,* y)

the auxiliary

cost function

~:

X + Rp

(x) _A Sup ~ ( x , Y). Definition

2.1 is then equivalent

x* in X is Pareto

optimal

to :

if for all x in X :

by

:

:

186

In order to characterize be

:

used

Pareto optimal elements,

scalarization process

The

[i] - [ ~

based on an extension of the theorem of Kuhn 3. The main results

THE SCALARIZATION

the scalarization

rectly on the cost functions tion : i ~

and Tucker

[I]

cannot,

~.

PROCESS

in optimization with a vector-valued

around the process of scalarization perturbations

two approaches will

and a direct method

[6].

criterion revolve

In the presence

of

in general, be performed di-

as for ~j > 0 , j = i,..., p

the condi-

X and :

P Sup { Z ~j ~j(~, y) j =I

P : y ~ Y} < Sup{ Z ~j ~j(x, y) j --I

: y ~ Y}

Yx~X

does not imply that i is Pareto-optimal! T)"" The scalarization must be applied to the auxiliary cost function First we state the following well known result. Theorem 3.1 ~x

:

g X

Let ~. > 0, j = i,..., p and x* in X be such that ] P P Z ~j ,J~:(x*)_< Z ~j~_ (x) j=! j=l J

:

then x* is Pareto optimal. Corollary 3.! VxaX

Let ~j > 0, j = i,..o, p and x* in X be such that : Sup yl ~ Y

P Z aj ¢~j(x*, y j) _< j=l

<2 YpCY Sup

yl~Y

P Z c~, ~j(x, yj) j=l 3

yt Y Yp Y

(T)Consider as an example x 2 - y(x-l)).

Thus

only Pareto optimal which attains

:

~(x)

X =A R , = (x z +

element is

y A= {-l, l},~(x, y) ~A ( x ~ y ( x - l ) , !x - !I,

x* = ~.

x z + Ix - i])

and the

But ~l(X, y)+~2(x,

its unique minimum at i = 0.

y) = 2x 2

t87

then x* is Pareto-optimal. :

Proof P

Z aj j--i =

Sup Cj(x, y ) ygY

P

=

Z Sup aj ~j(x, yj) j=l yjaY

P Z

Sup ylgY,... ,yj£Y

~j Cj(x, yj)

For a necessary scalarization tions are needed.

•

j=l

condition to hold some convexity assump-

The following lemma indicates what kind of convexity

must be met. Lemma 3.1 :

x* in X is Pareto-optimal

iff the following holds :

(~(x*) - Rb n (~(x~ + Rb = { ? ( x . ) } Proof

:

Clearly

x* i s P a r e t o - o p t i m a l

Obviously

(3.1)

implies

L.H.S.

(3.1)

then,

of

thus FOx) -- T ( x * )

(3.2).

for

(3.1)

iff

Conversely

some x i n X, ~ '

let

~ be an e l e m e n t o f t h e

a n d ~" i n Rp one h a s

- (~' + ~ " ) e ~O(x*) - RP

and the L . . . S .

:

o f (3.Z)

contains ~ (x) . Thus

(3.2)

implies

~(x*)

--~(x)

, ~'

= ~" -- 0

and f i n a l l y

o~ = ~ ( x * ) ,

that is (3.1). I Remark 3.1 : If X is a convex subset of a linear space and e a c h ~ j convex on X, then~(X) Remark 3.2 : ~ ( X )

Vx,

x'

~

÷ R p is convex, (but~j

is not sufficient)

¢ R p is convex iff :

x,

Vx

i.e. there exists a mapping

Vx, ~, ~ x ,

quasi-convex

V~[o,

[o, l]

3z

~x

s.t.

T : X × X × E0, I~ ÷ X such that :

:Lq ~(T(~, x,, ~,)) __<~,~(~) + (1 - ~ , ) ~ ( x , ) .

Theorem 3.2 : I f ~ ( x ) + R p is convex and x* is Pareto-optimal then there exist ~j _> 0 for all j and ~Z > 0 for some Z such that : x ~ X

is

P Z ~j ~j(x*) j =I

S

P Z ~j~j(x) j =i

188 Proof

:

Direct

AssumRtion

3.~

application

of lemma 3.1 and the separation

:

X is a convex

subset of a Banach space U

Y is a compact

space

: U x y ÷ R p is continuous x + ~x(X,

~,

Theorem 3.3 (i)

the following

and Lemaire

:

theorem using

If x* E X is Pareto-optimal

where

aj

Yj(x)

results

of Danskin

Max y ~Yj(x*)

:

3.1

then there exist a. Z 0 for all j, J :

aZ > 0 for some % such that P

Z j=l

in x E U.

family.

~,

Under assumption

Min x(X

and convex

y) is an equicontinuous

Then~ we can prove Dem'Yanov

theorem, m

< ~ j'x

(x*,

~ {y ~ Y : * j ( x ,

y)

'

x

- x*

y) = Max ~ j ( x ,

>

=

0

(3.3)

y)}

y~Y (ii)

Conversely, holds,

Proof

:

[12_]

[14

:

(i)

is d i r e c t i o n a l l y

differentiable

with

(see Refs

: ; h) =

P ( 2 j=l

Max y 6 g j (x)

e~j L~j)'

We c a n u s e t h e o r e m get

(3.3)

then x* is Pareto-optimal.

For each j, ? j

H~j' ( x Since

if there exist ~j > 0 for each j such that

and

It is clear

(x ; h)

=

(x,y),

P j =2l

h >

c~j 'h0' ' j(x

," h)

2.4 o f Dem'Yanov [13] and t h e o r e m s

3.1

and 3 . 2

to

(ii). that the scalarization

to be applicable necessary

< ~ j' x

it requires

approach

convexity

is the simpler one, but

assumptions,

even to get

conditions.

In the next section, optimality,

we will develop necessary

which do not require

convexity.

conditions

for Pareto-

189

4. In this

section,

A LAGRANGE

the results

MULTIPLIER

of D a n s k i n

THEOREM

[12] and Bram

[15] will

be used

extensively. Assumption

4.1

:

Y is a compact

space

: R n x y ÷ R p is c o n t i n u o u s

and its d e r i v a t i v e

~xv (X , y) is

continuous. X ~ R n is d e f i n e d

by a set of c o n s t r a i n t s

X = {x ~ R n : gi(x) gi Theorem

: Rn ÷ R

4.1

:

is

_> 0

i = 1 .....

C I, i -- I,

Under A s s u m p t i o n

...

4.1,

m}

, m.

if x* in X is P a r e t o - o p t i m a l

if the Kuhn

- Tucker

there

~i• ~ 0 for i = i, "" . ' m and aj >- 0 for j = i,... ' p

exist

aZ > 0 f o r

s o m e ~,

P Z aj j=l and

such

Max y~Yj (x*)

then

Proof

qualification

is s a t i s f i e d

and

at x*,

then

: , y),

h >

~

m Z hi< i=l

(x*) gi

h >

(4 1) "

'

= 0.

if the gi's

(4.1) w i t h

that x*

that

< ~jx(X

~k > 0==~gk(x*)

Furthermore,

(4.2) J

are concave

~i ~ 0 , i = I,~..,

and the ~j s are c o n v e x m and aj > 0, j = i,...,

in x p implies

is P a r e t o - o p t i m a l .

:

Let x* ¢

every x E X

this

constraint

:

implies

X be a P a r e t o - o p t i m a l

either

~ j s.t.?j(x)

or

V J

that

but

that

This means

that

for

j(x) :

jcx*)

:

Vxex (Note

point.

> ~j(x*)

this

Max

J is a n e c e s s a r y

it is a n e c e s s a r y

0 condition

and s u f f i c i e n t

(43)

for P a r e t o - o p t i m a l i t y ,

condition

for w e a k Pareto-

optimality). Let F be the set of v e c t o r s such that

there

a tangent

vector

exist

admissible

an arc issuing

at x equal

to h.

at x, from x,

that and

is the set of h ~ R n lying

in X, w i t h

190

Thus

(4.3)

becomes

:

j Under

the

(4,4)

J

constraint

qualification

F = {h ~ R n : < gi(x*), where

I = {i : gi(x*)

Let

Wj ~= ~ x ( X * , Max WEW.

h > _> 0,

one

i E

has

:

I}

: 0}.

Yj(x*))

< w,

Max j = I.-, p

condition,

, then (4 .3 )becomes

h > _> 0

~h

£ F

< w, h > _> 0

~h

E F

:

]

or

Max ~$~UW~

and,

if W is the c o n v e x

hull

of

U

W.

j:l . . . . . p Max wEW Consider

the cone

following Let ~ £

Bram~s

F*N

W

,

J

< w, h > _> 0

~h

F* ~ {z ~ R n

: < z, h > ~ 0

proof

~

it can be s h o w n

, t h e n there

exist

~ r

~h

that F * ~

Ii ~ 0 , i ~ I

~ F}

then

W is not

such

that

?

=

Z iEI

ki gi (x*)

and there

exist wj ~ Wj,

such that

: P

j:l therefore, ~h

]

aj ~ 0 , j : 1 .... , p

,

one has

:

]

setting

Xi = 0

P Z ~j Max j =i wj~Wj

~ Rn

if

i ~

< w

I h >

J'

m

_>

that

is

(4.1) o

Z j=l

i i < gi(x*),

h >

P Z ~. = i j=l ]

:

empty.

191 Let us remark that if each set Wj reduces each ~j is differentiable Kuhn - Tucker result

m

X mj VJ'.x(X*, Y j ( x * ) )

j--i

classical

then

:

p

Finally,

to a single element,

at x* and (4.1) yields the generalized

=

J

X

i=l

t i g~Cx*).

the proof of the sufficient case.

condition

is the same as in the

•

5.

CONCLUSION

The notion of Pareto-optimality The scalarization

process

which have been obtained

has been extended

to perturbed

and the extended Lagrange multiplier can be used to characterize

systems. rule

and compute opti-

mal decisions. A promising

field of application

theory without boundary

side payments

of the Auman's

corresponds

of these results

[16]

characteristic

game

function for a given coalition

to the set of all Pareto-optimal

defined perturbed

is "n-player"

; the reader could verify that the outcomes

in an adequatly

system. REFERENCES

[i]

A.W. STARR et Y.C. HO : Nonzero-Sum

Differential

Games, JOTA,

3 (1969),

184-206.

[2] Further Properties (1969),

[3]

[4]

of Nonzero-Sum

Games, JOTA

207-219.

T.L. VINCENT

& G. LEITMANN

Control

Space Properties

(1970),

91-113.

A. BLAQUIERE

Differential

: of cooperative

games, JOTA,

4

:

Sur la g~om~trie

des surfaces

de Pareto d'un jeu diff~ren-

tiel ~ N joueurs, 744-747.

C.R. Acad.

Sc. Paris S6r. A, 271 (1970),

192

[5]

A~ BLAQUIERE,

Lo JURICEK &

K.E. WIESE :

Sur la g~om~trie des surfaces de Pareto d'un jeu diff~rentiel ~ N joueurs; th~or~me du maximum, C.R. Acad. Sc. Paris

A, 271 (1970), i030-I032 [6]

A. HAURIE

:

Jeux quantitatifs

~ M joueurs, doctoral dissertation, Paris

1970. [7]

M.C. DELFOUR

& S.K. MITTER :

Reachability of Perturbed Linear Systems and Min Sup Problems, SIAM J. On control, [8]

D.P. BERTSEKAS &

7 (1969), 521-533

I.B. RHODES :

On the Minimax Reachability of Targets and Target Tubes, Automatica, [9]

J.D. GLOVER &

7 (1971), 233-247.

F.C. SCHWEPPE

:

Control of Linear Dynamic Systems with Set Constrained Disturbances, [I0]

A. HAURIE

IEEE Trans. on Control, AC-16

(1971), 411-423.

:

On Pareto Optimal Decisions for a Coalition of a Subset of Players, [11]

H.W.

IEEE Trans. on Automatic Control, avril 1973.

KUHN & A.W.

TUCKER :

Non-Linear Programming,

2nd Berkeley Symposium of Mathema-

tical Statistics and Probability,

Univ. Calif. Press,

Berkeley 1951. [12]

J. DANSKIN : On the Theory of Min-Max, J.SIAM Appl. Math., Vol. 14 (1966), pp. 641-664.

[13]

V.F. DEM'YANOV & A.M. RUBINOV : Minimization of functiona!s

in normed spaces, SIAM J. Con-

trol, Vol. 6 (1968), pp. 73-88. [14]

B. LEMAIRE : Probl&mes min-max et applications au contrSle optimal de syst~mes gouvern~s par des ~quations aux d~riv6es partielles lin~aires, Th~se de doctorat, facult6 des sciences, Universit6 de Paris, 1970.

[15]

J.

B~M

:

The Lagrange Multiplier Theorem for Max-Min with several Constraints,

J. SIAM App. Math. Vol 14 (1966), pp 665-667.

193

[161

R.J. AUMANN

:

A Survey of Cooperative Games without side Payments, in Essays in Mathematical Economics, ed. M. Shubik, Princeton 1969.

A UNIFIED THEORY OF DETERMINISTIC TWO-PLAYERS ZERO-SUM DIFFERENTIAL GAMES Christian Marehal Office National d'Etudes et de Recherches Atrospatiales (ONERA) 92320 - Chhtillon (France)

Abstract This paper is a shorter presentation of "Generalization of the optimality theory of Pontryagin to deterministic two-players zero-sum differential games~ [MARCHAL, 1973] presented at the fifth 1FIP conference on optimization techniques. The very notions of zero-sum game and deterministic game are discussed in the first sections. The only interesting case is the case when there is "complete and infinitely rapid information". When the minimax assumption is not satisfied it is necessary to define 3 types of games according to ratios between time-constant of a chattering between two or several controls and delays necessary to measure adverse control and to react to that control ; it thus emphasizes the meaning of the "complete and infinitely rapid information" concept. In the last sections the optimality theory of Pontryagin is generalized to deterministic two-players zero-sum differential games ; it leads to the notion of extremal pencil (or bundle) of trajectories. When some eanonieity conditions generalizing that of Pontryagin are satisfied the equations describing the extremal pencils are ~ery simple but lead to many kinds of singularities already found empirically in some simple examples and called barrier, universal surfaces, dispersal surfaces, focal lines, equivocal lines etc...

lntroduction Many authors have tried to extand to differential game problems the beautiful Pontryagin's theory used in optimization problems, but there is so many singularities in differential game problems, even in the deterministic twoplayers zero-sum case, that a general expression is difficult to find and to express. A new notion is used here : the notion of "extremal pencil (or bundle) of trajectories". This notion allows to present the generalization of Pontryagin's theory in a simple way.

t. Two-players zero-sum differential games Usually two-players zero-sum differential games are presented as follow : A) There is a parameter of description t that we shall call the time. B) The system of interest used n other parameters xt, x2. . . . xn and we shall put : (1)

X =

(~IzXtA./. . . . )Xn)

= state vector.

We shall assume that t, the performance index of interest (called also cost function or pay off) is a only function of the final values--~¢, t~ ; if necessary, if for instance I is related to an integral taken along the described trajectory X(t), we must add into X a component related to that integral. C) Ther~_~ two p i e r s

that we shall call the maximisor M and the minlmisor m, each of them chooses a measurable

control M(t) and re(t) (respectively in the control d o m a ~ , g~)M(t) and ~)m(t) Borelian functions of t) and the velocity vector V = dX/dt is a given Borelian function of X,t and the two controls :

195

D) There is a "playing space" ~ subset of the Rn+l ~,t space and a "terminal surface" ~ , "terminal subset" ~ along the boundaries of ~ .

or more generally a

We shall assume that the set ~ is open.

E) The control equation (2) is defined everywhere in ~

and the performance index I(Xf, tf) is defined everywhere

in ~ ; the game starts in at a given initial p o i n t ~o, t o ; themaximisar tries to maximises I at the first arrival at and the minimisor tries to minimizes it. 2. Zero-sum games Since there is only one performance index I(Xf, tf) it seems that the above defined game is zero-sum, however it is possible that both players have interest to avoid the termination of the game (e.g. the cat, the mouse and the hole around a circular take, when the cat blocks the hole) and in order to avoid diffieutties of non zero-sam games the value of the performance index must also be defined in non-finite cases. 3. Deterministic cases of two-players zero-sum differential games

A first condition of determinism is that both players have complete informations on the control function (~,M, m, t), the control domains ~f)M(t) and

re(t), the performance index

I(Xf, tf),

the playing space ~ , the

terminal subset ~ and the initial eonditions-Xo, t o. It is possible to imagine some particular eases of detem~inistie games such as : A) One of the two players, for instance M, has more or less complete informations on the present and past state v e c t o r ~ t ) and choose a pure strategy based on these various informatinns :

B) He must indicate his choice to the other player. C) The second player choose then its own control function m'~t). Hence : When (3) is given the problem of m is an ordinary'problem of minimization and then M chooses its strategy (3) in order that this minimum be as large as possible. However, if the informations of the first player are incomplete, the eouditions A and B are generally unrealistic : the choice of a good mixed strategy improves very often the possibilities of the first player and the real problems is thus not deterministic. The only realistic and deterministic cases arc the following : A) Both players can measure the present value of-~at an infinite rate ; we shall call T M and T m the infinitely small delays necessary to obtain this measure and to react to it. B) In some eases the optimal control requires a chattering between two or several controls, we shall assume that these chatterings can be made at an infinite rate and we shall call ~"M and

~m the corresponding infinitely small

intervals of time. C) There is then 3 deterministic cases : CI) Case when "~m + Tin<< ~'M It is the maximin case or Mm-case, everything happens as if the minimisor m could choose its control after the maximisor M at any instant. C2) "~M + TM<<'~r'm"

It is the minimax case or raM-case symmetrical to the previous one.

196

C3) ~-'M~< ~-m * Tm

and

~'m < < ~ M + TM

It is the neutral case or .N-case, both players choose their own control independently of the opponent choice. The determinism of that Iast case requires some more conditions (see in chapter 5.1 the condition of equality of H 1 and t"12). Of course the assumption of determinism implies that both players know if the game is a raM, Mra or N-game. A simple example of these 3 tapes of game is given in the following example : Initial conditions : xo = t o = 0 ; terminal subset tf = 1 ; performance index I = xf ; control function d x / d t = P-~I2 + Mra - 2I'n2 ; control d o m a i n s : Hence

]MJ <. 1

;

lint .<

1.

:

Maximin case : M = .#-+ 1 ; m = - s i g n M ; xf = - 1 Minimax c a s e : m -

4. 1 ; M = s i g n m ; x f =

+1

Neutral case : M and m chatter equally at very high rate as in a Poisson process between + 1 and -1, it gives xf = 0. We shall see that these 3 types of games are equivalent (and the comparisons of TM, Tm, ]~"M' "g'm are not necessary) if the control function has the form :

and i f ' ~ M a n d / o r ~ m a r e bounded.

4. The upper game and the lower game Another reason of undeterrainism appears when there is discontinuities of the performance index (e.g. : [ = 0 if xf5 t$ 0

and I = 1 if xf = 0) or when the terminal subset has particular forms (such as the two sheets

of a cone) and when an infinitely sraall change of the control, e s p e c i a l l y near the final instant, gives a large change of the performance index : it i s indeed impossible to follow the opponent reactions with an infinite accuracy. In order to avoid that kind of undeterminisra it is sufficient to give to one player an infinitesimal right to cheat, i.e, to add to the velocity vector V = d ~ d t

a component " ~ ' w ' h o s e integral j ~ l ~ ' ~ l l . d t is as small as

desired by its opponent in any sufficiently large bounded set of the"~,t,t space. We obtain thus the upper game and the lower game according to the player who has the right to cheat (cLassification independant of the minimax, raaxirain and neutral types), The upper and lower values of a game will be major elements of appreciation of that gmne in a given situation since an infinite accuracy is never possible.

5. Extension af the Pontryagin's theory to deterministic two-players z ero-sura differential games 5. L The adjoint vcctor]5~and the Generalized Hamiltonian H*(P, X, t) We s h a l l use the ordinary notations and first the adjoint vector of Pontryagin P which will be c l o s e l y related to the strategy of each player.

197

As usual the Hamiltonian will be the scalar product :

(5)

H = ~

~"'/M~

=

"~..V"~"

and by a direct generalization of the notion of "Generalized Hamiltonian" (MARCttAL 1971, ROCKAFELLAR 1970) the new "Generalized Hamiltonian ~ is : A) For a game of "maximin type" :

I"l (P,)~.~).=

'6)

SU-[~

irlf

P.V(X, MIFrI~

B) For a game of "minimax type n :

{71

=

C) For a game of "neutral type n let us define H1 and H2, the

)~i. Is being positive and their sum being one :

(8)

It is easy to verify that always :

but these two quantities are note necessarily equal (if for instance, for given P, X, t, the scalar product P.V is equal to M - m, M and m being arbitrary real positive numbers). The determinism of a game of "neutral type" requires that the two functions H 1 (P, X, t) and H2(P, X, t) be identical, they of course are then equal to the Generalized Hamiltonian H*(P, X, t) of the game. A sufficient condition of equality of H I and tt 2 is that at least one of the two control domains ~ M ( t ) and ~m(t) be a compact set of a Rq space and that for any (X, t} the control f u n c t i o n " ~ = - ~ ( ~ , -M,,-~, t) be uniformly continuous with respect to the corresponding control parameter. Of course the maximin type being the most favourable to the minimisor the corresponding Generalized Hamiltonian is always the smallest and conversely the Generalized Hamiltonian of the minimax type is always the largest. It is easy to verify that the 3 Generalized Hamiltonian are identical if the control has the form (4) and if the velocities VM and/or Vm are bounded. It is possible to see now how the adjoint vector P is related to the strategies of both players, let us assume for instance that locally, between the instants t: and t; + ~ I:, the maximisor want to maximises the scalar product K.X and the minimisor want to minimizes it, they will both choose the control corresponding to P = K in (6), (7) or (8) according to the type of the game (with a chattering if necessary) and, if H* is continuous in terms of P-~,X-~and t, they ~vill obtain :

=

+ o@

198

5.2. The conditions o~ eanonicity

In order to avoid the difficulties coming from the discontinuities of trajectories X~t) we shall assume that the control function V = V(~, M, m, t) (with-~£ ~)M(t) and m ~ ) m ( t ) is bounded in any bounded set of t h e - ~ , t spa~e, it implies that the trajec:ories X(t) are Lipehitz functions of t: and that the Generalized Hamiltanian H*(P, X, t) is bounded in any bounded set of the P, X, t space. On the ether hand let us note that the part of the terminal subset where the performance index is very small is considered as a forbidden zone by the maximisor and conversely the part of the terminal subset where the performance index is very large is cansidercd as a forbidden zone by the minimlsar. Thus, in order to obtain a generalization of the Pantryagin theory to differential games, it is necessary that the conditions of application of that theory to problems with forbidden zones be satisfied, that is (MARCHAL 1971, page 151). A) The problem must be canonical in the generalized meaning of Pontryagin for the admissibility of the discontinuous type, i.e. here, since velocities V are locally bounded : The Generalized Itamihanian tt (P, X, t) must be a locally !

Lipschitzian function of P--~,X-~and t.

This severe condition has uhe advantage to involve the equivalence between chattering and relaxation, which is necessary especially for neutral type games. B) The terminal subset ~' must be union or intersection of a finite number of closed or open "smooth" sets (i.e. manifolds with everywhere a tangent subspace Lipschitzian function of the position) these manifolds must be limited by a finite number of "smooth" hypersurface themselves limited by a finite number of "smooth" hyperlines et c .... For instance the terminal subset can be the surface of a polyhedron or of a cylinder and this condition is satisfied in almost all ordinary problems, however it is not satisfied at the origin for the function y = xn when either 0.5 < n < 1

or l < n < 2 .

C) For any value lo the parts of the terminal subset defined either by I > I o orby I / > I o

orby I < I o orby

t ~ Io must satisfy the same condition o~ "smoothness". 5.3. The Generalized Pontryagin~s formulas In ordinary problems of optimization, when the Generalized Hamihanlan H*(P, ~ , t) is defined and when the conditions of canonicity are satisfied it is possible to adjoin to each cxtremal trajeetory'-~t) an absolutely cantinuans adjoint function P-~t) different from zero and such that, with H*[P(t)j X(t), t] = H*(t), either :

or, more generally (when H (P, X, t) is locally Lipehltzian but not continuously differentiable in terms of P, X and t). I Thevect°r

~

(~(~)]

-'~(~) / ~*(~))

~ ~elongs [or almost all t to the domain

"~F

~ ]D~)~'X--'~(~)j ~)

199

This domain DHt(~, ~ , t) being the smallest closed and convex set (of the R 2n+1 space) containing the gradient vectors ~H / 9 ( P , X, t) obtained at points (P'~+ ~

X*+ ~

t) where :

A) H* is differentiable in terms of P, X and t ; B) ~:~ and

~

are infinitely small (i.e. of course DHt is the limit for fz-----.~-0of domains DHt E

tl$'gll and II g ~ II can vary from 0 to E

obtained when

; DHt is a particular "convex hull").

When there is forbidden zones with "smooth~ boundaries the adjoint function P(t) becomes the sum of an absolutely continuous function and a "jump function" (i.e. a function with a finite or denumerable number of discontinuities and which is constant between these discontinuities) and the equations (11) and (12) become more complex. Let us now try to generalize these equations to differential game problems. We shall only consider upper game problems with a bounded playing space and we shall decompose them into the different "games of kind" corresponding to either I ~ I o

or l ~

I o.

Let us note that some people consider that the part of the terminal subset ~ corresponding to I ~ lo. in the upper game problem, part that we shall call ~ + , is only the closure of the corresponding part of the initial game, some other people add to that closed set the points where the local upper limit of the performance index is I o and thus obtain all points where that local upper limit is larger than or equal to Io. Anyhow in both cases the set ~ + is closed and thus the two cases are similar. We shall call

~_~,_the subset

~-- ~+.

It is easy to demonstrate that the part of the playing space corresponding to [ < I o (in the upper game problem) is open, we shall call that subset Oio and we shatl call ~ cially interested by the closed set ~

the remaining part of the playing space. We are espe-

intersection of the boundaries of Oio and ~IIo'

The generalization of Pontryagin's theory lsee demonstration in MARCItAL, to appear)leads to : If the playing space is bounded, to each point (~o' to) of the boundary ,J~ corresponds a pencil (or bundle) of -.-> absolutely continuous trajectories Xi(t) belonging entirely to :~ , each of them being associated to an adjoint function'~i(t) defined on the same interval of time and sum of an absolutely continuous function of t and a jump function of t (with a bounded total variation). We shall call extremal pencil or extremal bundle the union of the trajectories Xi(t) ; this extremal pencil satisfy the following generalized conditions : A) The pencil begins at the point (~o' to) of interest with at least one trajectory. B) Each trajectory Xi-'~t) is defined on a closed interval of time (tio, tif) and ends at the terminal subset ~ (and not only at ~ + as written erroneously in MARCHAL 1973). ¢>

C) h point ( ~ , t) of the extremal pencil can belong to one or to several trajectories Xq(t) and there is always at least one corresponding adjoint vector Pi(t) different from zero. D) Wish H* (t) = H*(Pi(t), Xi(t), t), the equations (11) become :

-_ (13)

H

J J

200

with :

(14) E) In the same way the generalized equations (12) becomes : For any given i we have for almost all t :

(~)~

--~

-9, ~

~V

ae k F) The functions Pi(t) and H~ (t) can also have jumps in the directions given by the infinite values of the positive factors )~ij" G) As usual when there is forbidden zones, if a point (~,t) of the pencil belong to the terminal zone ~ one can add to the derivatives of the vector ( ~ , -H*) given in (13) and (15) a component normal to ~ and directed outward (if ( ~ , t)t~'~-) or toward the playing space (if ('~, t)~ ~e+), one can even add a jump provided that exist connecting absolutely continuous fnnctions P i ( ~ ) , Hi (tp) leading from P(t-), tt*(t-) to "~t+), H*(t+), verifying for any t.~ the relation (6)(or (7) or (8)) and having their derivatives with respect to ~o in the directions given by these outer or inner normal components. H) Finally for each trajectory Xi(t) of the extremal pencil the final values [Pi(tif), H~(tif)l ,must satisfy the ordinary final conditions of Pontryagin. (also called *transversality conditions").

A simple way to obta;n the directions normal to ~ is to use a Lipsehitzias penalty function f(X, t) equal to zero on ~ and negative in the playing space ~ ; the local gradient of f ( ~ , t) with re~peet so (~,t) gives the outer normal direction (or directions, for instance at a corner of ~ ). On the same way, for the condition H. the final vector [Pi'(tif) ; -I~* (tif)l,. with (n + i) components, must b'e parallel to and in the direction of the local gradient of a Lipsehitzian function f+(X, t) equal to zero on ~ + and negative anywhere else (or antisymmetrically wdth respect t o " ~ - if (Xif, tif)~-~-). Let us note that around points (~,, t) belonging to ~'+ but not to ~e-~_ it is useless to consider the function

f+(x~ t) out of ~ + ~ ( ~ d conversely for L(~, t) if (~,, t)e ~-). On the other hand if the direction of grad f+(X, t) (or grad f ( X , t)), it is not continuous at the final point (~i{' tif) of interest, the vector ~ ,

-H'ill may be into an), direction of the corresponding conic convex hull. Conclusion

The generalization of the optimization theory of Pontryagin to deterministic two-players zero-sum differential games leads to the notion of extremal pencil and to the above equations and rules which are sometimes sufficient to determine these pencils (see for instance the two examples of MARCHAL 1973). The main remaining question is to improve the conditions of backward construction of extremal pencils outlined in that reference.

201

References

ATHANS, M .

-

The status of optimal control theory and applications for deterministic systems, A, 8 differential

games. IEEE trans, on automatic control, (April 1966). BEHN, R.D. and HO, Y.C. - On a class of linear stochastic differential games. IEEE trans, on auto-control, (June 1968). CttYUNG, D.H. - On a class of pursuit evasion games. IEEE trans, on auto-control, (August 1970). HO, Y.C. and BARON, S. - Minimal time intercept problem. IEEE trans, on auto-control, (April 1965). HO, Y.C. , BRYSON, A.E. and BARON, S. - Differential games and optimal pursuit-evaslan strategics. IEEE trans, on auto-control, (October 1965). tSAACS, R. - Differential games. -John Wiley and Sons, (1965). JACOB, J.P. and POLAK, E. -On a class of pursuit-evasion problems. IEEE trans, on -auto-contro|, (December t967). MARCHAL, C. - Thearetlcal research in deterministic optimization. ONERA publication n° 139, (1971). MARCHAL, C. - The hi-canonical systems. Techniques of optimization. A.V. Balakrishnan Editor, Academic Press, New York and London, (1972). MARCHAL, C. - Generalization of the optimality theory of Pontryagin to deterministic two-players zero-sum differential games. ONERA, tir$ ~ part.n° 1233, (1973). MARCHAL, C. - Theoretical research in deterministic two-players zero-sum differential games. ONERA publication, to appear. MESCttLER, P.A. - Ou a goal-keeplng differential game. IEEE trans, on auto-control, (February 1967). MESCHLER, P.A. - Comments on a linear pursuit-evasion game. IEEE trans, on auto-control, (June 1967). PONTRYAGIN, L.S. , BOLTYANSKII, V.G. , GAMKRELIDZE, R.V. and MISCR~KO, E.F. - The mathematical theory of optimal processes. Iuterscience Publishers, John Wiley and Sons, Inc. New York, (1962). ROCKAFELLAR, R.T. - Generalized Hamiltonian equations for convex problems of Lagrange. Pacific J. of Math. 33, 411-427 (1970). ROCKAFELLAR, R.T. - Dual problems of optimal control. Techniques of optimization. A.V. Balakrishnan editor. Ac. Presse, New York, London, (1972). WARGA, J . 1, (1965).

-

Minimax problems and unilateral curves in the calculus of variations. SIAM Journal on Control A, 3,

ABOUT OPTIMALITY OF TIME OF PURSUIT

M. S.NIKOL 'SKII STEKLOV MATHEMATICAL INSTITUTE, MOSCOW, USSR

I~ll tell about results of me, Dr. P.B.Gus~atnikov and V.I.Uhobotov in the problem optimality of pursuit time. I have studied with Dr. P.B.Gusjat-aikov this problem in 1968. There is the article about this results in Soviet Math. Dokl. (vol. 184, N 3, 1969). After this article was the article of N.N.Krasovskii and A.I.Subbotin about this questioao Their result is more general. I'll not tell about result of N.N.Krasovskii aud A.I.Subbotin (see Soviet Journal of Applied Math. and Mechan., N ~, 1969). Recently Dr. P.B.Gusjatnikov and I have got some results in this field in cooperation with Dr.V.I.Uhobotov. Let the motion of a vector

~

in Euclidean space

~be

descri-

bed by the linear vector differential equation

(1)

Q =

where

~£ 6. ~

, C

E QC~

~. t~ and ~

is a c o n s t a n t

square matrix , i~6. []>C-

are control vectors. The vector

to the pursuer, the vector

I~ to the evader.

compact sets. Let a terminal subspace

~

P

and

~

corresponds Q

are convex

be assigned in

pursuit is assumed to be completed when the point

•

~:

The

reaches

for the first time~ It is in the interest of the pursuer to complete the pursuit. The information of pursuer is the equality (1) and ~(~), V(~)

for all present

~ ~ O

. The functions

are meserable functions. The pursuer don't know ~)

~)

~ V(~)

~(~)

in advance,

can be arbitrary meserable function. P~utrjagin have constructed the method of pursuit from initial

point ~@ and gave estimate

%(~o)

of time of pursuit.

203 I'ii say some words about this method. Let

~

is complemental subspace of ~

of pro~ection of

~

onto ~

their geome~ical difference

~ g~.C~

1~f(~)= l[g

WO:,'-~d~

-W-~I-0

..rfg ~

~ -- l~g

~

and in-

.-_

, w h i c h i s a compact c o ~ v e ~ s e t ~

"~o in

The time of pursuit frnm

is operator

parallely ~ .

Pomtrjagi~ considers the compact sets

te~al

and ~T

the theory of Pontrjagin is the least

root of inclusion-

~C

Let

~(%o)

is such root.

Definition I. The optimal time of pursuit from ~o is the least time within pursuer can complete the pursuit from ~o. The Pontrjagin time tion ~

%(~o)

is optimal if the following Condi-

is fulfilled.

Condition A.

EQ

For all extreme point

such that for all

~E

~ 6 P

exists ~ =

[0,T]

~C

"cC

A

Theorem I. If the Condition A is fulfilled and then

~(~)

~(~o) ~ T ,

is optimal time of pursuit.

We shall give some sufficient conditions for fulfilmemt

the

Condition A. Let us

OE P

subspace and ~ , V

, 0

their support subspaces in Ye

restriction of mapp~g~T ~

~C mapping

q[~

Let us

is interior point of Q

on

~, ~(~)

o~ V . ~[%)

!

can be factored in such wa~

~h

in its support ~(~)

is

is re~triction of

204

where ~

is linear mapping ~

is linear mapping

g:}{1

and homeomorp~c ~or set

Q

i~[~ JV >/~ )

~to

L

~ ~ go,T]

sweep

Definition 2.

into

oo

,

,

L

1{~}

{~1~ ~)~( ~ ~. , ~1~(~) is analytical for ~{£o,T]

with the exception f i n i t e

pletely :he set

for

A convex closed set S

from space

set ]{;

~/

is strictly convex, if each its support plane has

only one common point with S . Definitio~

A convex closed set S

from space ~ ( ~ I )

is regular, if it is strictly convex and each its boundary point has only one support plane. Theorem 2. Let

here

~

~(~}

can be factored in the form

is linear mapping V-

analytical functiono Let

FQ

into

~

,

~(~)

is strictly convex in

-nonnegative ~

. In

these conditions Condition A is fulfilled. Theorem ~,

Le~

~

is regular in V

• In these

conditions t h e

equality (5) is necesssry for fulfilment the Condition A. The another sufficient conditions for fulfilment the Condition A are given by t~he following Theorem ~. Theorem @. If for

~ >I 0

q~

~C and

~

are commutative on

205

, ~

amd

~Q

sweeps completely the set ~ P

, rhea the Corn-

ditiom A is fulfilled.

Example.

The P o m t r ~ a g i n ' s t e s t e x ~ p l e :

"6

•

are positive comstamts,

~,~,~,~ ~ R K , K ~

{al~ ~ , frill , the~ the conditiom A is fulfilled

amd the time

~(~)

is optimal.

ALGEBRAIC AUTOMATA AND OPTIMAL SOLUTIONS IN PATTERN RECOGNITION

E.ASTESIANO Istituto

- G.COSTA

di Matematica

Via L.B. Alberti,4

- Universit~

di Genova -

16132 GENOVA (ITALY)

INTRODUCTION U. Grenander (1969) proposed a formalization of the linguistic approach in pattern recognition (see also Pavel (1969))o Though the main interest in this method is undoubtely for its pratical application to the construction of particular gr~m~nars of patterns,we think the theoretical questions worthwhile further insight. Therefore we devote our atten~on to the abstract formulation of the problem; however, in order to give a detailed model,we restrict ourselves to a definite class of decision rules. In the recognition system we consider here, the objects to be recognized are (represented by) terms on a graded alphabet and the decision rules are (impleme5 ted by) algebraic automata; moreover we assume that sample classes (in some sense the "training sets") and rules of identifications of objects in images are given. In this context we investigate the problems related to the definition, the existence and the effective construction of optimal grmmmars of patterns. This paper is a generalization and improovement of a previous work of one of the authors. The results can also be considered (but the point of wiew is completely diff~ rent) a generalization of well known classical results in algebraic theory of automa ta; these can in fact be obtained as a particular case of our results (when the sample classes are a partition). The first part is devoted to set up substantially well known definitions and results, in a language more apt to treat our problem. In the second the recognition model and a definition of the optimal solution are proposed and ex plaited. In the last section~conditions of existence for the optimal solution are g! vet and the problem of the effective construction is solved. Finally a few examples are presented to show that sQme seemingly incomplete results cannot in fact be substantially improoved. i. PRELIMINARY NOTIONS. We refer to Cohn (1965) and Gr~tzer (1968) concepts

and to Thatcher

cerns algebraic

and Wright

(1968) an~ Arbib and Give'on

for the algebraic

(1968) for what co~

automata°

If X is a non empty set, DF(X) is the set of all families of non ~npty,mutually joint,subsets

of X; we denote its elements b y ~ , ~ , . . . , O

Ao: U~'--~

~,..°

If~=[A

.

dis-

i, i aI} then

.

~/ A. and ~ : = ~ f A ~ . Conslder on DF(X) the relatlon ~-- defined by: i~l i ~ O; iff a (unique)map ~ : ~ - - - ~ e x i s t s s.t-~ A ~ , we also write

A~ f(A);

Propositio n i.I

~-~

is an order relation on DF(X)

and DF 1 (X) : =(DF(X),b--)

is a

complete

lattice.//

Remark.

We indicate by // the end of the prof; the proof is omitted whenever it is

This work was partially

supported by a G.N.A.F.A.

(C.N.R.)

grant.

207

straightforward or substantially known. Denote now by E(X) the complete lattice of equivalences on the set X, ordered by_. ~ as a subset of X ~ X • I f ~ D F ( X ) and ~ ~E(X),then @ i s an~-equivalence iff, A~, A = x~A~x~o For any P g E(X), let DL(X,P) be the set ~ ( ~ , ~ ) /~6DF(X), ~P , ~ is a ~-equivalence

Proposition 1.2.

and ~. be the relation on DL(X,P) defined by :

E_~ is an order relation and DLI(X,P) : = (DL(X,P),~) is acomplete

lattice iff P is a (complete) sublattice of E(X). // A graded (ranked) set is a pair (~,~'),where~-.is a non empty set and ~ a map fnto N

(the set of non negative integers); if n ~ , t h e n

from

~ := ~'-l(n). From now n

on we shall simply w r i t e ~ instead of (~,O'). A ~-algebra (or simply:an algebra) is a pair ~ = ( A , ~ the carrier of ~ , on A. If o J ~ n if n -0,then ~...

and ~ is a map that assigns to each element ~ i n ~ ' ~is

%is

an n-ary operator, that is : if

n~l, then

Given ~ =

then we denote free 5--algebra of~'-terms on X ( or

and its carrier by F~(X); if X = ~

(A, ~), we denote by rp~ ~is

connected iff rp~

denoted by C (~); C( ~ ) i s

the

unique homomorphism from ~ i n t o

is the set of

;we

DLI(A,C(~)) is

all ~-eongruences on ~ • C(~),

A graded (ranked) alphabet is a finite graded set ~

, s.t. ~

o will be a graded alphabet that we consider as fixed and given. is a pair M = ( ~ , ~

D L I ( ~ ,P). ~ ~. From now on

),where : ~ - - ( Q , ~ ) is a ~" -algebra and ~

DF(Q). The response function of M is just r p ~

of M is ~ M

~

then an ~-congruence is simply an ~-equivalence

We write : C,C(~), DLI(P), instead of C ( ~ ) ,

M is connected iff ~

and F~.

is onto. The set of all congruences on ~ will be

which is a congruence on ~; C ( ~ , ~ )

~-automaton

we write ~ .

a (complete) sublattice of E(A), hence

a complete lattice. If o ~ D F ( A ) ,

A

~ : An--P A ;

to indicate ~ - a l g e b r a s .

simply :terms) by ~ _ ( X )

on A

an operator

an element of A. We use capital German letters,~p~,...#~,...~

If X is a set disjoint from ~ ,

say that

),where A is a non empty set,

; we shall denote it also by rPM

•

is connected~and M is finite iff Q is finite. The behaviour

.= IB/B = rPMl(F), F e ~J; the equiresponse congruence of M

the canonical congruence on ~

ER(M), is

associated with rPM ; eventually ,~(M):----(ER(M),~M).

Lemma 1.3. For any ~- -automaton M, ~A.(M)e DLI(C).// If ( ~ , ~ )

~ DLI(C), then " ~ ( ( ~ , ~ ) )

is the ~-automaton ( ~ / ~

, ~/~),

where

- =

Lemma 1.4. If = (@,03).I1

(~,~)

e DLI(C), then q ~ ( ( e , ~ ) )

is connected and ~ ( ~ ((~,0~)))--

208

Let M

= (~[) ~ ) be a ~-automaton, i =1,2 ; an homomorphism from M into M is a i ....... I 2 pair ( ? , ~ ) , w h e r e ~ : ~ , - - ~ ~ Z is an algebras homomorphism, 9 : ~I--~ ~2 is a map and, ~

F @ ~i' ~ (F) ~ ( F ) "

(~,~)

is an isomorphism, and we write M I~M2, iff :

is an algebras isomorphism, ~ is one to one and onto and ~(F) = ~(F). If Mland M 2 are connected and there is an homomorphism ( ~ , ~ )

from M I into M2, then ( ~ , ~ )

is uniquely determined by the properties : rpM = ~ o r P M a n d ~ ( F ) - ~ (F) ~ F F - ~ , too2 1 reover ~ is onto, we indicate this by writing Ml--@~M2.These definitions and properties allow Lermna 1.5.

us to state the following lermna.

If M and M' are connected automata, then : i) ~ (]k(M))=M ; ii) M--~M'

iff ~ ( M ) 4.~(M~).// We denote by ~ .

the class of connected ~ -

lance, by [M]~ the class of ~ _

automata mod. the isomorphism equivac corresponding to M and by ~ the relation on ~ d e -

fined by : E M I ~ [ M ~ - iff M 2 ~

M I. By the above lermna, ~ is correctly defined and

it is order relation. Theorem i°6.

(~,~4)

is a poset (i.e. partially ordered set) anti-isomorphic to

DLl(C),therefore ( 9 Q ' ~ )

is a complete lattice anti-isomorphic to DLI(C).//

We recall that similar results~ for monadic algebra automata without final states~ can be found in B~chi (1966). Given

~=

(A, ~ ), a symbol x not in A IJE. and S~--A, then for each "Itin F~(SU{x~)

we can define the unary operator o n ~ i) if • = s @ S,

then

i~(a)=s;

iii) if ~ = ~i++. ~nOO ,then if~ =O~e~o Remarks.

,then

I~I=H~

, as follows : ~ a

ii) if ~ = x , then

e A,

ll~ll~(a) = a ;

~I'~I~(a) = ~o~( II~--iII~ (a), .... ~l~nll~ (a)); clearly

ll~II~(a) = ~

•

i) the above operators correspond to the "polynomials" in Gr~tzer (1968);

ii) One can verify that if ~ = I ~l~li~, ~: e F~ ~x~ )

;

is connected, then, ~ S ~ A ,

iii)

the ~ - t e r m on Y obtained from ~

if ~ =

~(Y)

[ II~:|I~,~eF~(SU~x}~

, for any set Y , then

l l ~ ( a ) is

by replacing each occurrence of x in ~: with the

term a+ Consider

~

e DF(A) and ~ connected.

Definition i+I. N ~ i s eA.a~Fz

,

I[~;~(a)

the relation on A Sot., if a, beA,

@ A~ i f f

llz'l$(h)

(a,b) e N ~

iff~'6~(Ix)),

e A i.

Theorem 1.7.

i) N~

iii) C ( ~ , ~ )

is a (complete) sublattice of the complete,lattice C(~).

Proof. of N ~ ,

Hint :

is the maximnm of C ( ~ , ~ )

; ii) G(~,~)----I~'C(~)/e_~N~}

use a modified, but equivalent (see above remark),definition

considering F~ ( A ~ x ~ )

instead of F~( ~x~ )°//

209

For any

~

@DF(F~[ ), set C ( ~

): = C ( ~ Z , ~ ) ;

minimal 5"-automata for ~(i.e.

2. THE RECOGNITION MODEL.

having ~

by theorems 1.6 and 1.7 a class of

as their behaviour) exists: the class

We need a few other definitions before we can give our

recognition model.Consider on DF(X), X non empty set,the relation M-M-- ~

iff

~ ~

~

defined by :

and the map ~ is a bijection. It is soon verified that

W-- is an order relation.

If ~ =

(A, c()

and

~C(~)

then ~

is an O~-separa-

ring congruence iff each congruence class intersects at most one element of ~ . denote by SC ( ~ )

the set of all d-separating

Consider now the free algebra i = I,...,N~ ; we denote by

congruences on ~[. FZ

__~ and a set H = [(ti,t'i)I (ti,t')_ ~(H)

We

the congruence on ~ g e n e r a t e d

X

by H , seeGr~tzer

(1968). One can verify that ~(H) coincides with the reflexive~ symmetric and transitive closure of the relation R(H), defined by :(t,t')~ R(H) iff there is a pair (ti,t')i in H s.t. t' is obtained from t replacing in t t i by t'i " We can now give in detail our recognition model; see Grenander (1969) and Pavel (1969) for the motivations,

for part of the terminology and a wider discussion on the subject.

- The "objects" to be recognized are coded on a graded alphabet - W h a t we actually recognize are (structural)descriptions of the "objects"(analogous to the "configurations" in Grenander (1969)) i.e.

-terms. One "object" may corre-

spond to mauy different descriptions. - We have some information about the corrispondence descriptions - "objects" ,i.e. we are given a finite set H

F

x F

: (t,t')eH means that t and t' correspond to

the same "object" and can thus be identified. If appears quite natural that we identify also all the pairs in R(H) and that we extend the process by reflexivity, symmetry and transitivity. This eventually amounts

to consider on F

the identifica-

tions given by the congruence ~(H). We can restate all this, by saying that we are given a finitely generated congruence (mod I ~ )

and the images now become,

- We are given

~DF(F~),

h~- ~ -

and

An admissible,

for I ~ I~

on ~ Z " we call imases the classes

for us, the objects to be recognized.

the family of examples , i.e. a family of sets ofalrea

dy classified descriptions, o ~ a n d - An admissible,

I~

I~

must be such that I ~

is -~-separating.

and O~ , family of patterns is a family

~ 6 D F ( F ~ ) s.t.

is a ~-congruence.

for I ~

and ~ ,

decision rule is a map

r : F~.

~

~

, such

that : ~r: = -l~r'i(A)' A~)--~ is an admissible family of patterns and a (connected) -automaton M e~ists such that ~r----~ M

(we say that r is implemented by M ).

210

Usually a decision rule can be implemented by several (even non isomorphi¢)~'-aut. Definition 2.1.

A solution of the recosnition problem,with I ~

is a E-automaton implementing an admissible, for I ~ Consider

~=(A,c~), ~EDF(A) and P _ ~ S C ( ~ A ) ,

and

~

~given,

and o~,decision rule.

then : EDL ( ~ , P):= [ ( ~ , ~ )

(~,~)

DL( A ,P), Remark • The order ~ (~,~)~(~,~)

induced on EDL(~,P) by the order < we have on DL(A,P) is sot.

iff ~

and ~ H - - ~ °

We call quasi-complete lower semilattice (q.c.l.s.-lattice~ any ordered set in which all non ~ ={0~DF(F~.

subsets have a g.l.b. ) I~H-~}

Thus~ if we consider the set EDF(~): =

, the set of extensions of ~

,ordered byH--

; it is ea-

sy to see that (EDF(~),~--) is a q.c.l.s.-lattice but not a lattice. From this we have also that,if P is a q.c.l.s.-lattice, then E D L ( ~ P )

is a q.c.l.s.-lattice, but,

even if P is a complete lattice, EDL ( ~ P ) is not in general a lattice. This fact I is of great importance as+we shall see now. Set SC(~): = S C ( ~ K , ~ )

and, if P ~_ SC(~), Ip 1 = ~ ~ ; ~ 6 P

/~2-I~

then from

theorem 1.6 and the remark about <~ we have the following theorem. Theorem 2.1.

M is a solution of the recognition problem, for given

I~

and ~

,

iff ~ ( M ) E EDL ( ~ , ISC(~)).// Remark. We can use this theorem for a new

definitionof "solution of the recognition

problem". Indeed from now on we shall refer to EDL ( ~ , i SC(~)) as the set of solutions of the recosnition

problem, (for given

I~ andS).

We can now characterize different kinds of solutions. Let :

~6DF(F~_), I~ & SC(~)

and P _~ SC(o~) be given. First of all we observe that considering P~_.SC(~) instead of S C ( ~ ) means that we are using only a subclass of all the admissible decision rules: exactly the class of rules implemented by automata which, for their algebraic structure, correspond to P

(see theorem 1.6). For instance if P=FSC(c~):=

=

+~

{ ~{ ~ S C ( ~ ) ,

ind ( ~ ) <

then we consider only the decision rules implemen

ted by finite ~ -automata whose equiresponse congruence is o~-separating (now we are not taking account of Definition 2.2.

I~

).

EDLE(~'!P) : =

I (~'~):

(~'(~) ~ EDL(O~, Ip), ~ =

max.of

(ipn c(~))} This is the set of "economical" solutions: for each admissible family of patterns ~,

we consider the minimal, i.e. 'Imore economical", ~ -automata for ~

class of automata corresponding to P ( if they exist).

in the

211

Definition 2.3.

If ~ =

(A,O~), ~6DF(A) and

~II~SC(~O~), then ~T: =IBT ,B&~I,

where B ~ := a ~ B [a]~ Definition 2.4.

EDLJ(~'IP) : = I ( ~ ' ~ )

/ M~G Ip~ •

This is the set of ~ustified solutions: for each ~ in Ipwe extend each A i n ' b y adding to A only elements of F~ which are ~I/-congruent to at least one element of A. It is now quite reasonable the following definition. Definition 2.5. The set of $ood .solutions of the recognition problem~ for given O~, I@ I I I I

and P is CS(~, P)

=EDLE(e{, P) t% EDLG(~, P)= l(T,~T)/T=max( P~C(~l)) .

I

The three above sets are ordered by<< ,as subsets of EDL(~, P)° Definition 2.6. The optimal solution , for given ~ , I ~

and P, is the maximum of

CS(~,IP), if it exists. We denote it by o.s. (~,Ip). Consider now the two conditions : I

[o(] ~ ( ~ , ~ T ) , E~]~(T,~)

~,, Ip , l ' u ' b ' / c ( ~ T ) (Ip~, c(,AT))e P . £EDL (~,Ip),

Propositien 2.2.

l.u.b.lc(~ ) (ip n c( 8)) el P .

i) EDLJ(0~, Ip)~-- Ip (as posets); ii) if max EDLJ(~,IP) exists,

then max GS (~,Ip) exists and the two coincide; if condition~dJholds, the converse is also true. // Proposition 2.3. (Stability property)

Ip) then, ~'l~t" s't'~tt"~ll~-~,

Proof.A'~-~ = ~ = ~ T

=max

--max

If condition [ ~ holds and ( ~ , & ) = o.s.(~,

(4 I, IPl°t SC(~t))= (~[/,(~).

o.s.

~= SO(4)

andlas ~ _ l e , Is ~ SC(~)° Since by prop.2.2

P and c l e a r l y ~ - ~ H - - ~ :--

EDLJ

Remark°

(1~', Ip 1"~SCI~')).

= (~_t~,~) =

implies

Thus by prop.2.2

One can easily verify that condition ~

= G.S(~I,IP ~ SC(~I)).//

:=~ max

, then ( T , ~ )

,and therefore E ~ ~ is met, for

example~ when P = SC(~) or P = FSC(~), 3. EXISTENCE OF OPTIMAL SOLUTIONS.

Let M =(~, ~) be a ~-automaton; if ~eSC(~,~),

then M/~ is the ~'-automaton ( 9 / ~ , ~ / ~ ) , F & ~.

where ~7~:={F'/F'=I[q]~ , q~F) ,

Recalling def.2.3, we shall write I~ instead of ~ I ~ •

Proposition 3.i.

If ER(M) = I~ and ~ M

I~, then : o.s. (~,Isc(~)) existsiff

max SC(g,~) exists; if max SC(~,~) = ~ then o.s.(~,Iso(~))=]~(M/~ Proof.

Set ~ : = I ~

).

; from well known isomorphism theorems (see, for instance,

Gr~tzer (1968)), we have the lattice isomorphisms : C ( ~ ) ~ _ _ C ( ~ ) - - - ~ / ~ 6 C ( ~ ) , ~};

this implies the poset isomorphisms: S C ( ~ , ~ ) = S C ( ~

~@Isc(~M)

-- I S G ( ~ )

, ~/~)~-I~ /

= ISC(~). Thus by Prop.2.2 the first part of the proposition

is proved. As for the second, if ~== : = max ISC(~), remarking that ~ / ~

corresponds

212

to ~ i n

the isomorphism

SC(B,~)= ~SC<~),

we have

~19=~1e1~,1~'-~19.

:

Hence and fro~ the definitions of M/@ andp.,we have :>(M/@ )=(~, ~ Proposition

=

3.2,. If o.s. (~,Isc(~)) exists and ~ ( M ) e EDLJ(o~,Isc(~)), then

exists s.t.~ = max S C ( ~ , ~ ) and ~ ( M / ~ ) = o.s. ( ~ Proof.

)

ISC(~)).

Setting ~ "= ER(M), by propo.2.2, o.s, ( ~ , I s c ( ~ ) ) = m a x

and thus : (I~, ~ ) ~ < ~ ( M )

= (~,~M)~o.s.

EDLJ(~,Isc(~))

(~,Isc(~)). Since ~ - - ~ M i m p l i e s

SC(~ M) ~--_SC(~) , recalling the stability property (Prop.2.3), we have : o.s.(~,Isc(~)) =

o.s. (~M, I S C ( ~ ) ~

yields : B o,s. (~M, ISc(~M)) ~

~

S c ( ~ M ) ) = o.s.(~M ISc(~M))" Prop.2.2 max I S c ( ~ M ) ~ o.s.(~ M, ~SC((~M)) =

=

o.s. (~M, ISc(6M)). Thus by Prop.3.1 ~ exists and # ( M / ~ ) = o.s.((~M,~sc(~M))

=

o.s.

(¢~, ! sc

Consider

(A)).II

~--- (A, ~ )

and ~ e D F ( A ) ;

we now i n v e s t i g a t e

on t h e n a t u r e of S C ( ~ ) .

'~ L'I'A, recalling the definition of the operators

If x ~

llzll ,'=

ix} ), we

have the following definition. Definition 3.1. ~

V' is the relation on A defined by : if a, b6A, ( a , b ) G ~

F~_(~x~), ~ B ~ ,

ll~ll~ (a)~ B ~

iff

II~l~(b)~ B ~ B ° and II2:l~(b) ~ B

II~l~(a)@ B V B .o This relation is a modification and generalization of a relation defined in Verbeek (1967).

It is clear that V ~

is reflexive and symmetric but examples show that it

is not in general transitive (e.g. when ~ - - ~ _ a n d Let V ~ be the transitive closure of V~

Proposition 3.3. ii) V~

i)

(a,b)@ V~ ~

is a congruence o n e ;

B~

B is a finite set).

; we have then the following proposition •

(llZtl~(a) , Ilrll~(h))ev$ ~=~Fe ¢×~ ); ----[76c(~)l~=_v,~} ; iv) sc(gg(~)

iii) S C ( ~ )

is a (quasi-complete) lower sub-semilattice of the complete lattice C(~); if V ~ = V~ Proof.

then V~ = m a x S C ( ~ , ~ ) and S C ( ~ ) i) follows from the property :~C,~'~ F

----]l~l]~a), # where ~: =[l~ll~(~)--~ and ~ . ' = ~ x } ) ~

is a complete lattice. (~x~), ~ a S A , ll~[l~(l|i?{~(a)) = ii) consider~ (V~6~); by i) and

the definition of @ (V'~), it is easy to prove that ~ ( V ~ iii) again is based on property

i)

) = V~ . The proof of

and the fact that the same property holds for

any congruence. Eventually it is quite easy to verify the first part of iv)~ while the second is rather trivial given ii) and iii). // Remark.

If V ~

~ V~

it is not true, in general, that V~

is the l.u.h, of

SC(~,~), as we shall see with an example later on. The following proposition is strictly related, as we shall see,to the stability property.

213

Proposition 3.4. If V ~ = % , G(~)

s,~ ~ = ~ %

; then : V~ = N ~ = V~

-- S C ( ~ , ~ ) = SC(~,~), hence S C ( 9 ~ )

V'~

, denote BV~ by B, and~11~H~ byIl~II, ~ - F

; so N ~ V ~

= V~ • Moreover : ~ll'--~

; hence : V ~ _ V ~ N ~

o

(~x})o As V ~ is a congruence

(a,b) ~ V~=~ (HZll (a), I~ZlI(B))6 V~ , now, ~ B @ ~

i.e. (a,b)~ N~

and

is a (complete) sublattice of C(~).

Proof. It is easy to verify that ~I~-~ implies V ~ If B e ~

= V~

, we have : |IZlI (a) 6 B

, hence : (+) V ~ = V~ = N ~ which implies also V ~ =

yields SC(~,~)

~. SC(~, ~ ) 9 - - C ( ~ )

; by (+),

Theor.l.7 ii) and Prop. 3.3~ iii) , we have the coincidence of the three sets.// Consider now ~(i) and "~ :=

~(1), ~(2) ~ SC(~,~),

Proposition 3o5,

Proof.

~(1) =- ~(2) ; if ~ (i).=~/,~,

~(i)/ ~(i) , i = 1,2, then the canonical epimorphism ~ : ~ ) ~ ( l J

is such that ~ B

if f

where

6 ~

, B / ~(i) ~'~

B / ~ (2)

and ~'I(B/ ~ (2)) = B / ~ (i)o

If u,v ~ A/ ~ (1) , the carrier of ~

(~(u), ~(v)) ~ V~tL)

(where obviously V'~(~)

(1) , then : (u,v) ~ V~I,)

is on A / ~ L&) ).

It follows from the definition of the V'-relation and of the epimorphism

and from the following property, (see GrEtzer (1968)): ~ q~&F~.( ~x~ ),~u 6 Q(1)

~f (II'~ll~w(~)) Setting ~ =

~,

= ll"cll~c,~ ,f(u)).// if M = ( ~ , ~ ) , we have the following results.

Corollary3.6. ~f ~ = ~ q , i , ~ then v& = %

and ~ = ~ I B i ! ~ i

if__~v~ = v~ (whereV %

-i rp,

is on ~ Z and V~

(F.), i~l~ ,

one)//

0or.3.6. and Prop.3.3~ together with Prop.3.2~ give the following theorem. Theorem 3.7.

If ~<M) g EDLJ (~,ISC(~)) and V'~

exists and o.s. (~,ISC(~)) = ( V~ , ~ V ~

)= ~

(

= V~ then o.s.(~, I SC(~))

~Iv~).11

This theorem gives a sufficient condition for the existence of the optimal solution, and can be used to obtain an effective construction for it. This problem is investigated in what follows. Consider ~

= (Q, ~) and ~ ~ DF(Q); on Q we define the following relation Z'~: =

=

' where, if a,b@Q, i) (a,b)~Z'o --iff a @ F ~

(~ nT/O

Z'n

b@F~Fo

and b & F ~

a @F kTF ~ F ~ " ii) (a,b) ~ Z' iff (a,b)~Z' and ~ O ~ , if O~ @ ~ then, o ' m÷l - m ~al,... , ap ~ Q, the elements ~(al,... , aj,a, aj+l,...,ap) an_~d~(al,...,aj, b, aj+l,... , ap) are Z'm - related, j = O,...,p. Lemma 3.8.

then z ~ = v ~ Proof.

i) Z ~ V ~

; ii) ~_~ S C ( ~ , ~ ) ~ & Z ~

; iii) if Z~is transitive,

v~

We use a modified, but equivalent, definition of V %

, considering F~ ([x}%/Q)

214

instead of F~([x~ ) and set [[~i~ : = ] ~ . i). Obviously (a,b) 6V'~ ~ then ~ 0 ~

e ~ptl

' ~

(a,b) ~ Z'o ; if ( a , b ) ~ V ~ al'~''a p ~ Q

==)

(a,b)~ Z'k ' ~ k ~

consider, for j = O,®..,p and u = a , b

m, :

~j : =

~ ( a l , ~ . o ~ a j ~u, aj+ l..... a ) and ~.:----a o..a x a + . o . a ~ @ ~ x ~ L f Q P 3 I J j I p Then, also recalling P r o p . 3 . 3 , w e have , (a,b) @ V ~ (%%ZE~ (a) , ~ 3 ~ ( b )

)o =

^

= (~.,B)&j J V~

and thus by induction hypothesis

(~3,bj). ~ ZT'm This shows that

(a,b) ~- V~ ----~ (a,b)e Z'

m+l

ii) is quite obvious as Z %

is an ~-separating relation.

iii). We show that if Z ~ i s transitive then it is an ~ -separating congruence, so Z ~ ~- V ~

;hence, by i), Z ~ = V~

.If Z~ is transitive, clearly it is an ~-sepa-

rating equivalence; we have only to prove it is a congruence. First of all we observe that : ( a , b ) ( - Z ~ @ ~ i

, ~-p~O

,~a

I.... , a p e Q, ( ^aj,hj)~ Z ~

, j =O ..... p.

Then, if u~ e ~

and ( u . , v ) % Z ~ , j=l,.o.,p and ~ : = ~-'(mod Z'~ ), ~(Ul,..,Up) P 3 J ,~. OC (Vl,U 2 ..... Up) o.. :-" 4~(vl,..O,Vp).//

Remark. It is also quite clear that : a) b)

if Q is finite then

if Z'

m exists, m ~--IQ~ 2

= Z' then Z' = Z' ,~ m,p~O ; m m+l m m~ , s.t. Z ~ -- Z' • (The transitivity is r~

not involved here.) Theorem 3.9.

If a finite automaton M = ( ~ , ~ )

is given s.t. ~.(M) ----(I~, I~),

there is an algorithm for deciding whetaer o.s. ( ~ , I s c ( ~ ) )

exists and, if so, for

finding it. Proof.

By the above remark Z %

then by lermna 3.8 and M/V~

ZY~ = V ~

= V~

can be actually eoraputed. If ZT~ , o.s. ( ~ , ! S C ( ~ ) )

is transitive,

= (V~ , ~ - V ~ ) -- ~ ( M / V ~ )

is obtained by an effective construction. Otherwise, we can by inspection

on all the congruences contained in Z'~ ther S C ( ~ , ~ )

(and they are a finite number) verify whe-

has a maximum or not and so decide whether o.s ( ~ , I s c ( ~ ) )

or not ; if it exists, by Prop.3.1., o.s. (~, ISG(~)) ----~ (

M/~

exists

), where ~ : =

= max SC(~,~)./I

Theorem 3.10.

If a finite automaton M = ( ~ , ~ )

is given , s.t. ~ ( M )

G EDLJ(~,

ISC(~4)) and Z'~. is transitive, then o.s. (~,Isc(~4)) exists and there is an algorithm for finding it. Proof.

It follows directly from Prop.3.7ar~d lemma 3.8//

Theorem 3oil. ~=

~

Let @ : = ~ (H) ~ C(~Z), H finite set, and a regular family ~

(i.e.

, for a finite ~-automaton ~) be given. There is an algorithm for deciding

whether o.s. (O~, ~ SC(~)) exists and if it exists, for finding it.

215

Proof. Recalling from section 2 the definition of ~ ( H ) we can see that if D is the maximal depth of the terms in H, then ind ( ~ ) < -congruent to some term of depth ~ D to construct a finite~ -automaton ~ ~----(~,~))~ b e i n g _ ~x

~

q-~iff

. If ind ( ~ )

each term od depth D+I is 4-. q- ~

it is not difficult

s.t. E R ( ~ ) ----~ • If ~---- ( ~ ,

the automaton for ~

and ~ : = ~ i , i ~I~

~)

let M :=

and (~x~,

) and M c be the connected subautomaton of M (whose construction is effective).

Mc = .here ,}c= [ Fi c/.c i: = - ix ~ ' Ai:---- rp'l(F*)1 . ~is -separating i f f

i

,, ~Fi,= I .Q~. / rpi l ( q ) a , i,, and DF(~)~ e,----~ i , i 6 I J ; M~. is

Ai#

cJ.

fi-

M

nite, then w~ can decide whether its elements are non empty and mutually disjoint. If it is so, define

M : = (~,~)

" then ,

~M

= ~ @

and

ER( -" M ) ----~. Hence by

Theorem 3.9 we have proved our assertions. //

to CONCLUSIONS AND EXAMPLES. The above results show not only that, as it was be expecte~ the existence and the nature of the optimal solution for a recognition problem depend on the devices we use to recognize, bout also that when the o.s. does not exist every maximal element in the set of good solutions is an o.s. in respect to a restricted class of recognizers. Thus supplementary "goodness" criteria are needed to define

a

unique o.s.. Moreover we have shown the essential role of the images ; for instance, if we are able to recognize the images, then for a regular family of examples ~ know all about the optimal solution. Finally if u ~ Z ~ = V~ = N$ and the algorithnfor computing Z ~

we

is a partition, the V ~ = N~.

is exactly the one used by Brainerd

(1968) for minimizing tree automata. Example I.

Let ~

where

qo;

q2 ~

~=

= 5"o L2 ~ i = I~I l)la,b~ ; Q = lqo,ql,q2, q31 and

~a:qo ~-9 ql,q I ~

q2' q2 ~

q3' q3 ~

q3 ;

~

=(Q,0(),

~ b: qo e--)q3 'ql ~--2q3'

ql' q3 ~ q3 " is therefore a monadic-algebra ; it can be seen that in a monadic algebra the

Z' and V' relations coincide.

If

~ = llql~ , ~q~l

then we have V ~ = Z ~ = Z' '

1

and the "relation classes" are lql,q3 ~ , lq2,q3, qo~ ; hence V ~ * by inspection that the only

~

-separating congruence on ~

gruence ~

.Therefore : max SC( ~ , ~ ) = ~

Example 2.

Let

~

and ~bviously

be as above; then the terms in

F~-

elements of ~'2~ • So consider the congruence ~ on ~ so=[Z%~ , Sl= ~a~ , s2= s7= ~--"- ~ s .

a2b(ab) *, s3= a(ab)*a

V~ . One can verify

is the diagonal con ~ = N~ # V~ .

can be viewed simply as whose classes are :

, s4= ab(ab)*a, s5= ab(ab)*,s6=b(ab)~ $

216

Set

~ = I SI~

s2

'

s31

It is easy to verify that on

~ then ~/~

~:=

isobviously

/.~(M) ~ E D L ( ~ , ~ S C ( ~ ) )

, [a2bT®~ ,L[a2j~l. are :

; we also have : l.u.b. S C ( ~ _ / ~ , ~

congruence whose classes are ~so~ and

different from V ~ .

max S C ( ~ / ~ , ~ )

=[~[a]~

• the "relation classes" of V ~

~sl,s2, s4, s5,s6,sT$ , {s3, so,s4~s5,s6,s7} is the (non~-sep.)

~/~

Moreover, setting

Isl .... , s7~

M = (~,~)

)

) and which

- see ex.l- ,

; then , as we have seen, max, S C ( ~ ,q ) exists, while

does not exist, which implies that also o.s. (~,~)SC(~))

does

not exist. Example 3. of

F~ °

Let Z = ~ o

~/ 5" ! -

i~

V l a , b,c~ ; then we can consider ~ ~ instead

Set : D: = ta~AJ ~b~ U cc*a LTcc*b, A1 : = c*bD*, Al:=C*bD*cc * ,A2:=c*aD* ,

A2 := c*aD*cc*, A ~ =

c* , ~ =

AI'A2

, I~

A 3 ' A I V AI' A 2 V A2' so that o . s . ( ~ ,

= A

. Then V a

= ~

, with classes

SC(~)) = (V~, ~ A I V A 1 , A 2 V A 2 1 ).

RE FE RE NC E S ARBIB,M.A° and GiVE'ON, Y.(1968) "Algebra automata I : parallel programming as a prolegomena to the categorical approach" Inform. and Control 12,331-345. ASTESIANO, E. (19"/3) "A recognition model", to appear in Pubblicazioni dell'Istituto di Matematica dell'Universit~ di Genova. BRAINERD,W.S.

(1968) "The minimalization of tree automata", Inform.and Control 13,

484-491. B~CHI,J.R. (1966) "Algebraic Theory of feedback in discrete systems- Part I " in Automata theory ,edited by E.R. Caianello ,Academic Press. COlIN,P.M. (1965) "Universal ~Igebra~,, ~arp~,New York. GP&TZER, G. (,1968) ~Universal algebra" ~Von Nostrand,New York. GRENANDER,U.(1969)

UFoundations of patterns analysis" ,Quart.Appl.Math. 27, 1-55

PAVEL,M. (1969) "Fondements math~natiques de la reconnaissance des structures", Hermann, Paris THATCHER, J.W. and WRIGHT, J.B.(1968) nGeneralized finite automata theory with

an

application to a decision problem of second order logic",Math. Systems Theory,Vol.2. N. 1,57-81.

217

VERBEEK,

L.A.M. (1967) " Oongruence separation of subsets of a monoid with

application to automata " , Math. Systems Theory, VoI.I, N.4, 315-324

A NEW FEATURE SELECTION PROCEDURE FOR PA~EIhN RECOGNITION BASED ON SUPERVISED LEARNING by Josef K i t t l e r Control and Systems Group, D e p a r t m e n t of E n g i n e e r i n g U n i v e r s i t y o f Cambridge, E n g l a n d

I.

Introduction Linear m e t h o d s of feature s e l e c t i o n are c h a r a c t e r i s e d b y a linear t r a n s f o r m a t i o n

or 'mapping' o f a p a t t e r n v e c t o r

x

from an

N

dimensional space

X

into a

n < N

m

d i m e n s i o n a l space transformation if successful,

Y .

The feature v e c t o r ~ e Y w h i c h is o b t a i n e d from ~ b y T T ~ i.e. ~ = xJT , has a r e d u c e d n u m b e r of components and should,

contain all of the i n f o r m a t i o n n e c e s s a r y for d i s c r i m i n a t i n g b e t w e e n

classes p r e s e n t in the o r i g i n a l v e c t o r

x .

Many methods h a v e been s u g g e s t e d for d e t e r m i n i n g the t r a n s f o r m a t i o n m a t r i x r e q u i r e d for linear feature selection.

T

But most of these can be c l a s s i f i e d in one

of the f o l l o w i n g t w o categories: a)

m e t h o d s b a s e d on the K a r h u n e n - L o e v e expansion

b)

m e t h o d s u s i n g d i s c r i m i n a n t analysis techniques.

In the first p a r t o f the p r e s e n t p a p e r a n e w m e t h o d of feature s e l e c t i o n b a s e d on the K a r h u ~ e n - L o e v e

(K-L) e x p a n s i o n is proposed.

S u b s e q u e n t l y a relationship

b e t w e e n the s u p e r f i c i a l l y d i f f e r e n t K - L e x p a n s i o n and d i s c r i m i n a n t analysis approaches is e s t a b l i s h e d and in so doing a more u n i f i e d a p p r o a c h to the p r o b l e m of feature s e l e c t i o n is introduced. 2.

A M e t h o d of Feature Selection for Pattern R e c o g n i t i o n B a s e d on S u p e r v i s e d L e a r n i n ~ The m e t h o d of feature selection d i s c u s s e d in this p a p e r is b a s e d on the proper-

ties of the K-L expansion.

Since the d e t a i l e d t r e a t m e n t of the K a r h u n e n - L o e v e ex-

p a n s i o n of discrete and continuous p r o c e s s e s Fukunaga, K i t t l e r and Young) here.

can be f o u n d elsewhere

(Mendel and Fu,

only a b r i e f d e s c r i p t i o n of the m e t h o d will be given

A l s o for simplicity, we s h a l l confine our d i s c u s s i o n to the case of discrete

data. C o n s i d e r a sample of r a n d o m is a s s o c i a t e d w i t h one of and denote the noise on

~i = ~

m

N

d i m e n s i o n a l p a t t e r n vectors

p o s s i b l e classes

--Ix by

~

~i "

~ .

Let the mean of

Each vector -xs~ i

be

~i

, i.e.

+ zi

ll)

W i t h o u t loss of g e n e r a l i t y we can assume that the overall mean

! = E{~} = 0

since

it is c l e a r l y p o s s i b l e to c e n t r a l i z e the d a t a p r i o r to analysis b y removal of the overall mean.

Suppose that the p r o b a b i l i t y o f o c c u r e n c e of

i-th

class is

P(~i )

and let m e m b e r s h i p of p a t t e r n s in their c o r r e s p o n d i n g classes be known. Suppose that we w o u l d like to e x p a n d the vector d e t e r m i n i s t i c functions

~

~

and a s s o c i a t e d coefficients

l i n e a r l y in terms of some Yk

' i.e.

2!9

N

(2) k=l subject to the conditions: ~)

~

are o r t h o n o r m a l

~)

Yk

are u n c o r r e l a t e d

y)

the r e p r e s e n t a t i o n e r r o r m

:

[ i=l

c i) {Lh

(3)

2} ^

incurred by approximating

x

with

x

c o m p o s e d of

n < N

terms in the expansion

(2),

i.e. n k=l is minimised. F u and Chien have shown that the d e t e r m i n i s t i c functions satisfying the p r o p e r t y through

B

are the e i g e n v e c t o r s T

T

. . .

of the sample covariance m a t r i x

C

d e f i n e d as m

c

=

~ {xx T}

=

---In order to satisfy condition ~l,...,tN

~ i=l

P(~)

E{xx

l

T}

--l--l

~ , it is necessary to arrange the eigenvectors

in the d e s c e n d i n g order of their associated eigenvalues, l I ~ 12 ~ ... I n ~ ... ~ 1 N

Chien and F u a l s o s h o w e d that the eigenvalues features

Yk

Ik

are the variances o f the t r a n s f o r m e d

and that the expansion has some additional favourable properties, in

p a r t i c u l a r , the total e n t r o p y and residual entropy a s s o c i a t e d w i t h the t r a n s f o r m e d features are minimised. The t r a n s f o r m a t i o n

T

of the pattern vectors into the K-L coordinate s y s t e m

results in c o m p r e s s i o n o f the i n f o r m a t i o n contained in the o r i g i n a l N - d i m e n s i o n a l p a t t e r n vectors

~

into

n < N

terms of the K - L expansion.

This latter p r o p e r t y

has b e e n utilised for feature selection in p a t t e r n recognition b y various authors and a few of the p o s s i b i l i t i e s are listed below, i)

If the features

Yk

are to be u n c o r r e l a t e d then

T

should he chosen as

the m a t r i x of e i g e n v e c t o r s a s s o c i a t e d with the mixture covariance m a t r i x as

(Watanabe)

cI

=

E {x_ _xT}

C1

defined

220

ii)

If ~he transformation

is required to decorrelate

noise vector

z then the m a t r i x of eigenvectors T -% averaged within class covariance m a t r i x C 2 F i.e.

the components

should correspond

of the

to the

m

c2

:

[

P(~i ) ~

{hzi T}

i=l It should be noted that method ii) selects ory information the mean vectors

m

=

Ik

m

Yk

m

2 + ~ P(~i ) Yki i=l

of the discriminatof information

about

This can be seen from

of the new features

2 = ~ P(~i ) Oki i=l

~ P(~i ) E(Yk ) i=l

where

irrespective

in method i) is not optimal in any sense.

a detailed analysis of variances

Ik

features

contained in the class means and the utilization

' i.e. "2 0

=

+

(4)

~k

2 ~ki

=

E { (Yki - Yki )2}

(5)

Yki

=

E {Yki }

(6)

and

In an earlier paper, discriminant provided

Kittler t J. and Young,

P.C.

that the averaged within class covariance

The magnitude information

(1973) have shown that the

power of a feature against class means is related to the ratio

of the first term in

at all.

It is therefore

matrix is in the diagonal

(4), however,

desirable

~ k / [2 form.

contains no discriminatory

to normalise

the averaged within

class

variances

to unity and thus to allow selection of the features on the basis of the

magnitude

of the eigenvalues

kk .

It is this norraalising transformation

the essence of the proposed new feature iii)

In order to normalise

within class covariance

matrix

selection

the noise, we first have to diagonalise

the averaged

C2~

m [ P(~i ) E {z_~ --~zT}

~2

which is

technique.

(7)

i=l b y transforming

x

a s s o c i a t e d with

C 2. y

T

into a new feature vector

:

will have a diagonal

Thus the feature vector x

T

(81

eovariance

matrix

T

~ X P(~-) E {u z { h y

i= !

using the system of eigenvectors

y__ where

U

m

c

y

l

--

T

u}

:

ili I

12

O i

1

, I

[

. 0

" 1

N]

{9)

221

!

where

lk s

are eigenvalues associated with

C2°

The matrix C can now be transformed into identity matrix by multiplying Y each component of the feature vector ~ by the inverse of its standard deviation. In

matrix form this operation can be written as T

T X S

=

(i0)

where

s =

".

(11)

Once the averaged within class covariance matrix is in the identity form, by solving the eigenvalue problem, C B = BA g g we can obtain a new K-L coordinate system, mixture covariance matrix

Cg

C

E {g --

=

g

(12) B , which is optimal with respect to the

, where

T} g

(13)

_

A

is the matrix of eigenvalues of the matrix C g g Note that the class means are included in this case. in which

the

Thus

B

is a coordinate system

square of the projections of the class means onto the first coordinates

averaged over all classes is maximised. The eigenvalues

lgk

which are the diagonal elements of

Ag

can now be

expressed as

kgk

=

1

+

m ~ P(~i ) ( E { ~ g i}) 2 i=l

=

1

+

"%k

(14)

It follows that the features selected according to the magnitude of

kg k

will now

be ordered with respect to their discriminatory power. The feature selection procedure can be summarised in the following steps: l}

Using K-L expansion diagonalise the mixture covariance matrix of the original N-dimensional data

x .

C = E {x ~T}

Disregard those features which are

associated with negligable eigenvalues and generate a feature vector T = xTw of dimension N < N where W is the system of eigenvectors of C.

x 2)

Find

the K-L coordinate system

covariance matrix 3)

C

Y Normalise the features

U

in which the averaged within class

defined in (9) is diagonal. Yk

to transform the matrix

C

into identity Y

matrix form. 4)

Determine the final K-L coordinate system mixture covariance matrix

B

which is associated with the

C

g If we ignore the first K-L expansion, which does not affect feature selection but only reduces the computational burden of the following two K-L analyses, we can

222

view the overall linear t r a n s f o r m a t i o n ture c o v a r i a n c e m a t r i x TTc2 T

C1 =

=

as one that decorrelates the mix-

subject to the condition that

I

(15)

The resulting feature vector fT

T = USB

f

is then given as

xTT

(16)

m

Note

that the p r o p o s e d feature selection technique is applicable to s u p e r v i s e d

p a t t e r n r e c o g n i t i o n p r o b l e m s only b e c a u s e the m e m b e r s h i p of the t r a i n i n g p a t t e r n s in the i n d i v i d u a l classes m u s t b e k n o w n a priori so that the a v e r a g e d w i t h i n class covariance matrix

C 2 can be computed.

In o r d e r to show the r e l a t i o n s h i p b e t w e e n the K-L e x p a n s i o n techniques i), ii), iii)~

and the d i s c r i m i n a n t analysis techniques w h i c h we describe in the next section

it is n e c e s s a r y to derive the results o u t l i n e d above in an a l t e r n a t i v e manner.

This

r e q u i r e s that the p r o b l e m is v i e w e d as one of s i m u l t a n e o u s d i a g o n a l i z a t i o n of the matrices

C2

and

C1 °

suTc2us Now b y u t i l i s i n g

F r o m the p r e v i o u s discussion, we k n o w that

=

I

(17)

(17), the eigenvalue p r o b l e m

(12) can be w r i t t e n as

suT(cI - I g C 2 ) U S ~ = 0

B u t since

SU T

is nonsingular,

ICl-XgC2i

=

the c o n d t i o n

(18)

(18) will be s a t i s f i e d only if

o

(19)

-I It follows that

l

g

are the e i g e n v a l u e s of the m a t r i x

(CIC2 -I ~ I g I) -t =

ClC 2

~ i.e.

0

(20)

E x p e r i m e n t a l Results

2.1

The results of an e x p e r i m e n t a l c o m p a r i s o n of the three K-L p r o c e d u r e s o u t l i n e d above is given in Fig. I.

Data for the e x p e r i m e n t were g e n e r a t e d d i g i t a l l y accord-

ing to the rule,

where

(class A) :

x I ~ N ( 2 , 2 ) , x 2 . . . x 9 ~ N(O,I) ,xlO ~ N(O,O.25)

(class B) :

x I ~ N(1.95,2) ,x2...x 9 ~ N ( O w l ) , X l o ~ N(O.5,0.25)

N(Uto)

defines a normal d i s t r i b u t i o n w i t h s t a n d a r d d e v i a t i o n

F r o m these results, it will appear that the m e t h o d

~

and mean

~ .

iii) p e r f o r m s s u b s t a n t i a l l y

b e t t e r than the o t h e r two K-L procedures. 3.

D i s c r i m i n a n t Analysis Let us now formulate the p r o b l e m of feature s e l e c t i o n in a d i f f e r e n t way.

Suppose that we w i s h to f i n d a linear t r a n s f o r m a t i o n matrix

T

w h i c h m a x i m i s e s some

223

distant criterion space. a)

d

defined over the sample of random vectors in the transformed

Two of the most important distance measures

The intraset distance

dln

are as follows:

between the n-th feature of all the pattern vectors

in one class averaged over all classes, which is defined by m

N.l

Ni

dln = ½i=l [ P<~')~~.~2 jZ I1 ~t~ (~i~ - ~il)(~i~ -

~il)T t

(21)

l where

N. is the number of vectors l transformation matrix T .

b)

The interset distance

d2n

x E ~. -l

and

t --n

is the n-th column of the

between the n-th feature of all patterns, which is

defined by m

i=l

i=2

m

Ni Nh

h=l

i h j

It has been shown, Kittler, J.

1

(1973), that these distance critera can be ex-

pressed in terms of sample covariance matrices. and

d2n

dln

become =

dln with

In particular the distances

C2

tT C2t_n -n

(23)

given by (4) and d2n

=

t~T~ t_n

where m

= cl -

[ P2(~ i) E {zihT}

(24)

i=l By analogy, the sum of the interset and the intraset distances m d3n = ½

m

N~

Nh

~ P(~i ) ~ P ( m h ) N 7 h i=l h=l

~ j=l

~ i=i

d3n , is

tT(x_ij - ~hl ) ( ~ j --n

- ~l)Ttn

(25)

can be written as =

d3n

t T (C2 + M) t_n --n

=

<

Cltn

(26)

where m

M =

[ P<%) AA

T

i=l Clearly a number of different distance criteria could be constructed;

but, in all

cases

it would be possible to express the distance in terms of a covariance-type

matrix

C , i.e. d

n

=

t T Ct -n -n

(27)

224

U s i n g these r e s u l t s carried

out b y m a x i m i s i n g

some a d d i t i o n a l

=

subject

to

constant

criterion

vector

some d i s t a n c e

s

d

can be

t , subject

purposes,

The s o l u t i o n

multipliers.

s = const

to

where (28)

can b e w r i t t e n

the first d e r i v a t i v e s

s

might,

for example,

for this k i n d of p r o b l e m

In case o f a s i m p l e

f = d-l(s - const)

Setting

distance

to the t r a n s f o r m a t i o n

e.g. h o l d i n g

for c l a s s i f i c a t i o n

distancep

method of Lagrange

o f a chosen

tTc t s-n

is i r r e l e v a n t

as the i n t r a s e t

with respect

constraintsr s

which

the m a x i m i s a t i o n

constraint,

as the m a x i m i s a t i o n

of

be defined

can be o b t a i n e d b y maximisation

f

with respect

d

f , where

= tTct - I(tTc t - const) -%q --~ --n s--n

of

of

(29)

to the c o m p o n e n t s

of

t --n

equal

zero y i e l d s (C - ~C )t = O s -n B u t if

(30)

(30)

is then p o s t m u l t i p l i e d

b y the inverse

of

C

s

, we get an e i g e n v a l u e

problem (CC -l s W h e n there since the

f

special

~I)t --n

is m o r e

function

the s o l u t i o n

-

than

=

a single

(31)

constraint,

is m o r e c o m p l i c a t e d

(32)

m u s t be o b t a i n e d u s i n g g e n e r a l

TTT

the s o l u t i o n

now becomes

d - ~ ~ I ( S I - const) i

=

optimisation

are only two c o n s t r a i n t s

of o r t h o n o r m a l i t y

the p r o b l e m

0

under consideration

case w h e n there

condition

=

of the m a t r i x

T

techniques.

However,

and one of these is simply

, ioe.

I

(33)

of o p t i m i s a t i o n (C - ~C

in the

the

can be p o s e d as the e i g e n v a l u e

- ~I)T

=

problem

0

(34)

s A n d since in the d e s i g n of p a t t e r n classification mum error 3.1

performance~

ficially

w e can d e t e r m i n e

systems ~

o u r chief i n t e r e s t

experimentally

is an

to achieve

the m i n i -

rate.

Experimental This

recognition

Results

latter approach generated

data.

for two c o n s t r a i n t s The classes

were generated

(class A) :

Xl, ..... ,xlO ~

(O,i)

(class B) :

Xl, ..... ,x 8

(O,l),x 9 ~

~

has b e e n

t e s t e d on four class,

according

(4.2,1),xi0

~

to the rule,

(-4.2,1)

arti-

225

(class C):

Xl, ..... ,x 8 %

(O,l),x 9 ~

(2.2,1.3),xi0 ~

(2.2,1.3)

(class D) :

Xl, ..... ,x 8 ~

(O,1),x 9 %

(2.2,1.3),xlO %

(-2.2,1.3)

The summary of the results obtained using classifier with linear discriminant function is in Fig. 4.

2.

The best error rate corresponds

to

~ = 0.7 .

Discussion There are many possible ways of defining the distance b e t w e e n the elements of a

sample and there are even more combinations that could be maximised in any particular

of constraints

and distance

feature selection problem.

criteria

Consequently

we shall restrict our discussion here to a few specific methods that have been suggested in the past. First, distance

let us consider the method p r o p o s e d by Sebestyen.

criterion

to be maximised is

of the intraset distances is the solution

remains

of the equation

d3n

In this procedure,

the

, subject to the condition that the sum

constant.

In this case the transformation

matrix T'

(31), i.e.

!

(CI C'l-z - II) t --n Comparing

the relationship

=

O

(35) with

(35)

t' --n system obtained by the method iii) described in

colinear with the coordinate

(20) we see that the column vectors

are

section 2, i.e. t'

=

~n t

(36)

But in contrast to the method iii), where

TTc2 T

=

T

was chosen to satisfy

I

(37)

the columns of the transformation

matrix

T'

in the present case must be such that

N

TC 2 t'

=

const

=

K

(38)

n=l And from

(36) it follows that N

2 {n --ntTC~tz--n =

K

(39)

n=l Thus, we can choose the there any particular From

N-I

coefficients

~n

and evaluate

choice of the coefficients

~n

the last one.

But is

which w o u l d give better features?

(35) it follows that t~ T Clt ~

- Int'Tc2t~

2 ~n

+

2 ~n IM

2 - In~ n

=

O

(40)

n where 1

and

T'TMT '

M

= n

t 'T Mr' --n --n

is diagonal matrix.

(41)

Using

(40) and

(36) the m a x i m i s e d distance

d3n

226

can be w r i t t e n d3n

However,

2 In6 n

=

(42)

from ~0)~ the e i g e n v a l u e In

=

1

can be expressed in terms of

n

1M

as n

1 + lM

(43) n

Thus we can conclude that although the distance d i s c r i m i n a t o r y p o w e r i n h e r e n t in of

~n "

Thus

~n 2 ~n

and

T~

d3n

3n

2 ~ n ' the

is p r o p o r t i o n a l to

is a function of

1

n

and therefore i n d e p e n d e n t

can be chosen =

2 ~i

=

const T

=

const

n,i.

(44)

becomes T~

(45)

A p a r t from some c o n s t a n t of p r o p o r t i o n a l i t y the features o b t a i n e d in this m a n n e r will be e x a c t l y the same as those o b t a i n e d b y the m e t h o d iii) the o r d e r i n g criterion

(13).

and will satisfy

It is i n t e r e s t i n g to note that the d i s t a n c e

o n l y p r o p o r t i o n a l to the d i s c r i m i n a t o r y p o w e r of the n-th feature. as an o r d e r i n g criterion, therefore,

If

d3n

d3n

is

is used

any i l l - c h o s e n coefficients c o u l d r e s u l t in a

s u b o p t i m a l o r d e r i n g of the features. If the i n t e r s e t d i s t a n c e straints,

~ .

It can be shown that the m a t r i x

(31) d i a g o n a l i z e s b o t h m a t r i c e s T T ~ T - AA O

where

i0

is m a x i m i s e d i n s t e a d of

d3n , w i t h the same con-

then the s i t u a t i o n is rather more c o m p l i c a t e d since the m a t r i x

is now r e p l a c e d b y of

d2n

TTc2 T

and

=

A

~

and

C2

T

O

(46)

is the m a t r i x of e i g e n v a l u e s of

m [

in (31)

and this m e a n s that

from (24) the f i r s t t e r m on the left h a n d side o f

AO + T T (M -

C

o b t a i n e d as the solution

p2 (t0i)E{~zl})T T - AA 0

~C21 .

S u b s t i t u t i n g for

(46) can b e r e w r i t t e n to y i e l d

=

O

(48)

i=l The term in the middle o f the left h a n d side of the e q u a t i o n must also b e d i a g o n a l and it is, in fact, this t e r m w h i c h determines the optimal coordinate s y s t e m T . m D e p e n d i n g on the relative d o m i n a n c e o f M o r ~ p2(~i)E{z_izT }__~ the axes T may i=l coincide w i t h the coordinate s y s t e m in the p r e v i o u s case o r lie in the d i r e c t i o n d e t e r m i n e d b y the t e r m

m

P 2 ( w i ) E { z zT} --l--i °

However, in general,

i=l c o m p r o m i s e b e t w e e n these two. Let us denote the s e c o n d t e r m of

(48) b y

AM s i.e.

T

w i l l be a

227

AM

=

m [ P2(w i) E {ziziT})T i=l

TT(M -

Then by analogy to (43) the elements expressed

i

n

(49)

of the matrix of eigenvalues

A

can be

as 1

=

n

1 + ~S

(50) n

Now even if we select the features instead of the distance the magnitude of

1M

d2n

according

to the magnitude

of the eigenvalues

the ordering may not necessarily be satisfactory

is proportional

n-th feature b u t als n to noise,

not only to the discriminatory

represented b y the second term of ~ M

In,

since

power of the , i.e.

n

~M

t T Mt - t T ( ~ p2 (~.) E{z.z.T}) t -~n --n --n l --l--l --n i=l

= n

Thus the method m i g h t in certain circumstances tained b y maximising Finally,

the criterion

yield inferior

features to those ob-

d3n .

a few remarks are necessary in connection with the special

feature selection with two constraints experimental

(51)

discussed in the previous

results obtained where the distance

d2n

section.

case of From the

was maximised subject to the

condition

N

tnC2tn

=

const t

(52)

n=l we can conclude that the optimal coordinate vectors of the matrix

M

(since

the within class scatter matrix

axes

T

almost coincide

with the eigen-

~

is such that the constraint matrix cancels out m C2 - g? p2([~i)E{ziT})_ . This fact is even more i=l

obvious from the experimental

results of Nieman, who originally

suggested

He assumed that the a priori

class probabilities

of ten numerals were equal.

The m i n i m u m error rate was then obtained for

Now if

P(m i) = 0.i, Vi,

when

~

=

0.9 .

=

(M - II)T

=

~ = 0.9

.

(34) becomes

O

(53)

This result is only to be expected since we cannot possibly de-

correlate both within and between transformation.

associated with the classification

then the eigenvalue p r o b l e m defined b y

(C 2 - C2/IO + M - ~C 2 - II)T

this approach.

class scatter matrices by a single orthonormal

And it is quite reasonable

therefore,

that the most important

features will be selected with some degree of confidence only in the coordinate system where their means are decorrelated.

Thus, in practice,

we can only hope that

our choice will not be affected by the projected noise. These same remarks case when

v = 0

apply when the distance

the p r o b l e m reduces

d3n

is maximised.

to the K-L method i).

However,

But in this f r o m the above

228

discussion we see that better results can be obrained for

~ + + 1 o

Consequently

Nieman~s method will yield superior features. 5.

Conclusion The most important result of the comparative study described in this paper is

the correspondence and, in particular cases, the direct equivalence that has been established between some statistical feature selection techniques developed from the Karhunen-Loeve expansion and of a distance criterion.

some

alternative techniques obtained by maximisation

This allows us to extend the properties of features obtained

by linear transformation derived from the K-L expansion to the distance optimisation methods,

and vice versa.

Thus we know, for instance,

that the features obtained by

Sebestyen's method will be not only maximally separated but also uncorrelated. In a previous paper we have shown analytically and confirmed experimentally that the K-L procedure iii) is particularly favourable for feature selection applications.

Some additional experimental results supporting this conclusion are pre-

sented in section 2 of the present paper.

Moreover the comparative study described

here reveals that this procedure retains its advantages for an even larger class of linear transformation techniques which include methods based on separability measures. The coordinate system used by the K-L method iii) can be obtained by a successive application of the K-L expansion,

as described in section 2.

Alternatively it can

be obtained from the system of eigenvectors associated with the product of two matrices, one of them being the within class covariance matrix as described in section 3.

Howeverw the designer may have some difficulties with the latter approach,

particularly if the matrix being inverted is not well defined.

Beth for this reason,

and also in order to have a greater control of the analysed data, it seems better to use the first method.

Although this implies two eigenvalue analyses, the matrices

involved are symmetric and the problem is computationally quite simple. References l.

Chien, Y.T.~ Fur K.S.~

2.

Fukunaga, K.:

Kittler, J.:

Inf. Theory,

IT-15, 518 (1967)

Introduction to statistical pattern recognition, The Macmillan

Company, New York, 3.

IEEE Trans.

(1972)

On Linear Feature Selection.

University Eng. Dept., CUED/B-Control/TR54

Technical Report of Cambridge (1973).

4.

Kittler, Jo; Young~ P.C.:

A new approach to feature selection based on the

5.

Mendel, J.M., FUr K.S.:

Adaptive, Learning and Pattern Recognition Systems,

Academic Press, New York

(1970).

Karhunen-Loeve expansion t Jnl. Pattern Recognition

6.

Niemann, H.:

(to be published,

An Improved Series Expansion for Pattern Recognition,

Nachrichkentechn,

Z, pp. 473-477,

(1971).

1973).

229

7.

Sebestyen,

G.S.:

Decision Making Processes

Macmillan

Company,

8.

Watanabe,

S.:

N e w York

Computers

in P a t t e r n

Recognition,

(1962).

and I n f o r m a t i o n

Science

If, A c a d e m i c

Press,

(1967). Fig.

1

70

----........ ~._

60

Unordered Method i M e t h o d ii M e t h o d iii

5O

4O

30

• . . , , , : .-~...: . . , - , , v"~'~.:,."rr~... "...-e-'~.,.,':~ 20

10

.....

O

I

1

!

|

1

2

4

6

8

I0

N u m b e r of Features

Fig.

20

o o

a

2

a ------ b

~ = -.5 ~ =-.i

~,\

........

~:

~), \

-.-

~

.7 :

1.o

15

o q4

~4 o

IO

O

|

2

the

,

.

I

4

I

I

I

6

8

iO

N u m b e r of F e a t u r e s

New York

A CLASSIFICATION

PROBLEM

IN M E D I C A L

Georg IBM H e i d e l b e r g

RADIOSCINTIGRAPHY

Walch Scientific

Center

I. I N T R O D U C T I O N

In this

paper

developed

we present

in the

a classificatlon

framework

to do the c l a s s i f i c a t i o n normal

pattern

procedure

of h y p o t h e s e s

testing

of s c i n t i g r a p h i c

and that

of pattern

with

which we have

and d e c i s i o n

images

into the

anomalies

theory

class

of

in an i n t e r a c t i v e

way.

In nuclear in human

medicine

body

gamma-camera

the d i s t r i b u t i o n

is m e a s u r e d to receive

the base

of d i f f e r e n t

roundi n g

healthy

image

in the d e t e c t i o n

These

storage

of emitted

scintigrams

are

low spatial

resolution.

produced

damage

ters

the

of Wiener

Both

of the d i f f e r e n c e

in the

sense

mise

between

of least noise

lesions

and the

is called

and on

sur-

scintigraphic are

reduce

to improve

of applied

between

squares,

involved

suppression

stage. the

result

applied

With regard

Therefore,

and

to radiratio

digital

fil-

to m i n i m i z e

the expec-

and o b j e c t

distribution

to give

and r e s o l u t i o n

both

of small

signal-to-noise

the goal

filter

noise, nature,

the d e t e c t a b i l i t y

activity.

with

were

statistical

of statistical

in an early

constructed

tation

scanner

and m e t a s t a s e s

in crystals

by high

being

facts

by tumours

amount

data

compounds

y-quanta.

and y - d e t e c t i o n

type,

like m o v i n g

in those

accumulated

radioactive

tumours

scintillations

it is not allowed

by increasing

devices about

effects

characterized

decay

ation

The

because

radioactive

anomalies

by imaging

information

tissue~

or s c i n t i g r a m

of applied

an optimal

enhancement

compro-

(Pistor et

al). They may be c o n s i d e r e d optimal

estimation

for the i n v e r s i o n process.

as two

stage operators,

of the u n d i s t u r b e d of the

linear

signal

the

first part

response,

transformation

involved

the

for the

second

part

in the imaging

231

In the filtered noise

ratio,

improved.

and thereby

whether

nature,

classification

RATIO

fluctuation

anomalies

and a b n o r m a l

ratio

are pattern,

is of s t a t i s t i c a l

inspector.

or at least to base

a likelihood

signal-to-

or bio-

To do this

the human d e c i s i o n

test

is adapted

to this

TEST

In our case

the null

hypothesis

image d i s t r i b u t i o n

The a l t e r n a t i v e The normal known

HI

H0 is:

is a known

normal

intensity

pattern.

is:

pattern

is changed

at known

position

by an anomaly

of

shape.

The p a r a m t e r s

size and strength

Therefore,

ratio

is a composite

one by estimating

likelihood

likelihood

of the anomaly

the a l t e r n a t i v e

to a single

the m a x i m u m

The

of small

normal

left to the human

measures,

Remarks

reduced

an observed

automatically

2.1 General

vary.

between

the

case.

2. L I K E L I H O O D

The

the contrast,

the d e t e c t a b i l i t y

is still

on q u a n t i t a t i v e special

the resolution,

But the c l a s s i f i c a t i o n

the d e c i s i o n logical

images

those

are unknown

and may

hypothesis.

parameters

It is

according

to

method.

test rejects

the null

hypothesis

H0 if the ratio

L0 (X) A

-

<

K

(1)

Ll (X) or T = -- 2 log A > K' where

X is the given

joint p r o b a b i l i t y ively. is most

According powerful,

error which

sample

functions

(2) of observations, of the

to the N e y m a n - P e a r s o n i.e.

there

has a smaller

L0 and Ll denote

sample under lemma

a likelihood

is no test with equal

type

II error

the

H0 and HI, r e s p e c t -

(Lindgren).

ratio

or smaller

test

type

I

232

2.2

An

Specialized

image

image i-th

area

cells. cell,

Formulae

containlng If Pi

the

n observed

is the

expected

frequencies

x i are P o i s s o n be

may

probability

frequency

butions

and v a r i a n c e s .

joint

probability k L (X) = ] i=I

for

or

counts

a single

in this

cell

with means

by n o r m a l

distributions

The

x. b e i n g l L of the s a m p l e 1

is d i v i d e d

count

is nPi.

distributed

approximated

means

quanta

to be

The

independent,

X =

x k)

...,

distri-

the

statistically (xl,

in the

observed

nPi w h i c h with

same the

is

(xi_nPi) 2

(3)

e (2~nPi) I/2

in k

2np i

and - 2 log L(X)

If H0 (4),

is t r u e if Hi

expressed

TO m a k e

use

and HI,

i.e.

the

of this

errors

of

2.3 D i s t r i b u t l o n

the

filtered

image

to get

for

! and

to k n o w

only

This

small

first

term

anomalies,

in

the

in eq. T is

hypotheses

and we have

to k n o w

H0

to

fix

the p r o b a b i l i t i e s

and D e c i s i o n

test

areas

normal

by

sum Then

II.

if the

calculating

last

(5)

p0 i and pli,

image

(5), w h i c h

is n e g l e c t e d ) .

the

the

But

between

probability

the d i s t r i b u t i o n

of i m a g e s

points

Asymptoti-

randomly.

difference

is m a x i m u m ,

is small

H0.

is c h o s e n

therefore,

simulation T at the

St r a t e g i e

of T u n d e r

area

where

for T i n c r e a s e s ,

is g a i n e d

the

(X2) I

we w a n t

Statistic

at t h o s e

(4) 2nPi

to f o r m u l a t e

the

type

(xi-nPi)e

respectively.

the d i s t r i b u t i o n

and e x p e c t e d values

distribution, (the

we h a v e

explicitely

(X2) 0 for

(X2)l,

(X2) 0 -

K ~. F u r t h e r m o r e

of the T e s t

test

greater

T is c h a n g e d .

ences

( ) + P1i

it is X 2 d i s t r i b u t i o n

applying

sity

P0i

log

type

To fix K ~ we h a v e cally

pl i and

formula

to give value

p0 i for Pi and

we w r i t e

k T : Z i=I

critical

of the

we w r i t e

is true by

k k + Z log Pi + E i=] i=I

= k log(2zn)

with

of t h o s e

compared

to

given

maximum

of

intendiffer-

(X2) 0 -(X2) ~

233

Figure

I shows

in the

tic T is greater to d e t e r m i n e versa

the type

to fix K'

TO d e t e r m i n e anomalies figure

I error

for given

type

8 further

size and

strength

in spite

for

critical

level

the

statis-

value K'

or vice

~.

simulation

of images

are needed.

The

containing

right

curve

of

Pl that T is less than. X in case of

from it the p r o b a b i l i t y

of HI being true which

only

P0 that

X if H0 is true. It allows

~ for a given

I shows the p r o b a b i l i t y

is valid

the p r o b a b i l i t y abscissa

significance

II error

of known

HI. We can read K'

left curve

than the given

fixed parameters

that T is less than a fixed

is the type mean

count

II error. rate,

(This curve

size,

and strength

of the anomaly).

For d e f i n i t i v e l y are possible, minimizes

fixing

e.g.

the total

tal risk R which

the critical

to take K' error

is the

for a fixed

of the test,

is false,

is shown

2 as function

Figure

3 gives

2.4 I n t e r a c t i v e

To apply

K' to m i n i m i z e weighted

probabilities

p

the to-

by risk

(H0) and p

funct(HI) for

respectively: ~ + Rl p(HI) which

B

in the receiver

the power

(6)

is the p r o b a b i l i t y operating

of ~ for a given p a r a m e t e r

for a confidence

strategies

~, or to take K' which

sum of the two errors

R = R0 p(H0) The power

K' d i f f e r e n t

~ + 8, or to choose

ions R0 and RI and by the a-priori H0 and HI being true,

value

as function

of rejecting

characteristic

H0 if H0 in figure

set.

of the strength

of the anomaly

level e = 5 %.

Application

the test p r o c e d u r e

for clinical

images we have

to overcome

two difficulties:

I) We do not know the that

a sphere

tumours

shape of the anomaly.

is a good

in an early

approximation

But we assume

for the

shape

of

stage.

2) We do not know the normal

pattern with

sufficient

precision

234

because have

there

are

to make

the test

large

is executed.

Therefore

in the n e i g h b o u r h o o d least

There

remains

squares

of the organ with

The

surrounding under

first

shape action

profiles

arrows

marked

with

ence b e t w e e n

lines¢

position

this

the

crossing

curves.

inner

within

the

is d e m o n s t r a t e d

the

We first d i s p l a y

in-

suspicious point

which

image

border

values

as points,

and

the

is

light pen at

the test

area

where

and

the differ-

The next display

of this m a x i m u m

two profiles

of the trend

the value

marked

used for a p p r o x i m a t i o n

is maximal.

the p o s i t i o n

region,

of the profiles

at the p o s i t i o n

and trend

crossing

we know the

but we take

We d e f i n e w i t h

points

and two p r o f i l e s

coordinates

size and shape

image where

of anomalies,

knowledge.

is e x e c u t e d

shows,

of o b s e r v e d

as wavy

procedure

limit the n e i g h b o u r h o o d

test

filtered

6) then

by

its surrounding.

of the anomaly

4 is a simulated

4. The

two

which

The

one because

interactive

5) c r o s s i n g

in these

curves

points

of the trend.

{figure

(figure

I in figure

each of these two outer

files

This

and the p o s i t i o n s

a star

polynom

figures.

in figure

trend

the trend

as q u a d r a t i c

the test area and

on the p o s i t i o n

as in the case w i t h o u t

tensity with

example

of the

But we

only at the area w h e r e

we a p p r o x i m a t e

area

is an i n t e r a c t i v e

depend

of some

of this

to d e f i n e

examination.

the help

differences.

fitting.

the p r o b l e m

The best way to do this

individual

use of this k n o w l e d g e

of the filtered

as parabolae.

of the test

two pro-

statistic

image

In a d d i t i o n

are d i s p l a y e d

at the right.

While

the anomaly

ing cases

at the

first

at the p o s i t i o n

The

appearance

of the profiles

tic

is greater

than

position small

The

of the true

next e x a m p l e the

effects.

Figure

figure

anomaly

shows

search

10 shows

9 and the result

true

a clinical

is made

value

(figure

without

similar,

7) and the

anomaly

case

the p r o f i l e s

the two follow4 are doubtful.

but the test

statis-

K' = 16 for e = 5 % at the statistic

(figure

of a liver

for m e t a s t a s e s

in figure

is obvious,

2 and 3 in figure

is very

the critical

at the p o s i t i o n

9) where

test r e g i o n

of arrows

with

through

is very

8).

scintigram

negative

{figure

storage

the p o s i t i o n m a r k e d

11 is a c c e p t a n c e

of an anomaly.

in

235

3. CONCLUSION

While digital image processing

in nuclear medicine until recently

consisted in a quality improvement by various filter methods,

but

left the recognition and classification fully to the human inspection and experience,

the proposed test is a step towards automatic pattern

recognition and classification.

Its practical application is still in

the beginning stage, but the success with simulated images lets us assume that it will be helpful in the clinical work.

REFERENCES

(1) (2)

B.W. Lindgren:

Statistical Theory, Macmillan,

P.Pistor,

G.Walch,

P.Georgi,

H.Luig, P.Schmidlin,

H.G.Meder, W.A.Hunt,

ing in Nuclear Medicine,

(1968)

W.J.Lorenz, A.Amann,

H.Wiebelt:

Kerntechnik

London

Digital Image Process-

14, 299 - 306, and 353 - 359

(1972)

(3)

P.Pistor, Processing

P.Georgi,

G.Walch: The Heidelberg Scintigraphic

System, Proc.2.

gram s and Technology in Nuclear Medicine , Laboratory

(1972)

Image

Symposium on Sharing of Computer ProOak Ridge National

236

~)tcJTRI~UTI~I

0.8

gE ST,~T[ST[C

T

"~

0.$

Fig. I: Distribution of test statistic

0.4 DO

=

=J

A~l~ = 2 5 LEV = 5e

0.~

~e

~5

RECfi[VER a ~ R ~ T ~ N G

2e

:tg

X

C~ARACT£RI~T~C

J

DO

=

Fig. 2: Receiver operating characteristic

5

A~P = L ~

0.4

LEV = .{~

e.~:

W

a.:t

B.3

e.4

t.g

I

OpERAT 10¢~ C~A~ACTER [ ~T ~C

f

0.8

/ / / ~"

DO

=

5

Fig. 3: Operation characteristic

LEV : g e 0,4

e.2

/

CDN =

5 "~

/ 18

2.e

30

48

A~P

237

~,¢7:.:"

-:7~*•

:.%-'-

;:.,

~+*t

z~tLr~

+s-.~

l__-+

AL~

e I~A~I'INITT

"~AECh~ST~'s 8 ILD

READY

3

D~OC~SSED

Fig° 4: Simulated image after filtering

+ /~,..•.+

m

I"

~'tltE

gt

,•*'•*,.. ,'-'.

,

' .....

'*++• i

i'

-\, l

Is , ,

.•"

$

m

N

m

+m

I

' +.~6~&~&6~.~6~:.~

Fig. 5: Profiles at 1

238

F ~T

)

S9

~4

~3

~l

@@

)

@

i

lP~LT(

Fig. 6: Result at I

f

WA~.

~O

i~

-

Fig. 7: Result at 2

239

ZEILE 100

D~IUCICff

~I

~-

R£TUR+I

TE~T DCHIO

98

g@ $ i

$

IIm$

IS

Y 25

Fig. S: Result at 3

~a,×=

-H.;7=~;~;~:~;~==

&4

-=H~. x = -, : ........

Zg~LE

:

~:::':

:7._~

~=,,~;~;,._:-.

W :.7-

M--

-7 '

:'

~'r~

~s,-~

. '"

l.., i"

. I -

:-I•

H--' :

.! -

'--

=l-

,

7 _~- .

w t~E~-~..E

=I-'

=I-" I-"

|ILD~ ~$~$&

f~IN~LZI~

IA~IINIA~

~2II$~12~2eee

Fig. 9: Liver scintigram after filtering

240

ZI~

1~,

I

IIII

.'

i

¶

"

-'-

l

'IU~r,.I'E

I,I

o

,

%

t

..%

i

" ~J

.

.

.

.

.

.

.

.

•

•

%_.

.

.

Fig. 10: Profiles of scintigram at marked position

I r

TnT ~ g~

~f I

"I

Fig. 11: Results of trend fitting and testing

THE D Y N A M I C OPTIMIZATION

CLUSTERS

IN NON

METHOD

AND

HIERARCHICAL-CLUSTERING

E. D I D A Y I.R.I.A~

Rocqueneourt

(78)

FRANCE

Abstract Algorithms partition optimum.

which

and

efficient operates ve

into

After the

set,

aim

of

in spaces

take

for

on g r o u p s

of points

having

that

that

study

centers" or

which

give

are not

is a s y n t h e t i c a l

by p a r t i t i o n s

of

and

solutions

paper

formed

a model

interesting

efficient

produce

this

techn~uesof"clusters

computer

cial

operationnally

of a finite The m a i n

of o p t i m a l i t y malize

are

necessarily

study

of p r o p e r t i e s

of a f i n i t e

a family type.

set.

We

for-

of p a r t i c u l a r i l y

The p r o p o s e d

"kernels"lthese

a good

kernels

algorithm

adapt

and

evol-

clusters.

developed

the n o t i o n

aspects,

we

of " s t r o n g "

illustrate

and

"weak"

the d i f f e r e n t

patterns,

results

and

by an a r t i f i -

example.

I/ I n t r o d u c t i o n 1.1

- ~ e _ p 5 ~ e

In v a r i o u s vastsets

scientific

of o b j e c t s

quently

appear

; for

and h o m o g e n e o u s " a set

constitutes approach

techniques

such

that

than

the o b j e c t s

terion

under

(medecine,

with

the

which

one

of

more

forms

objects

terms

for

of

of his by

its

the p r o b l e m

such

data.

cluste-

a finite

within

: considering

fre-

"natural

elements

is p r o v i d e d

a partition

to the

In m a t h e m a t i c a l

the f o l l o w i n g

the g r o u p i n g s

in the u n d e r s t a n d i n g the p r o b l e m

economy,etc)

of p a r a m e t e r s

representative

of f i n d i n g

resembles

outside. of

stage

archeology,

number

obtaining

the most

solution

consist

object

biology,

by a finite

specialist,

an i m p o r t a n t for

ring

borated

the

together

A good

each

areas

represented

set

group

can be ela-

a certain

W cri-

:

A - Find

the p a r t i t i o n

of E w h i c h

optimizes

W.

B - Find

the p a r t i t i o n

of E w h i c h

optimizes

W among

all

the p a r t i t i o n s

in K classes. The

family

(~)Institut

of m e t h o d s

to w h i c h

de R e c h e r c h e

we will

d'Informatique

refer

concerns

mainly

et d ' A u t o m a t i q u e .

problem

B,

242

but

it w i l l

also

be h e l p f u l

for

the user

in r e s o l v i n g

the f o l l o w i n g

C

problem. C - F ~ n d _ a m. ~. n. g. each

class

a!l

the p a r t i t i o n s

in K c l a s s e s ,

the p a r t i t i o n

for w h i c h

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

will

have

the m o s t

representative

kernel

(kernel

is a g r o u p

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

of

points

from

In p a r a g r a p h namic del

the p o p u l a t i o n

1.2.,

clusters

for

the

true

takes

a function

several

ple,

from

which

will

we

this

technique

BALL

(1965),

FREEMAN

as a m o d e l

a)

allow

us

much

SNEATH

b)

These

DIDAY 900

more

(1970)

the dyas a m o in & 1.3.

our

of

us

and D I D A Y

study

for

storing

two by

important

the

table

This

(|969),

fast.

For

the p r o c e s s i n g

the f and

is i n s u r e d

We

• The

took

this

family

on an IBM

classical

in

of a

techniques.

LERMAN

the v a r i a n t

360/91

by 35 p a r a m e t e r s

of

N = c a r d (E))

processing

(1968),

instance,

and

:

the

more

formafor

of H A L L

of N . ~ - ~ ( w h e r e

ROUX

sim-

g func-

variations

the m e t h o d s

reasons

permits

than by o t h e r

JOHNSON

characterized

(1970).

is

some h y p o t h e s i s

numerous

cases,

the p a s s a g e

this m e t h o d

manner

criterionW

numerous

two.

of

; providing

to o b t a i n

of a p a r t i t i o n

f permitting

principle

as p a r t i c u l a r

are v e r y

allows

each

of

be u s e d

transformation

a function The

decreasing

(1963),

techniques

items

properties will

will be d e v e l o p e d

in an a l t e r n a t i v e

allows

to a v o i d

(SOKAL

and

the

of k e r n e l s

(1969),

of

objects

population and

the

and n o t a b l y ,

of

study which

permits

choice

giving

methods They

the m a i n

of m e t h o d s

method

of a p p l y i n g

are

similarities

this

of k e r n e l s

be given,

which

give

family

to a p a r t i t i o n .

an i n i t i a l

lism

of

g which

set

kernels

it c o n s i s t s

tions

briefly This

Clusters

classified)

. . . . . . . . . . . . . .

of E in a f i n i t e of

shall

purpose

1.2 - The Dynamic . . . . . . . . . . . . One

we

method

to be

(1970). studied

of a p o p u l a t i o n

three

and

a half

in of

minu-

tes . c)

These

(|96~)). apart

In o t h e r

closer

points d)

techniques

closer

HILL

nor

(1967)

a) The u s e

to each

to stop

empty

suffer

from

they do not

other

if

these

the c h a i n

tend

effect

to b r i n g

two p o i n t s

are

(cf.

two p o i n t s joined

JOHNSON that

are

by a line

of

other.

necessary

to d e f i n e

the p r o c e s s

arbitrary (cf.

thresholds

SEBESTIEN

(1966),

to d e t e r m i n e Bonner

the

(1964),

etc...). of k e r n e l s

agglomerations marginal

not

words,

to e a c h

It is not

classes

do

points

classes.

with

the r e a l i z a t i o n

a high density

(cf. And

favors

figs

finally,

i4 &

15).

and

of p a r t i t i o n s

attenuates

It f a v o r s

let us u n d e r l i n e

also that

the

around

effect

of

the

the a p p a r i t i o n the use

the

of

of k e r n e l s

243

permits

All

us

of

to p r o v i d e

the r e a l i z a b l e

criteria

W,

optimal.

Yet,

status VAN

of

extended

studies

(1971),

tain

part

tribute

type

techniques.

attained

their

goal

(]970), of

to m i n i m i z e

proves

carried

study

that

out

they

the

are

on the p r e s e n t

(1969),

CORMACK

FISHER

(1971))

the s o l u t i o n s

is a p a r t i c u l a r l y

hypothesis,

stable

solutions

and

for

are

applied

results

One b u i l d s structure

for m u l t i p l e comparison

of

b) We d e f i n e with

a random

and

empha-

obtained

for

types

which

the

could

called

respect

V k.

to a certo at-

shown

that,

roots

so-called

"impasse

into

a finite

"non

biased"

members".

The v a r i o u s

:

permits thus

the a t t a i n i n g

obtained

which

evolving

an idea

of

is i n t e r e s t i n g

timewise

and

a

techniques. sets" w h i c h

a n e w kind tree

he

leads

of r e a l i t y

rooted

analysis

will

with

tree.

ourselves

This

the d a t a

"fuzzy

publishing

f r o m one

for

of

of

facets

rooted

some

is

this

solutions

can be p a r t i t i o n e d

for

invariant

limited

It is p a r t i c u l a r l y

as f o l l o w s

the e f f i c i e n c y

the v a r i o u s

switching

leaves

: notably

several

herewith

space

have

variable

of V k. An reasons

this

which

of

is o p t i m a l

to the V k space.

trees

c) We are

set

a structure

of r o o t e d

We h a v e

; but n a t u r a l l y , The

of V k w h i c h

some

attained

study.

by an a l g o r i t h m

under

by

as

(see B O L S H E V

BALL

.to this

of a l g o r i t h m

number

user

optima".

nothing

recently

of a s y n t h e t i c

is d e v o t e d

to o t h e r

solution

the

having

for w h i c h

in " c l u s t e r i n g "

WATTANABE

paper

Each

a)

"local

algorithm.

present

to a p a r t i c u l a r be

techniques,

the v a r i o u s

(1971),

c) w i t h

solutions

the n o n e x i s t e n c e

a given T~e

provide

the r e s e a r c h

NESS

size

problem

will

that

he w a n t s

of

technique

to a n o t h e r ,

truly

provide

the

to g r a s p . which

an a p p r o a c h

will

allow,

to the g l o b a l

optimum. The

example

forms"

to e x t r a c t Finally, ments,

will

which

are

from

particularly a very

his

restraining and

tool

population

let us m e n t i o n

understanding

underline

useful

~he m o s t

that we

ourselves for

have

and D e f i n i t i o n s

E

: the

objects

~(E)

: the

set of

ek

: the

set

the

interests

of

the p r a c t i t i o n e r significant

skipped

to r e s u l t s

the c o m p u t e r

2/ A F e w N o t a t i o n s set of

for

that

the

to be c l a s s i f i e d ,

groups

subset

of p a r t i t i o n s

him

develop-

interesting

for

employed.

it w i l l

be a s s u m e d

finite. the

"strong

of p o i n t s .

theoretical

are b o t h

techniques

the

allowing

of E. of E in a n u m b e r

n
parts.

to be

244

Lk

C {L

=

(A] ~. ~o ~Ak)/A~cA~ }

where,

A will

represent,

for

instance,

E or R n. Vk

= ~k

x Pk" +

W an A

injec~ive

local

If

application

optimum

C = V k one

on C ~ has

: V k+

V k will

a global

~

be

.

an

element

Let x

the

I. Let

set

us

the

I ~.xx T

choose

'

"x~,'

W(v

Euclidian

L~

=

of

E that

the

(

= Z I=I

,A 2)

is

form

~ is u s e d

shall

l

The

global

in

fig°

and

P~

: the

to r e p r e s e n t

write

~ Pi x ~ A i

shown P|

2.

us

v =

(Pl'~°~'Pk)

P of

E~

shall

R

in p r a c t i c e

take

component

will E.

: A x T x Pk ÷

~+

This

mapping

instance, The

will

the the

(f,g,w)

express

(where

be

R(x,i,P)

triplet

17 p o i n t s

as

shown

in

take

used

v~

dotted

where

=

points

line

14K~ k P~'s

which

: L = are

by

the

the

points

M form

constitute

the

classes

mappings

Al ;

A2.

the

set

be

of and

a

of

with the

A.CAI and partition

:

R+

similarity

aggregate

can

(A 1 , , . . , A k )

the

: E x ~(E)÷

constructed

: Vk÷

where

indicates

of

one

element

integers

between

separate

the

of E w i t h

| and

classes.

k). For

chosen° as

follows

: k

+

W

d is

(L a , P~)

identified

following

T is to

= D(x,Pi) is

V 2 = ~ 2 x ~2'

(f,g,w)

where

also

of

one

The

two

D which

of

d(x,y)

optimum

(L,P)~V k where

: P =

~ Y~Pi

3 points

the

Triplets

PKPk

We

W(v)

L 2 = { L = (A I , A 2 ) / A i = E , card (A I ) =3, card (A2) =2}

card

distance.

3/ C o n s t r u c t i n g _ t h e

We

= min v~C

,

x

and Let

W(v ~)

l :

E be

fig~ x

that

optimum. Example

xx

v~such

: v =

(L,P)~

W(v)

=

~ i=l

~ R(x,i,P) x~A. I

f

: ~k +~k

: f(L)

P'l = { x ~ E / D ( x ' A i ) in c a s e

of

equality,

x is a t t r i b u t e d g

: ~pk-> ~Lk

: g(P)

= P

with

~ D(x,Aj) to

the

= L

part with

for

j¢i

},

of

the

smallest

index.

245

A i = {~he

n i elements

n. w i l l d e p e n d i In [9J we t o o k Remark The

: If R

dynamic

function from

upon as

the v a r i a n t

clusters

is n o t shall

our

the

choice

a)

For

by

which

R

one

if f u r t h e r m o r e

R(x,i,P)

ter

(~)

WATANABE b)

A

A

and

thorough

zation

of

L k must

be

chosen.

in a p p l y i n g the

all

the

to F.. i constitute

study

of

attained

the

possible

by

f~

Let

reader

to d r e a m

consists

in

chosing

j#i}

note

i<j

that

this

up

Rather

simply

varying

others).

A i is

in E 2 5 ]

the

cen-

.

and

R(x,i,P)

the A . ' s i

are

= R(x,j,P),

identical

to

of E. found

by FREEMAN

that

by

n. = ! Vi ; i = L is s u c h that

, if

can b e

proposed

us

and

variants.

interesting

is o b v i o u s

case

result

:

a partition

this

the m e t h o d

replaced

It

alternatively

to b e

: A ~ R n,

where

i th c l a s s .

at r a n d o m .

= D ( x , P i ) , g(P)

x is a f f e c t e d Fi's

has

n i = card(Fi)

F i = {xKE/R(x,i,P)

the

: Vk÷

of P. in the s e n s e of D. l an h i s t o r y of this k i n d of m e t h o d

gives

~ E and

the

g

of

2). of

g on

appear

value

"kernel"

or d r a w n

(allowing

variant,

of g r a v i t y

function

to e x p l o r e

those

of g and

this

the

then

} . The

A. the i

consists

estimated

intention

explore

Lk,

R(a,i,P)

(cf

to c a l l

method

either

minimize

chosen

a convention

f followed

It

A which

: A x T x Vk+

L(O)C~k

we

a ~

in E 9 ]

in

[12]

an i n t e r e s t i n g

(which where

variant

is a g e n e r a l i -

Lk = ~ of

and

g is

this m e t h o d !

: i~{!

2 .... k}

n i = ~ . c a r d ( F i) w i t h

~

= ~

as

an

example. c) A ~ E and chosen wise,

by he

n i is f i x e d

the u s e r can

let

A ~ E,

as

being

xed

and

n i is

for

idea

fixed

=

~.card k

in the

i ~ {l,...,k}

the

contents

; n i will

of h i s

data,

be other-

or

equal

for all i (see

and [,O] )

to

~.card

; A. is d e f i n e d I W h e n n. is f i i kernel become supe-

n. e l e m e n t s of P. w h i c h l I the c a s e w h e r e the n u m b e r

to

take

for i n s t a n c e ,

number

x is c a l l e d

all of

E

the

rior

(~

and an

: ni

d)

once

if he h a s

of

elements

n i = card

~he

center

Pi

of

the

P. w i t h O < ~
elements

per

corresponding

if c l a s s

Pi

of g ~ a v i t y of Pi in the inf = xKR n D ( x , P i)

D(x,Pi)

class,

one

will

is c o n c e r n e d .

sense

of D if

246

3.3-Construction

of

triplets

that

make

the

sequence

u n decreasing

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Definition that

v =

of

the

sequenc~

(L,P)~Vk~

u n and

h(v)

=

vn

sequence

{v n}

is d e f i n e d

b y v o and

A

sequence

{Un }

is d e f i n e d

from

of

h be

the V k ÷

V k mapping

such

(g(P),f(g(P))).

A

Definition

: let

the

V n + | = h ( v n) sequences

~ n } by U n

= W(Vn)

S : +

Let

S : ~k

x Lk ÷

~

:

S(L,M)

k

= i$ ]

R(x,i,Q)

x~A.

Q = f (M) .

where

l

Definition R will

be

of

a square

called

function

square

if

(~)

:

S(L,M) Theorem If R

is

t~ose

4/

The

Let

The

square,

cases

following

graph

properties

L = h(v)~

is a r o o t

c)

e = g(P);f(L)

The

d)

have

of

and F =

are

v =

v

non-biased

triplet

.~mher

of ~ k , ~ k , V k the

b)

We

(f,gpW)

the

~

and

~< S ( L , M )

makes

elements

Optimality

(Vk,

the per

u n sequence kernel

Then

appear

e)

of

a looped

and

particular

characterize

ele-

a non-biased

E Vk

tree (m~m~)

of

F .

= e.

properties

elements

shown

respectively

of ~ k ( r e s p .

an

example

allow

the

characterization

Appendix

! for

of

the

~k ) .

of

a square

the

results

function of

this

in

a definition

of

[9~

paper

(~i~) T h i s n a m e c o m e s f r o m the f a c t that the k e r n e l s s u c h a n e l e m e n t a r e in the c e n t e r (in the m e a n i n g of they determine (in the m e a n i n g of f). cf.

for

Properties.

h).

equivalent

(L,P)

(m~) T h e d e m o n s t r a t i o n s o f all " T h g s e d ' E t a t " of the a u t h o r .

(~m~)

decrease

is f i x e d .

:

a)

(~)

the

e.here

consider in V k

element

=) S ( L , L )

:

Structure

us

ments

]~

~ S(M,M)

a "looped

and are

in

[10] the

corresponding to g) of the c l a s s tree".

247

d)

g(f(L))

= L.

e)

f(g(P))

= P

The

properties

element

a)

a)

v is a leaf e # f(L)

ly,

are

equivalent

and

characterize

an i m p a s s e

v = ( L , P ) ~ V k.

b)

Let us

and h)

of

r .

or f - l ( g - l ( L ) )

point

out

that

=

the c) and

permit

us

to c h a r a c t e r i z e

the

c)

g-! (L)

= ~

d)

f-I (p)

= ~ or g-! (f-1 (p))

The

following

d)

properties

impasse

which

elements

follow

respective-

of L k ( r e s p e c t i v e l y

~k)°

proposition Theorem

theorem

2j

(cf.

R

a)

Each

b)

There

connected exists

with

5/ S ~ a r c h i n g ~

_

One will

For

~

£

component

the

is

random (~,B)

S

~

the

_

~

_

that

X(v)

= W(w)

containing

the d e f i n i t i o n s

of

1.

is a looped

the v e r t e x to the

element,

set

tree.

element. of a tree

C,

of v e r t i c e s

w belongs

then v

of C.

to a looped

tree

element.

that

X

w is

triplet

trees

of

if C ~ ~

(f,g,W)

<W(x)).

The

is d e f i n e d by

makes

are

u n decreasing

probality

space

as f o l l o w s

the p a r t i t i o n

~ which

is the u n i o n

:

of ~ in looped

the u n i o n s

of

VUo,

(~,~,P)

looped

of n looped

trees}

trees).

trees

n

Borelean a

X is a c t u a l l y

card

i~ I

(so called

B is the

where

:

the

= card(~)

P(C)

~

generated

the parts

variable

v.

respect

W(h(x))

looped

is such

where

= (Vk,h)

a non-biased

1

in

F

one n o n - b i a s e d

v ~ V k is

with

VX~Vk,

of

CI, • . . , C n t h e n The

from

theorem

w ~ , and W(w) > W ( w ° ) .

assume

set of

: ~÷(O,I)

of

a non-biased

= Vk~ ~ = { the a l g e b r a

P

be d e d u c e d

from

Invariants

words

the f a m i l y

element

optimum

_

first

(in other

(i.e.

I) and

=

:

optimum

the root

The g l o b a l

-

immediately

in V k at least

If w~V k is not

e)

of

then

If a n o n - b i a s e d

d)

= ~ or f-! (g-I (f-l (p)))

appendix

is square,

is a local

5.1

can

=

2 :

If

c)

or f-I (g-I (L))

of

the

C.. l family

tribu~is

of

the

the m a p p i n g

non-blased

element

a random

variable,

of

the

for

trees) ~+~

of

such

looped

(~,~,P) that

tree

if I ~ B , X - I ( I )

is

the

248

union The

of

trees

of V k w i t h

distribution

obtaining biased

an

element

element

where

the

series

ble on in

x

size

5.2

us

of

a tool

one b e i n g

the

the

the

values

Z(V) Let V

of V k.

of

the r o o t s by W

C is

these

assume

a random

variable

of

random and

of

functions.

the d i f f e r e n t

of

the

largest

part

variables their

of

permits

respective

This

also

techniques

trees

that

varia-

(Vk,~,P)

the c o n n e c t e d

of

(see

not

define

In the

; the

correspond

6.1.).

: of the v a r i o u s . . . . . . . . . . . . . .

.

.

the n c o n n e c t e d

parts

C n ~ one d e f i n e s

= W(v|)

of C.).

H be

Let

the m a p p i n g

vector

+

.

.

t~2es of f o r m s . . . . . . . . .

of

the g r a p h

the m a p p i n g

.•. + W ( V n ) w h e r e

the

the

number

and F 1 be

us

E ÷~n

element ~ of

: V =

of

strong

the jth

~o

where ~i

belongs

each is

Z

:

(Vk,h)

and

: C ÷ ~+

by

(v|,...,v)¢C

in P I~. L e t

forms

properties

H(y)

: and v i ~ C i

the less .pn~

thin

=

such

the p a r t i t i o n

x CE of

that

class

to

and ~ (x,y)

x i - Y i = O.

on E such

pi

associates the

(~ l , . . . , B n )

= n } and F|(x)

that

be

Let F n

:

= { y~E/~(x,y)~l}

(@)

are

equivalent

P~ of E for w h i c h each c l a s s is a s t r o n g ~) p~= p l ~ p2 m ~ o.. ~pn ~ P~is pl~

of

element

functionsdefined

= {y~E/~(x,y)

class

the n u m b e r

i = 1,2,...,n

two m u l t i - v a l u e d

following

as

3 which,

indices

Fn(X) Definition

denote

(~l,..O,~n)

which

2)

one d o e s

The r a n d o m

components

a non-

an e m p i r i c a l

of V k is g i v e n .

also

introduction

of

= M i n Z(V) Let V ~ = (v] . . . . v n) and v ~ ~ (Li~pi~ VCC square C I is a l o o p e dt r e e ~ or a loop and v@ is the n o n - b i a s e d

(If R is

The

words,

where

taken

a loop

an e x a m p l e

of

: Z(V')

element

the

containing

can

a comparison

- C haracterization . . . . . . . . . . . . . . . . .

C I , . ~ . , C n be

or

inf W(y) = y~C

those where

smallest

tree

to a n - s a m p l e

connected

the v e r t i c e s .

the p r o b a b i l i t y

empirical distribution

to m a k e

C = C! x C 2 X . o ~ X

Let

of

I for

expresses

(in o t h e r

- ~$_~E~z_~H~ZZ~!~

5 • 2.1 Let

idea

by m e a n s

best to

an

that X ( v ) ~

< x)

In 7.1,

YUo),One

components

that X(v)

v belongs.The

to g i v e

gives

W(w) < x .

: W ( f ( x ) ) > W(x)

such

such

= pr (X

corresponding

connected

is

V k to w h i c h us

that

u n is d e c r e a s i n g

the

(R,B)

F(x

v ~ V k in a l o o p e d

w such

distributionfunction case

elements

function

~ ~) of

and

characterize

the p a r t i t i o n

form.

the p a r t i t i o n s

which

are

thinner

than

(~) The i n t e r s e c t i o n of two p a r t i t i o n s is the set of the p a r t s o b t a i n e d in t a k i n g the i n t e r s e c t i o n of e a c h c l a s s of one by all the c l a s s e s of the other. ~ ) A p a r t i c i p a t i o n P is said to be t h i n n e r than a p a r t i t i o n P ' ' o f E if e v e r y c l a s s of P' is the u n i o n of c l a s s e s of P.

249

3)

P @ is the p a r t i t i o n

defined

5y

the q u o t i e n t

4)

Pf

defined

by

the

Fn

is

the p a r t i t i o n

space

connected

E/H.

parts

of

the g r a p h

= (E,Fn).

Definition The

following

tion I)

is the pn~

Q~

is

More

of

hierarchy tance

partition

~

= (E,FI). if we

which

e q u i v a l e n t (f) class

induces

by

the

Fp(X)

parts

of

set

of

the

the p a r t i -

connected

• {ycE/6(x,~>p F

for

P subdominant

the

characterize

form.

partitions w h i c h are less thin than

the

impose

and

is a w e a k

defined

connected

(cf A p p e n d i x

Remark

of

the

the

are

each

thinner

generally, set

:

properties

Q*

the g r a p h

the

forms

Q ~ of E for w h i c h

p1 t 2)

of w e a k

parts

of

} and r p = (E,Fp),

p = O,l,2,...,n ultra-metric

constitutes

of a c e r t a i n

a

dis-

3).

:

It f o l l o w s

from

these

definitions

that

pt

is a thinner

partition

than

Qe. Definition They

are

of

the

over)apping

characterized

to a single

point.

by

They

the

a~E

is i s o l a t e d

a point

aGE

is o v e r l a p p i n g

5.2.2 The

they

a)

of " f u z z y

permit

intersection, To

profile

by

are

strong

points forms

the f o l l o w i n g

: 0<~ (a,x)

types

knowing strong

this

(for the

form hA

:

reduced

properties

:

that

we are

introducing

here

is

these

instance,

A can be

: E÷{O,I}

definition

h A in order

to have point

from

set o p e r a t i o n s

on the

strong

forms

etc.).

elements

overlapping (1) ~or

of Zadeh

forms

characterize

the m a p p i n g from

they

isolated

= O i~x~E.

if ~ x ~ E

sets"

new

(union,

Each

if ~ (a,x)

the

:

To o b t a i n

b)

thout

that

of

- ~ X X ~

interest

that

fact

and

are d i s t i n g u i s h e d

- a point -

points

an

by

the

that

property)

idea

of

of

this

without

calculating they

considered

or of another

a ~emonstration

forms

of w h i c h

such

(3rd

new

are

having of means)

that

the d e g r e e strong

(x~a~ n

=

set"

of

equivalence

even

wi-

characterized

where

hA(a ) = l

form.

and

their

constituted.

as a " f u z z y

hA(X)

to d e f i n e

similarity One cf.

can

a~A.

~a~A.

with also

Appendix

One

One

see

can use

A of an use

2.

by

the

250

mapping

F

: ~${O~i}

where ~is

the

set

of

the w e a k m

!

F(B)

where

the

Aj

expresses

are

the

are

dissimilar,

5.4

-

the

of

the

more

It

is a m a t t e r

and v 2 of ~/k_ a third

Suppose

R is

of

are

us

constructing

assume

that

Proposition If L #

that

e !, e ~

6/ E x a m p l e s

this

to

the

= D(x~Ci)

minutes

these

can

aid

of

forms

two n o ~ - b i a s e d

v 3 to

Li =

improve

(L~ ..... L~)

(which

F Aj

elements

the

criterion

and

is p r a c t i c a l l y

p i = ( p ~ ..... p~);

often

true)

z :

non-biased

values

prove

solutions

taken

. Let us

easily

P e e k,

of

ease

of

then

the

element

on

=

us

by

denote

and

that

z(x)

with ~" i" (P~_],.. .,P~k )_

P =

then t h e f o l l o w i n g

looped

v3

such

the

~_

CII

6 looped

of

the

obtained the

inputs

to o b s e r v e

the

tree

that

and

proposition.

containing

v =

(e,e)

W(v3)
trees.

about

one

that

which

= Lim U n ( C f

4 iterations. corresponds

of

figure

to

have

the 12.

have

3.3.)

in compa-

been ; the

defined

of is,

frequent

the

best

the each in fact,

in 5.1.

c~nvergence

The m o s t four

out

appearance graph

we

of L ( O ~ i n

pointed

This

,

distance,

the d r a w i n g

50 p a s s e s

in

the m e t h o d

First

K=4,n1=n2=n3=n4=n5

changing

frequency

variables

by u

at

The

indicated

of

taking

(cf fig.6).

d is the E u c l i d i a n

time

These

of R u s p i n i speed

in

where

(each

10 070.

random

nerally

the

d(x,y)

are

is r e p r e s e n t e d

on

; therefore,

y~:

abcissa

is

c)

the m e t h o d

6 solutions

a histogram

solution

the

two

~ 2 .... k }

of R u s p i n i

50 p a s s e s

existence

mapping

smaller.

element

the k s m a l l e s t j =

permitted

that

i,L)

This strong

of A p p l i c a t i o n s

of

of

v I and v 2 are

~2~ and

applied

all,

with

additive

a non-biased

We have

2,57

the

:

as a root

'

B.

more

:

non-biased

W is

~ .... L~Jk )° One

L = (L

ran

the

there exists a m a p p i n g + P(E)+~ such that k

are {V~l ,...,v~i k ~ i . J ' x ~ { ~ / ~ = 1,2 and

R(x

constitute

B but

that

z : ~(E)x

rison

which be

v i = (Li,pi)~v k with

presuming

otherwise

has

will

hAj(X))

square.

We will d e n o t e i i i vj• = (Lj,Pj).

Let

F(B)

of

(m j ~ l

g~)!~_!)~_~!~)~!_2~!~_~_~g~g!~_~!~

V~

We

forms

weakness

of E and

I

x~B

card(B)

m strong

degree

forms

classes

The

is ge-

; the

251

value

of u for

lutions,

this

which

solution

shows

that

The best

solution

which

satisfactory

for

in f i g u r e s

7,8,9

is

indicated given the

in figures

solution

"strong

corresponds

9,10

given

forms"

is c l e a r l y

it r e a l l y

to the root

the m e t h o d . and

and

better

than

corresponds

10.

can

carry

the

the b i g g e s t

The most

One

I I do not

of

for

frequent

easily any

other

to the best

see

looped

7. F r o m

these

solutions

corresponding

exactly

to the four

tree,

solutions

that

information

in figure

so-

partition.

one

classes

the

are

solutions

(of 5.2) obtains of

to

4

the best

solution. Let us r e m a r k

that

figures

9, one

tion

8 and

correspondong

One

gives

this

(table

time

taking

in a p p l y i n g brings

to I),

that the

K=6,n5=n6=5,

5 passes

of

tence

of

forms

and A 1 the

6 strong

proposition

out

the

of figure

table

of

without

the m e t h o d forms

and

strong

looped

whose

strong

changing

3 weak

forms

the

tree

solution root

forms

obtained

has

the other

This

table

forms.

Let

(cf Fig.|3).

sees

that

appear

B 3 is a strong

in fig. out

signify ler

that

B 3 is a r e l a t i v e l y

point

solu-

parameters

brings

out

taking

and

BI,B2,B 3 denote

One

in

the exis-

can m e a s u r e

the weak

the

"weak-

WxCB 3

: 13 F(B I) = ~-~, F(B2)

One

in

the

by

ness" of B i by using the F f u n c t i o n s (cf 5.2.2). As 3 4 4 5 j=l ~ hA'j (x) = I + ~ + ~ ~ x c B I ;j=4 ~ hA'j (x)=l+ 51 W x ~ B 2 ; and hA6(X)=l

One

given

is

7.

the

(n=5).

3 to

13 express

that

that

than

weak

in 4 of

form,

that

form.

These

quite

the

3 = ~

well

of classes

strong

these

of r e s e a r c h

is still

that

F(B 3) = I

B I is almost

5 solutions,

the number

and

values.

there

a strong

and weak

Finally,

appear

actually

empty

do

form

forms

and

that

let us classes

exist

must

one

should

be

which smal-

6.

Conclusion A wide

field

develop

the choice

develop

the

required variations niques survey larly

a priori) of

the c l u s t e r s

the p a s s a g e

of

the

as for

from

structure as

size

allowing

g (e.g.

allowing

; realize

of

elements, niques

of f and

techniques

of

the

a clearer

by m e a n s

an e x h a u s t i v e

centers

method

tree

the

space

trees,

: in p r a c t i c e

the choice

one

the r e l a t i v e of

open

number levels,

vision

of

of

of k

the

; develop

the

methods)

number

of c l a s s e s

of

with

elements,

forms

the

tech-

a statistical

is c o n c e r n e d

strong

;

the v a r i o u s

in d e p t h

; make

V k in r e l a t i o n of impasse etc..,

learning

(the

comparison

to another

:

E, p a r t i c u non-biased ; set up

table(e.g.of

techthe m i n

252

mum

spanning

the

strong

numerous

tree forms

Hstrong

lue

the

practical forms

W)

low

the

density

allow

procedure

Partitioning of

; use

weak

(by

us

forms

zones

applications).

table"

classification a)

type)

order

(obtaining

Let

to

in

us

obtain

also at

detect

"holes"

point

once

to

out

the

among

leading the

three

to

fact

types

that of

:

taking

the

partition

corresponding

to

the

best

va-

;

b)

Clumping

o)

Hierarchical

(using

the

overlapping

classification

points)

(using

;

the

"connected

descendant"

method),

BIBLIOGRAPHY (1)

BALL

G.H.,

(2)

BARBU

(3)

BENZECRI

1970

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M.,7968

M.S.H~ ~97|

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J.P.,

1970

(5)

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Co,

1967

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LoN.,

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des

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J,

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N,

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muni

et

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ses

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automatique de

Statis-

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Procedures.

Bio-

9]. discrimination

Recognition

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Statistique

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Infor-

2,p.319-350

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hi~rarchie

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G,DIDAY

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G.S,1966

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R,R,SNEATH

(25)

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(27)

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L.A,]965 C.I,

1971

t. 274,

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C.R.

Acad.

Multivariate

Data

Analysis

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Tasconomy.

W.H.

view

Inf.

Theoretical Gestalt C-20,

Freeman

and L o n d o n . of

ljubiana,

cribing

vol.

humaine.

464-467

pp.685-694

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ters,

en p a t h o l o g i e d.p.

des

Conference

San F r a n c i s c o

- Fuzzy

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les m o d i f i c a t i o n s

Computer

- A unified

Congress

entre

off-line

Joint

P.H.R,1963

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s~riques

Paris,

Proc. (24)

y,BARRE

correlations

clustering Booklet

Control methods

Clusters.

a l g o r i t h m s . IFIP

TA-2

8,pp.

338-353

for

detecting

I.E.E.E.

n ° I, J a n u a r y .

trans,

and

des-

on C o m p u -

254

(28)

McQUEEN

j,

1967

- Some

Methods

for

Multivariate on

Classification

Observations.

Mathematics,

I~ n °

I,

pp.

Statistics

and

Analysis

5th

Berkeley

and

probability~

of

Symposium vol.

281-297

APPENDICES Appendix

l

Let

B be

the

set

of

the

the

set

of

the

each

a finite

of

these

set

arcs

and

connected

: Each

will

h

be

components

components

lle~e!!!!~_!

a function

(h(x),x)

have

: B÷B.

noted of

F

F

graph

component

form of F

defined

=(B,h).One

constitute

a particular

connected

The

by

by

knows

a partition

B and that

of

B

;

:

contains

at m a x i m u m

one

circuit° One

gives

that

x

is

will

be

(fig°4)

an

a fixed

point

example

of

a connecte~

if x = h ( x ) . A t r e e

cOmponent

having

for

of F

its

.We

root

shall

a fixed

say point

+

called

a looped

tree(see

fig.5).Let

Proposition 2 : If W is i n j e c t i v e on .............. the property: W(h(x))<W(x) then : I)

Each

not

connected

contain

2)

Each

3)

If

component

another

connected

y~B

is

of F

entire

contains

a mapping sequence

a loop

and

B÷E v

only

and

n

one

verifies

and

does

such

that

circuit. component

not

the

W be

a fixed

of F

point,

is

a

there

looped

tree

exists

a fixed

or

a loop. point

x

W(x) <w (y) Appendix The

2

problem

is

to

show

to

characterize

I)

Q~

is

the

finest

2)

02

is

the

parittion

graph

F1 =

Append.ix

a weak

A be

and

E'

the

E' ÷ ~ ( E ' )

I)

-(~)

be such

two

properties

are

equivalent

the

partitions

defined

by

mapping

(~)

the

which set

are

of

the

less

fine

than

connected

P

parts

1~

,..o,

of

set

This

H

pn ~

the

the

quotient

that

Fp(X)

E x E÷ ~

s p a c e ~ ~) = {ycE'/

E/H.

such

If

~(x,y)>P

F

that is

P } a~d r p

the is

~

(x,y)

= n-~(x,y)

multi-mapping the

graph

E'

(E',Fp),

:

The

a hierarchy 2)

following

3, : Let

then

the

(E,FI).

Theorem let

of

that form°

of on

the

parts

of

F

P

for

p = O,I,2,...,n

constitute

E'

hierarchy ee is

connected

induces

e~itlon ot @ the mapping as

the

subdominant

in 5 . 2 . ] . defined in

5~2oI.

ultrametric

of

A

•

/t-

+

--~+~-+ ~

I

++~+

/

\ \

/

L "~T£

+

÷

+

+

+~

+++ + + ++4 - + / I

~'\

+

I

+

+

N

\

+

~-

+

+

+

/ /

\ /

+ ++

+

/

/

+

I {

\ +

x

+++ + + + + + +

\

÷

\ I

+ + ++

\

/ 1+

I /

| +

r

+

l

/

+++ + + + + + ++

L

/

+

\

/

/

~.

++.i

/

9 "ST~I

~L"~

•

~,

sl,

n~

%

/

0 L "BT£

i+

++ + +

+

•

++

+++~

"

,,

•,

+++++ +~,

+++ ++ + "~

I

6 ".~T~{ I+

I

~+++

.

+*

÷ I

.

.

+

+

+

~

•

•

.

.

.

< <

4

/ /

+ -++ ++ ÷+

~

•,

++

/

..,

xx

÷:I

x

++

+

s a

"--'"

x

I

++ ~ ++4"++ ++ +

++ -.

+ + ÷

, I

/

+ +÷

g

9q:

257

Frequency of appearance of each. solution after 50 drawings L (0)

Fig. 12

The value U=0.5 U= 2.11

2.22 U= 3.817

corresponds corresponds corresponds corresponds

to to to to

the the the the

solution solution solution solution

in in in in

B2

Fig. 13

Fig. Fig. Fig. Fig.

7 8 9 10

258

~O~S

! ,? 12 t~

2~

4~ 4~

47

~J ~4

59 6O 6~ 6~

'~

75

4~

Table

I: Strong rig°

forms

f )r the inputs

of Ruspini

rig.

14

75

0 X

X x X

X

The signs "x" represent the elements to be classified whereas the sign "0" represent the center of gravity of the 5 elements.

o~ K

The three closest elements of the population are represented by the sign "x"; the center of gravity of these three elements attenuate the effect of the m a r g i n a l ~7 ~

±

A MAXIMUM PRINCIPLE

CONSTRAINED

OPTIMAL

AN EPSILON

FOR GENERAL

CONTROL

TECHNIQUE

PROBLEMS

-

APPROACH

J e r o m e W. M e r s k y iZi5 S. Leland St. San Pedro, C A 90731 U.S.A.

W e wish to present an extension of the Epsilon Technique (Reference i) to general constrained optimal control problems with systems governed by ordinary differential equations.

The use of the Epsilon Technique provides a straight-

forward constructive approach %o the m a x i m u m

principle, and, in particular, to

the lagrange multipliers for a very general constrained optimal control p r o b l e m which s u b s u m e s the so-called "bounded phase coordinate" problem. Before formally stating the problem, w e establish s o m e definitions and notations. of M c S h a n e

Definition

We

shall be working in the class of generalized controls in the sense

(Reference 2) and L. C. Young (Reference 4):

Let U be a compact set in Rk.

A function u which, to almost every

t, assigns a probability m e a s u r e ~(- ,t), defined on the lebesque subsets of U, is said to be a generalized control.

Definition

If f(~, • ) is a function defined on U, it m a y be extended to be a function

of generalized controls f by the following: ciated with the generalized control u

if ~o is the probability m e a s u r e asso-

then O

f(~,uo) = /

f(~, u)d ~'o(u,t) U

Since in the following all controls are a s s u m e d to be generalized there will be no confusion in dropping the tilde over the symbol for generalized controls. Generalized controls m a y be considered as elements of the dual space ~F o(Q)"

Therefore, w e shall use the w e a k ::-"topology for the topology of Ug,

the class of generated by the compact set U.

260 We

Problem

are n o w in a position to state the p r o b l e m w e wish to consider: viz.

In the class of generalized controls, m i n i m i z e

P

T

f o g(t, x(t), u)dt

subject to

n

w h e r e x(t) ~ ~ , ~ ( t , x ( t ) , u )

£(t) = f(t, x(t), u)

a.e.

~(t,x(t),u) = 0

a.e.

~ ( t , x ( t ) , u) -< o

a.e.

* (t, x ( t ) ) ~ 0

a.e.

~ 1~p, ~ ( t , x ( t ) , u )

~ R r , ~ ( t , x ( t ) ) ~ R q, f, g, ~ and ~)

are continuous in (t,x,u) and continuously differentiable in x and, ~ is C " in (t,x). In addition, we r e q u i r e that~ f o r all a d m i s s a b l e x(t),

T

foil

~:(t)It

Zdt -< N < ~

w h e r e N is a fixed constant, and that

[x,f(t,x,~)] We

Problem

t ~ [0, T].

~ c(~+ilxll z)

replace this with the "epsilon-problem'.

P ¢.

irl the class of generalized controls m i n i m i z e h(¢,x(-),u) =

T

T

IA

11~(t) - f(t,x(t),u)II zdt + ~

z~ T

+~

II ~(t,x(t),~)ll Zdt

T

fo

Ifo

[re(t, x(t), ~), @(t, ~(t), ~)]dt + z7 T + fo

g(t,x(t),u)dt

[n(t, x(t)), ¢ (t, x(t))]dt

261

where

e > O, a n d

m(t,x(t),u)

n(t, x(t))

with the same conditions as in Problem

i0

~(t, x(t), u) < 0

( Qb(t, x(t), u)

~b(t,x(t),u) ~ 0

= {

=

o

,(t,x(t)) < o

¢ (t,x(t))

,(t,x(t)) ~ o

l

P.

The following t h e o r e m gives the existence of, and necessary conditions for, solutions to P r o b l e m P •

Theorem

i

Under the above conditions, there exists a solution pair Xs(. ), u

for p r o b l e m P ¢ .

I. I)

This pair satisfies the " e - m a x i m u m

principle":

Let

~'(e,t,x(t),u)

--

[ z ( , , t ) , f ( t , ~ ( t ) , u ) ] - [L(e,t),~(t,x(t),u)]

-[M(c,t), ~(t,x(t),u)]

- [ N ( e , t ) , ~ ( t , x ( t ) , u)] - g(t, x ( t ) , u)

where

x L(s,t) = ¼~(t, xe(t),u¢)

M(S,t)

= lm(t,

1

Xs(t),ue)

.T

N(e,t) - Z--e J t n(s'xe(s))ds ~V(t,x(t),u) = ~(t,x(t))~t + ~x (t'x(t))f(t'x(t)'u)

,01

262 Then,

Je(,, t, x (t),

1.2)

--

m a x JC(¢, t, x e (t), u) ueU g

If w e write Y ( % t ) = i (x ° (t) - f(t,x¢(t),u )) then w e have

: -Vf*Y +V~P>i'lL +V(~*M + V g * ( ~ n ) +Vg*

Y(%T)

a. e.

= 0

evaluated along (% x (t), u £ ). i. 3)

T h e multipliers Ni(e,t ) are monotonic non-increasing and constant on

intervals along which ~i(t,x e (t)) < 0.

Comments

The proof is omitted here and m a y

be obtained f r o m l~eference 3.

T h e basic idea is to observe first that T

[n(t, x(t)),* (t, x(t))]dt = 0

ol

t

I-

t

n(s,x(s))as,,(t,x(t))

IT 0

T h e n if r(a,t,~:,X) is used to denote the s u m of the integrands in the definition of h(%t,x(t),u)

as a function of × ~ c(t,x)

~ ~-~ ! f ( t , x , u ) , g ( t , x , u ) , ~ ( t , x , u ) , ~ ( t , x , u )

~ u

u I

then

d@dr-(e't':k ,X

+O(X-×e)) i

> 0 @=0

263 T h i s g i v e s I. l).

T o o b t a i n i . 2), l e t d(t) b e an e l e m e n t of t h e S c h w a r z s p a c e of

infinitely smooth functions with compact support, and Xd(t ) = xe(t) + @ d(t). Then

dh(e'Xd('d@ ) , u ¢ )

[

=0

0=0

gives I. 2).

Condition I. 3) is immediate.

We m a y p r o c e e d now to t h e l i m i t i n g f o r m of T h e o r e m

n e w existence theorem and m a x i m u m

Theorem

2

1 w h i c h g i v e s us a

principle for P r o b l e m P.

A s ¢ c o n v e r g e s to z e r o , x (.) c o n v e r g e s u n i f o r m l y to x (-) and u o P . If,

c o n v e r g e s w e a k * to u ° w i t h Xo(. ), u ° b e i n g a s o l u t i o n p a i r to P r o b l e m furthermore,

the matrix

~u

3u

has full rank along (t,Xo(t), Uo) then the following limits exist:

~(t) :

~r~z(~,t)

X (t) = l i r a L(~, t) ~0 ~(t) : l i m M ( e , t )

~(t) : l i m N ( ~ , t )

and w e have the m a x i m u m

2. t )

principle:

Let

~ ( t , x ( t ) , u) = [~ (t), f(t, x ( t ) , u)] - [x (t), ~o(t, x(t), u)]

264

[~(t),@(t,x(t),u)]- [~(t),,(t,x(t),u)] - g(t, x(t), u)

Then

Jd(t'x°(t)'u°) = um a¢x U g ~yf(t,x o (t),u)

z. z)

[ = -Vf*~ +V~*X + V @ * ~

+ V~;"'~ + Vg*

a.e.

~(T) = 0

evaluated along {t,Xo{t), Uo).

Z. 3)

The multipliers ~i(t) are non negative and ~i(t) = 0 w h e n ~i(t, Xo{t),Uo) < 0;

the multipliers ~i(t) are monotonic non increasing functions of t which are constant along intervals along which %i(t,Xo(t)) < 0 with v.l(T) = 0.

Comments

Again a detailed proof m a y be found in Reference 3.

The m a i n idea

here is that the matrix above, having full rank, has an inverse which allows us to write u

s

-u

o

as a function of the constraint t e r m s ~(t,x (t),u¢), etc. so that the

convergence arguments proceed in a straight-forward manner.

l~efe rence s

i.

Balakrishnan, A. V.,"The Epsilon Technique - A Constructive A p p r o a c h to Optimal Control" in Control Theory and the Calculus of Variations, A. V. Balakrishnan (ed), A c a d e m i c Press, N e w York, 1969

Z.

~vlcShane, E. J., "Optimal Controls, Relaxed and Ordinary, " in M a t h e m a t ical T h e o r y of Control, A. V. Balakrishnan and L. W. Neustadt (eds), A c a d e m i c Press, N e w York, 1967

3.

Mersky, $. ~V. ,An Application of the Epsilon Technique to Control P r o b l e m s with Inequali_~ Con____~straints,Dissertation, University of California, Los Angeles, 1973

4.

Young, L. C. ,Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Co., Philadelphia, 1969

OPTIMAL CONTROL OF SYSTEMS GOVERNED BY VARIATIONAL INEQUALITIES J.P. YVON (IRIA 78 - ROCQUENCOURT - FRANCE)

SUMMARY In many physical situations, systems are not represented by equations but by variationnal inequalities : a typical case is systems involving semi-porous mediums but there are many other examples (cf. e.g. Duvaut-Lions [4]). This paper (1)is devoted to the study of optimal control problems for such systems. As in the case of partial differential equations we are led to consider the analogous separation between elliptic and parabolic systems ; this is studied first and then we give two algorithms with application to a biochemical-example. I - ELLIPTIC INEQUALITIES Let us define - V Hilbert space, K closed convex subset of V, - a(y,z) bilinear form on V, continuous and positive definite, - j(z) convex %.s.c. functional on V, and -

U Hilbert space, N ad

-

B E

~(~;

V')

a closed convex subset of

.

Then we consider the following problem : Problem E O

For each

v EU

(I.|)

find

y E K

a(y,~-y)

where

f

solution of

+ j(~)-j(y) >i (f+Bv,@-y)

V ~ E K

is given in V'.

Theorem 1. I Under the following hypothesis Ii [1.2)

a(" , ")

is c°ercive : a(~,~) > - ~

f

~>

0

vuEV

or j(.)

is strictly convex

there is a unique solution y(v) to Problem E . O

For the proof of this theorem cf. Lions-Stampaeehia -

~ Hilbert space and

-

z d given in

C

[6I. We introduce now :

E ~(V;~)

and we consider Problem E I Find (1.3)

u E

~ad

solution of J(u) ,.< J(v)

(I) More details about definitions proofs etc.., will be given in Yvon E8] •

266 where

Theorem

1.2

If we assume (i)

Nad

li) If

(or v > 0

m --~m weakly

--

(1.5)

bounded vn

lim

v

in

in (I.4)) a n d V

and

v ---~ v

......

(B Vn,~n)

n

~ (Bv,~),

then nhere is at least one solution For the proof lows to take

one uses a minimizing the limit in (1.1).

Remark I The assumption case

(1.6)

ii) of (1.5)

weakly in 4, then . . . . . . .

sequence

to (1.3).

of J(v) and the hypothesis

is a property

of eompacity

(l.5)-ii)

of B. For instance

H ~__.V ~-~V'

(each space dense

in the next with continuousinjeetion) B v = ~ v

we may take

B E~(U;~)

so that

(B If the injection

v,~)vv,

=

(B v,~) H

from V into H is compact

then we obtain property

II - PARABOLIC We suppose now that we have - V,H Hilbert

spaces,

V ~ H

(|.5)-ii).

SYSTEMS

(as in (1.6)) dense with continuous

injection

so that

V~H = H ~ c_~V ~ - a(y,z) bilinear form on V, symmetric and coercive, - f given in L2(O,T;H), - j(z) a convex £.s.c. functional on V with domain D(j) = {~E V lj(~) < + ~ } - Yo

given in the closure

of D(j)

in H.

Then we have Theorem

2.1

With the previous

data, there exists a unique function

y E C (~,T]

(2.1)

;V) ,

y

such that

~"~ E L2(O,T; H)

and satisfying (2.2)

Idy z-y) + a(y,z-y) "dt ~

+ j(z) - j(y) ~ V z E D(j).

(2.3)

y(O) = Yo

Demonstration

in Brezis [2]

Now let us define - ~ H i l b e r t space, -

B E ~(~;L2(O,T;H))

~ad

closed convex

subset of

(f,z-y)

a.e.

in (O,T),

al-

in the

267

and Problem P

o For each

(2.4)

vE

Ua d find

y

satisfying (2.1) and

(~t,z-y) + a(y,z-y) + j(z) - j ( y ) ~

There exists a unique

(f+Bv, z-y) P O . Now let us introduce

y(v) solution of Problem

- ~ Hilbert space and C E ~(L2(O,T;V); ~) - z d given in and Problem P1 Find

u 6 Uad

(2.5)

such that J(u)
V v E Ua d

with 2 J(v) =~ Cy(v) - Zd[12 + v [ I v ~

(2.6)

v >10

Theorem 2.2 w-T{h the following hypothesis I i) The injection from (2.7)

ii) ~ d

is bounded

there exists at least one solution u

V

into H is compact,

(or v >0)

to Problem

P1 "

An example Let Q an open set of ~ n F its boundary and consider the system ~t

-f~y = v

O and (2.9)

= Yo

in Q

in

Q x ]"O,T [

Yo

given in L2(Q)

jO

if

r < h

kr

if

r >/ h

h,k > 0 .

¢(r)

In order to set properly the inequality associated with relations (2.8) le~ H=L2(Q) V = HI(Q) U = L2(Q) (2.10)

a(y,z) = JQI

i~i. ~Y(x) 8z--~-(x) dx •= ~ x i ~ xi ~0

(2.11) so that

j(z) =/F

F[z(y)]d Y

with

F(r) =

if

ri

h

Ik(r2_h2)

if

r >z h

y(v) is the unique solution of the variationnal inequality :

268 ~dd-~tv ,z-y(v)) + a(y(v)~z-y(v))

(2.

+ j(z) - j(y(v))

~ (v,z-y(v)) 2 e

•

12" I Y(V)It=o

= Yo

The cost functional (2. !3)

(~)

is J(v) =

- zdl 2 m~ dt + v

]

Then if ~ ; is bounded or v > 0 lution of ~ 5 ) with (2.13).

[[ V[l~

V ~0

there exists at least one optimal control

U

so-

Remark 2.1 As in the elliptic ease the main difficulty of the problem (theoretically and numerically) is the non differentiability of the mapping (2.14)

v ~y(v)

solution of

Problem

Po

(or

Eo).

So that the question of necessary optimal conditions from u is, as far as we know, yet opened. For some results towards this direction el. F. Mignot [ 7 ] III-

DUALITY APPROACH

Using the ideas submitted in a paper from Cea-Glowinski-Nedelec ~ ] one can obtain a dual formulation of the variationnal inequality. For reason of simplicity we suppose that we are in the elliptic case and that the variationnal inequality comes from the minimization problem : (3. 1)

Inf ½ a ( % 9 ) ~E V

+ j (9) - (f + Bv~9)

Then the fundamental assumptions are : There exists a Banach space L, a closed convex bounded set A o f and an operator G (non necessarily linear) from V into L such that (3.2)

j(~) = S ~ A

L' with 0 E A'

<~ , G(~)>(I)

Now we can re,trite ~he problem (3.1) in the form (3.3)

!nf ~p EV

Sup ~ EA

~(~,~;v)

with (3.4)

1

~(~,~9

v) = ~ a(~,~) + < ~ ,G(~) > -

(f+Bv,~).

The dual formulation of (3.3) is (3.5)

Sup

Inf

~E A

~EV

~(~,~;v)

Example 3 We take the analogue of example 2. The problem is in the general form of (3.1) with a(.,,) and j(.) given by (2.10) (2.11). So that we have L =

LI(F)

L' = L~(F)

!l = {~ 6 e~(r) I 0 ~ ~(~) 4 I

a.e. on

F} .

and ~(?)

= k~

(2(y)

_ h 2)

which is a non linear operator from V = H I( ~

into

L~(F).

(]) Here <.,.> denotes the duality product between L T

and

L.

269

Theorem 3.1 Under the following hyp0thesis IFor ea__ch (3.6)

k E A

the mapping

-+

L

is convex i.s.e.,

there exists a saddle point for ~(.,.;v) The proof is an application of the theorem of KY Fan-Sion. Theorem 3.2 If

G(~)

is continuous from

and if assumption (3.6) holds,

V- weak into L'-weake

then M(k) =;~(y(k),k;v) = inf ~(~0,k;v) ~6V is G~teaux diffenriatiable with derivative given by (3.7)

(3.8)

M'(k). ~ = < ~,G(y(k))>

Corollary 3. l A necessary and sufficient condition for (3.9)

< ~-k,G(y(k~))

> >~0

k ~ E A solvin$ the dual problem (3.5) is V

k E A

Corollary 3.2 If kW solve the dual (3.5) then y(k ~) solve the primal (3.3). Now we can state the problem corresponding to the paragraph ] : Optimal control problem A. (3.10)

I

B.

For each of

v E Uad

Sup ~EA Find

compute

Inf ~ EV

u E Uad

k(v)

and y(v) = y(k(v)) solution

;£ (~0,~;v)

such that

J(u) ~< J(v)

with

J

given in (1.4)

Then using the optimality condition (3.9), we can associate to the previous problem the Suboptimal control problem I A.

B. (3.11)

For k fixed in A compute

Inf i

l]Cy(k;v)- Zall2

v 6 ~ad this gives C.

y(k;v) solution of

Inf ~(~,k;v) ~EV Solve the optimal control problem

u(h)

+ v~l ~

and yO~) = y(k;u(k))

Then finally find ~ E A satisfying < k~-k

, G(y(k*))> ~ 0

V k 6 A

Remark 3.1 The previous technique which consists in permuting the determination of u and is due to Begis-Glowinski [i] . Notice that problem (3. ]I) is not equivalent to problem (3.10) this can be shown on very simple counter-examples (cf. e.g. Yvon [8] ). Theorem 3.3 Under assumptions of ~$| and theorem 3.2 there exists at least one solution k'to

270 problem (3.1~)~ IV - REGULARIZATION An other way to avoid the non differentiability of the mapping (2.14) is to approach Problem P! (for instance) by a sequence of problem more regular wich ensure the differentiabillty of cost functions~ We will expose the method in the parabolic case (§2). Fundamental hypothesis° (4.1)

There exists a family of functionnal js (.) of convex functionals on V twice continuously differentiable such that

(4.2)

j (~) + a(~,~) ~

(4,3)

V

lira j ~ ) e~O if

v

--~ r

5

(4.4)

~ (4.5)

y

~E V

= j(~)

~

V@

independant of g ~V

(I)

weakly in L2(O,T;V) as c -~O, then r

J 0 j(y(t))dt

4 lim

There exists a sequence

0 JE(Ye(t))dt z

bounded in

V

such that

j~(z s) = O

Then we define Problem

P

Find

y

(4.6)

oe g

solution of

i ( ~t

' z-Ye) + a(Y~'z-YE)

Ye(O) = Yo

+

Je (z)

- Je(Ys ) ~ (f+Bv,z-y~)

V z EV

Theorem 4.1 For each vE Nad

there is a unique solution

ye(v) to (4.6) such that

y (v) E C( [O,T] ;V). c Furthermore y (v) ~ y(v) in G where y(v) is the solution of Problem P . o With notation of §2 we introduce (4.7)

L2(O,T;V)

J~(v) = llCye(v) - Zdl]~ + V

as

s

~

O

Ilvll~

and the

PIE

Problem Find

ueE~ad

(4.8)

such that Je(ua) ~< Ja(v)

Theorem 4.2 There exists at least one u~ sequence ~E, } of {us} such that

where

(I)

u

solution of (4.8) and as

u , -~ u g is a solution of Problem PI"

For simplicity we assume that

V v ~ Ua d . g -+O

there exists a sub-

in I~

D(j) = D(j~) = V.

(Cf. notations of Th. 2.1)

271 Remark 4 As

je(.)

is in class

C2

the Problem Pos may be rewritten as

dY e ( ~ ,z) + a(ye,z) + (j~(ye),z) = (f+Bv,z)

V

z E V

and Problem PI~ is then an ordinary optimal control problem for parabolic systems. V - APPLICATION TO A BIOCHEMICAL-EXAMPLE The system represents an enzymatic reaction in a membrane with semi-porous boundary. The problem is unidimensional in space and the functions a(x,t), s(x,t), p(x,t) represent.respectively the concentration of activator, substrate and product in the membrane ~j#. In dimension-less variables the problem may be stated as Ida

(5.1)

~2a

la(o,t)

= Vo(t)

La(x,o)

= O

a(l,t) = Vl(t)

0s _ 02__s +o a s t Ox 2 l+a " l+s = O

~ (5.2)

]s(o,t) = So

0 > 0

s(1,t) = ~I

~o,~l E~

|

~

s (x,o)

t

= 0

0 x2

l+a " l+s

1

+ ~(p(o,t)) = 0

(5.3)

I ~x (l,t) + +(p(l,t))

0

p(x,o) = o where

~(r)

is real function given in (2.9).

Control variables are

vo

and

oo

v! :

co

U = L (O,T) x L (O,T) and O ~< vi(t ) ~< M

~ad = {v E

a.e. on (O,T),

i=1,2 }

The cost function is

(5.4)

J(v) =

J

(o,t)-

Zo(t)12dt+

z! (t) 1 2dt

0

Theorem 5.1 The system ( 5 . 1 ) . . . ( 5 . 3 )

a d m i t s a unique s o l u t i o n

a(v),s(v),p(v).

(I) For more details about enzymatic systems cf. Kernevez

[5] .

272

Theorem 5.2 There exist at least one $,iven bi

uE~ d

satisfying

J(u~

J(v)

V v E l~d

with

J(v)

(5.4). VI - NUMERICAL RESULTS

Example

I To give comparative numerical results on the two types of algorithm we have

considered first the example of § ll.Computation have been performed with = )O,I[ and

zd

so that

(61)

J(v) = 2

f~ l~x(l,t) - Zd(t) I2dt +

(solution is sy~netric by changing

Represents method

r = { O } U { I}

only function of time so that the cost function is

x

in

2 ~ !I~Iu

l-x).

the state corresponding to an "optimal control" computed by duality

(~III), The threshold is given by h=0.5.

Fisure 2 Gives comparison between regularization

and duality on the same example. The

suboptimality of duality is clear on this picture, Actually the "optimal" values of cost are : duality : 4. regularization

: O,36

10

-2

!O-2

Remark 6.1 In the previous examples ~ in (6.1) has been taken near zero so that the opti~ mal state may fit z d

as well as possible.

Example 2 (Bio-chemical example of ~V). As an example 1 the problem has been considered completely symmetric with a unique control

v(t) so that boundary conditions a(o,t)

Figure 3 and F i ~

= a(1,t)

in (5,1) are

= v(t)~

4

give optimal control and corresponding optimal value of the state computed by regularization,

Figure 3 represents also values of optimal control computed for two

values of the regularization this example.

parameter g . The only active constraint is

v ~ O

in

273

Example 1

desired state

0.5

I I

"optimal state"(duality)

I f

0.0

!

0.5

0.25 Fisure |

0.75

1.

Time

Optimal state

Example 1

desired state 0.5

!

! ! I

•

I 0,0

"

O.0

!

.i

i

0.25

regularization

:~ualit~ I I

7,

'

0.5 Figure 2

Optimal state

I

0.75

I.

•Time

274

IIv(t)

A /\\

O.03

Example 2

",\,{ :

s=

10 - 5

!'\

0.02

~ O

~ S

O"

#|

it"

iI |

0.01

:11

/.

i!t

i

10 -1

// i 0.75

0.5

0.25

Figure 3

.Time

t

Optimal control

Example 2

s

=

I0-5

0.5

0.3

J O. 0

i

. ~

U--

_

'Optimal state"

....

/.,,,!i ............................ ~'~

o!~

! Fi.$ure 4

Optimal sta~.

o'~

~ime

275

Vll - REFERENCES

(i)

D. Begis H. Glowinski

(2)

H. Brezis

(3)

J

"Dual num. meth. for some variational problem..." in Techniques of optimization. Academic Press (1972). "Probl~mes unilat~raux") Journal of Math. pures et appliqu~es 51, (]972).

tea

R. Glowinski

"Minimisation de fonctionnelles non diff~rentiables", Proceedings of the Dundee Num. Anal. Symp. (1972).

J.C. Nedelec (4)

G. Duvaut

"Les in~quations en m~canique et en physique", Dunod Paris (1972)

J.L. Lions (5)

J.P. Kernevez

"Evolution et contr$1e des syst~mes bio-math~matiques" Thgse, Paris (1972).

(6)

J.L. Lions

"Variational Inequalities", Comm. on pure and app. math.

G. Stampacchia vol XX, pp. 439-519 (1967). (7)

F. Mignot

S~minaire Lions-Brezis

(8)

J.P. Yvon

Th~se Paris 1973.

Paris 1972-1973.

ON

DETERMINING

THE

OPTI~L

SUBMANIFOLDS

VALUE

SURFACE

OF

HAS

Harold

AN

L.

STATE

SPACE

INFINITE

WHERE

THE

DERIVATIVE

Stalford

Radar Analysis Stsff Radar Division Naval Resesrch Laboratory Washington, D.C. 20375

ABSTRACT The

problem

control

process

surface

often

out of

of

which

the

solving the

is

investigated.

the

first

one

state

the is

dimension

examples

are

the

an

In

infinite

of entire

in

less

than

provided

An

set.

cost

produces

optimal

general,

the

classes.

a

Second,

there remain

are

from

consists

cannot Our

be

which is

of

We

is

to

In

submanifolds for

of

the

establishing

shall

points

determine

state

with-

necessity

submsnifolds state

space.

of

the

condition.

infinite

optimal at

approaching

at

is

least

the

optimal the

paper of

such

the

value every

surface.

point.

In

distinct

surface tangents

not

state

smooth.

is to The

approaching

smooth. the third

tangent

class.

first

This

calculate terminal

three

the

one

latter

derivative

equstions

the

to

to the above

smooth belonging

this

to

values

all

where

space.

cost

surface

where

is

state

where

the

solving

the

certain derived

The

the

points

where

investigate

an

is

for

problems

call

points of

where

without

practice~

set

points

those

cost of

established

utility

initial

necessarily

of

the

but

of

optimal

we

not

consists

derivative

problem.

dition

surface

those

only

the

control any

the

surface

bounded.

desire

infinite

plotting

bounded

be

optimal

INTRODUCTION optimal

have

optimal

problem.

dimension

an

points

can

but

illustrate

of

condition

control

the

to

transfer

we

necessary

theorem,

solving

surface

First~

surface class

a

value

A

in

and

the at

submanifolds

of

Calculating

space An

objective

optimal

surface

practice,

optimal

proved

value

derivative

such

I.

the

optimal

space.

equations

condition

having Three

obtaining

possesses

submanifolds with

of

value entire will

presents

surface optimal

occur

has

along

a necessary

subm~nifolds.

an

control smooth con-

277 We

shall

which

the

now

define

ensuing

II. The their

family

that

in

are

have

needed

Q.

elements

These

U,

optimal

by

where

f: E n x E TM ~ E n is a c o n t i n u o u s

state

is d e s c r i b e d

u(t)),

Constraints

the set valued

u(t)

motion

are

four

a function

space

dynamical by

of

elements

behavior

means

of

the

E Em

function.

The e v o l u t i o n

in the state

systems

(that

is,

in the

f an explicit

t itself. with

the two elements

of all L e b e s g u e whose

functions

of

space X, an open

set c o n t a i n e d

is e q u ip p e d

intervals

on the control

the basic

values

have range

in ~ are given

U

: X ~ set of all compact

where

U is a c o n t i n u o u s

in

implicitly

is p r e c i s e l y

subsets

control

A

set value

function.

the set of control

~

entirely

in

val

tf).

~(tf)

of the d i f f e r e n t i a l

function

trajectory

[to, is

terminating

and given

: [to,

the

state An

tf] space

initial

~ E n is X

admissible

contained or

by

of E m,

For each state x,

values

available

the set

to the controller

at the state x. A solution

Q

measurable

function

(2)

U(x)

The

have

equation

space ~ is the space

of time t on bounded

and

is modeled

of ~ is time

of the process

by

seven

®)

set ® is a closed

For n o n - a u t o n o m o u s

The f u n c t i o n

functions

state

by a point m o v i n g

The c o n t r o l l e r

for

differential

processes

(X and

r e ( t ) E En,

of t), one c o m p o n e n t

and U.

E m.

the

The terminal

of X.

control

hereafter,

f in

The

subsequently.

process,

ordinary

described

E n.

sets

~(t)

closure

processes

investigation

of

state

optimal

described

under

systems

of

(1)

funct i o n

PROCESSES

space

go ) , two

function

of E n.

CONTROL

evolution

velocity

subset

control

processes

state

= f(m(t),

optimal

Euclidean such

and

are

control

OPTIN[AL

control

define

fo'

of

governed

their

to

(f,

an

optimal

n-dimensional

functions

of

OF

behavior

and

a point

family

holds.

A FAMILY

of

dynamical

equations

the

theory

final

in

the time

to

all

times

trajectory

is

for

(i) for some m e a s u r a b l e

conditions

said

terminal for

equation

set

0.

a terminating

be

is called

admissible t contained

said

to

The

time

be

admissible

tf

a trajectory.

if

it

in

the

lies

terminating is

called

trajectory.

interif the

278

The

time

have

to

it

is

tf

belongs

be

the

constrained

A control has

at

such

to

same to

be

function

least

that

the

one

fixed

by

U(~(t)) $ corresponds

~);

for

the

tf]

corresponding

of

[to,

time

u:[to,

u(t)E

trajectory

interval

terminating

terminal

~ Em

is

all

the

not

set

9.

to

be

in

control

necessarily

trajectories

unless

admissible

trajectory

t contained

to

does

said

admissible

for

tf

distinct

~:[to,

[to,

tf).

function

u

if tf)

it

~ X

Here,

the

if

t

~(t)

= ~(co)

f(~(T),

+ f

u(~)) dr

4o for

all

only

t contained

continuous,

necessarily Let sible ing

unique x ° be

control

each

function The

set

(3)

J(Xo,

tf]

~ E m having

C(Xo)

emanating

, let

T(Xo;U)

satisfying

to

be

to

the

be

u(t)

transferred

initial

time

specified.

~

u)

the

domain

function ~

denotes

the

In

is

space

f

o the

by

continuous of

is

from

set

the

to

are

of

be

not

of

one

the

set

corresponding for

all

admis-

terminat-

control

Xo,

all all

Lebesgue

the

of

of

function

terminating to

the

con-

t contained

in

the

of

Septuple

functions, measurable

go

the

final

the

ter-

time

tf

is

criterion

u(T)) dr

go

is

a real

valued

a neighborhood

valued

bounded

of

The

family

of

(f,

U,

go'

fo'

optimal X,

X

9,

J(xo,

the

and

open

in

with

the

~,

tra-

u)

where

processes f,

differentiable, is

set

transfer.

control ~)

continuously terminal

function

C(x o)

number with

continuously

controls,

to

real

associated

the

the

continuous

u belongs

u).

is

in X to

o

function [(Xo;

while

fo(~CT),

on

performance

a member

fixed,

function

a real

x ° contained

performance

+ftf

defined

control

a member value

summary,

represented are

where

function

E n x E m,

jectory

(I)

least

For

C U(~(t))

t o is The

= go(~(tf))

minimized

differentiable Q,

from

the at

x o.

denote

~ emanating

and

denote

from

required

u.

u:[to,

t is

function

functions

is

9;

necessarily

control

f is

equation

u.

state

not

since

C(x o)

u, of

that,

differential

Let

trajectories

domain

minal

for

Note

the

in X.

in

admissible

tf]. to

trajectory

u contained

the

[to,

contained

admissible

trol

in

solutions

E n,

U, ~

and

is

and is

3 is

fo the

a

279

closed

set

optimal

sontained

control

in

the

III. Let tion

be contained o be contained in

in T(xo; all

u*).

The

control in

If is

the

pair

for

function

for

from

x to

the

space

X.

the

Some placed

x,

on

the

F denote

this

family

of

V

: X~

of

the

optimal

The

Xo,

func-

contained

optimal

at

for

trajectories

is

then

all

x ° if,

for

satisfied:

the

If an

optimal

state

space

denotes

the

function V

V

is well

value

needed

value

disjoint

be

value

J(xo,

pair

(u*,

X then

we

~*,

u*)

~*)

have

a

is

optimal

transfer

called

defined

function

on

above

cost

the

optimal

the

entire

the

state

state space

surface. in

order

function

V.

A decomposition of

at

V(x)

optimal

are

be and

inequality

the

that

value

to

C(x o)

control

~*

E1

set. suppose

the

numbers:

value

We

Let

trajectory

u)

in

real

optimal

i.

~,

X.

the

said

in

V(Xo).

the

terminal

collection

is

optimal

be

the

space

following

is to

definitions

Definition able

~*)

into

A plot

called

Let

FUNCTION

let

~*)

x o contained

a state

function.

is

(u*,

X

value

state and

~ J(x o,

defined

from

Thus,

the

u*)

every

the

(u*,

u),

~*,

arbitrarily

exists

X.

VALUE

u contained

T(Xo;

J(x o,

in

C(x o)

pair

functions

contained

of

OPTIMAL

x

u*

closure

processes.

D of

to

the

subsets

describe

state

whose

an

space

X

is

X.

union

assumption

is

a denumer-

This

is

usually

Y

written

as

D = .~Xo,

Xj

: j E Jl

where

J is

a denumerable

index

set

for

l

the

members

Definition

of

D other

2.

decomposition

than

A regular

D

X o.

decomposition

o,.

Xj

: j

D of

such

the

that

state

is

space

open

and

X is dense

a in

I

X and

such

submanifold

that of

for

each

X ° is o p e n submanifold Let

to b e c o n t i n u o u s l y

is

a continuously

differentiable

3

Since

3.

X.

E n.

ferential Definition

j E J,

in X,

it

follows

of d i m e n s i o n

B be

a subset

differentisble

that

X ° is a c o n t i n u o u s l y

dif-

n of E n. of

E n.

on B

A mapping if

there

F

: B -

is s n o p e n

E 1 is s a i d set

W

280

containing

3

tinuously

such

We

are

now

unresolved optimal

in

control

physical

I.

X such

ferentiable

in

examples

There that

exists

value

in nature.

control m o d e l

is s h o w n

surface

submanifo!d

tion can fail its place.

from

contained

type suing

an

to have

an of

satisfied

by

control

models

D of the state

V is c o n t i n u o u s l y

control

processes

Vincent

dif-

space. we

introduce

it is readily

that

[5] p r e s e n t s

crops.

Since

is c o n t i n u -

problems

problem where

a tear or split

is

theory tears

subset

and

Therein,

extending

the o p t i m a l

along

the c o n t i n u i t y

another

satisfied

an

the o b j e c t i v e

a

assump-

assumption

to take

if the optimal

value

S

to

$

a

function

S

to

C

if

neighborhood

: C is

closed

~

~

of X whose closure o V when restricted set

C

of

differentiable

V 1 V1

let X. be a m e m b e r of the r e g u l a r J on the s u b m a n i f o l d X.. 3

open

function

from

E1

is

on

to C

in

the

topo~

S has

a

such

continu-

contains

e

and

~.

be

and

~

contains to

that

on

said

continuous

X

of

a continuous Vl(X)

=

extension

V(x)

for

911

S. an

optimal I.

in

the

satisfies

value

Take

of

Then

manner.

along

problems smooth

which

It in

continuous

and the

resulting

2.

is

scissors deform

The

Assumption

encompasses running

surface

a pair

surface.

differentiable that

an

continuously

Assumption cuts

exists

open

a

in

! is met

value V!

V1

Consider

smooth

is

is

family

optimal

in some control

eat or destroy

Let ~ be a point

There

X. and j optimal

Here,

have

D.

2.

that

ously

function

of m o s t

of the state

Assumption

extension

fies

con-

is continuous.

Assumption

V

value

to be satisfied,

decomposition

of

as

It

the

decomposition

For example,

that

Incidentally~

Suppose

such

for It

well

of an a g r i c u l t u r a l

value

ous

is

of D.

function

insects

the

which

assumption.

herein.

discontinuous

is to control

of

function

theory

as

a regular

the o p t i m a l

on the m e m b e r s

processes

that

a

the

control

considered

model

logy

to

describe

processes

It is, however~

function

to

optimal

constructed

The o p t i m a l

smooth

extended

W.

a position

ous.

optimal

be

on

processes.

Assumption space

F may

conjecture

hypothetically of

that

differentiable

make

surface surface

is

introduced

which

the

submanifolds

the

and

satis-

number

in is so

optimal of

a

a of

continuthe

that

general the

value state

of

en-

surfaces

space.

x

281 IV.

THE

FUNDAMENTAL

EQUATION Let such

(f,

that

D =

U,

its

{Xo, .

Xj

fo'

go'

X,

optimal : j C

OF ®,

value

Jl

be

PARTIAL DYNAMIC ~)

DIFFERENTIAL

PROGRAMMING

be

an

function

optimal

control

V satisfies

a regular

process

in

Assumption

decomposition

with

I.

which

F Let

V

satis-

!

lies

Assumption

partial the

I.

It

differential

open

equation

(4)

is

shown

equation

and

dense

member

is

written

as

holds

for

This where

all

x ~ X

equation

the

will

optimal

D.

For

convenience,

manifold

be

some

a unit

X.. Let 3 row vector

denote

the

function

Definition at

~

4.

in

n-i

the

that

is of

The

must D

of

be

X.

met

on

This

in

determining

points

smooth.

STATEMENT

the

of

the

continuously a point

to

regular

M

at

differentiable

on

~.

decomposition

M

and

Finally,

sub-

let

N(~)

denote

let

N(~) T

N(~).

value

direction

Grad

fundamental

• f(x,v)}

not

~ denote normal

the

programming

member

denote

optimal

normal

limit

M

state

that

subsequently

dimensional let

[4]

decomposition

is

PROBLEM

the

transpose

dynamic the

+ G r a d V(x)

V. Xj

Stalford

. o be utilized

value

Let

of

X o of

0 = MINIMUM { f o ( X , V ) vcv(~

and

in

V(~

function

N(~)

+ h

if

. N(~))

V has

the

an

infinite

derivative

limit

- N(~) T

h~o + cannot

be

positive

bounded.

Suppose an

Recall

tive

that

infinite

at of

Problem.

the

notation

optimal

derivative

that

tions

The

h~o + denotes

that

h

takes

on

each

point

only

values.

an

each the

n-i of

in

at

value

function

least

one

dimensional its

points.

submanifold Vectorially,

of

V has the of one

at

normal E n has is

of

directions two

normal

course

of

M

to

M.

direc-

the

nega-

other.

Determine

the

equation

o f the

submanifold

M without

solving

282

first

the

optimal

control

problem

for

optimal

control

feedback

policies.

VI.

a.

ORTHOGONAL

Let

T(~)

denote

We

desire

to

nates

such

vector

by

(5)

y :

where

K(~)

The chosen

the

new

the

tangential

matrix

K(~) is

the

normal

orthogonal

vectors

linearly

system

vectors

COORDINATES

into

coincides

T(~).

Such

with

to

new the

M at

coordi-

normal

a transformation

is

equation

matrix

vector the

(7)

K(~)

holds

where

K(~) T

is

gonal

transformation

(8)

tangential x coordinates

coordinate

T(~),

that

, K(~) T =

K(~) T

equal

Equation

OF

, x

vector

such

of the

and

the

tangential

a matrix

transform

that

N(a)

given

TRANSFORMATION

to

(4)

is the

can

0 = MINIMUM

composed that

N(~)

of

the

normal

vector

N(~)

and

the

is,

and

the

tangential

vectors

T(ff)

can

be

Equation

(7)

implies

Equation

(5)

is

equation

Identity

the

Matrix

transpose

inverse

of

of

coordinates.

be

rewritten

of

K(~).

K(~).

Thus

an

that ortho-

as

i fo(X'V) + Grad V(x).K(e)T.K(~).f(x,v)}

v C U(x) where ing (9)

x E Xo .

zero

terms

Substituting

0 = MINI~T~ ~ l f o ( X ~ v) v C U(x) + [Grad

for all x C Xo~

Equation

(6)

into

Equation

(8)

and delet-

we o b t a i n + [Grad

V(x).N(~)T]-[N(~).f(x,

V(x).T(~)T].[T(~).f(x,

v)]}

v)]

283

VII.

Let

I Xkl

The

following

and

2.

be

I.

[Grad

This face

in

vation

In

the

is

8.2.1

case

not

to

the

invoking

x k converges

the

to

to

M

slopes

at

Stalford

the

case,

by

~,

state

~.

Assumptions

the

(i x

1

n-l)

bounded.

of

that

the

converges

verified

that

direction

Lemma

is

as

states

tangent

for

T]

X O that

be

limit

V(x k).T(~)

the

this

in can

the

observation

approached.

When

sequence

observation

Observation matrix

any

THEOREM

the

~ remain [3,

optimal

of

p.

finite

84]

value

Assumption

optimal as

&

asserts

function

2 can

be

value

used

sur-

is

this

obser-

is

continuous.

to

amend

the

lemma.

Let

C(~)

be

{N ( ~ ) . f ( c ~ , Note

that

since when

we

number

is

speak

1.

value

is

closure

of

the

set

: v ~ U(~)}.

composed n)

of

a

(i x

of

the

zero

is

a point

If

~

function

N(~),

then

dary

v)

convex

scalars

vector

and

vector

or f(~,

in

one v)

C(~)

dimensional is

a

we,

(n x

in

vectors i)

vector.

essence,

Belo~

mean

the

real

zero.

Theorem

of

Proof. the

C(~)

N(~)

the

has

it

is

an

of

infinite

necessary

the

submanifold

derivative

that

the

in

zero

M where the

vector

the

normal

optimal

direction

belongs

to

the

boun-

C(~).

We

zero

prove

vector

the

theorem

belongs

by

either

showing to

the

that

a contradiction

interior

or

the

arises

exterior

if

of

c(~). Suppose 5

be

some

interior

(I0)

that positive of

C(~).

the

zero

vector

number Let

xk = ~

such

I Xkl

+ hk.N(~

be

)

belongs that

5 and

the

sequence

to -8

the

interior

are defined

contained by

of

C(~). in

the

Let

284 where

the

sequence

to

infinity.

an

integer

~hk~

Since K such

contained

in

in

if

convex

In particular~

and

k ~ K then

v)

of

k

the

z K there

exist such

N(~)~f(xk~

Vl(Xk))

<

(12)

N ( ~ ) ' f ( x k,

v2(xk))

> 5.

the

sequence

(13)

IR(Xk) I of

R(x k)

for all and

=

xk contained

I S2(Xk)

~ be

Grad

real

controls

real

numbers

Similarly,

(15)

S2(Xk)

= N(~).f(xk,

v2(xk)),

the

sequence

and

(12),

IS2(Xk).R(Xk)} minimum

of

be

for

zero

terms

in

~.

The

and

since

bounded tion large

view

see

Definition

~ cannot

that

cannot the

of

I R(Xk) we

be

one

expression

in

first U(x) since~

1 holds. negative

the

curly

close

to

is bounded

since

the

for

each

compact

in

addition,

We

have

values

of

Equation

longing

to

the

bounded

the

already for

(9)

interior

and

are

v2(x k)

by

sequences

~Sl(Xk) }

brackets This

for

all

functions

C(~).

Next,

close

we

Equations

that

2rid U are

of show

to the that

~.

cannot

first

two

close

to

continuous

The

third

the

(9) the

continuous the

imply

1 and

Equation

because

in X.

f is

falsehood

invoking

x k sufficiently fo

that

(13)

implies

of is

x contained

function

and

I Sl(Xk)'R(Xk) This

~.

remarked

the

(I0)

Thus,

below.

x k sufficiently

proves of

Equations

sequences

from

are

diction

defined

the

bounded.

the

expression

is

4~

be of

bounded

x k sufficiently

that

-5

by

Vl(Xk))

(ii)

Vl(X k)

be

let

= N(~)'f(xk~

that

5 and

exists

-5

Sl(Xk)

In

numbers

k goes

there

that

(14)

respectively.

as

V(Xk).N(~)T

i n t Xk},

defined

zero

set

set

U(x k)

to

continuous,

.

(11)

Let

U are

the

: v C U(X k)

control

converges

f and

closure

for

the

positive

functions

that

the

N(o~).~(Xk~

contained

is

the

second and

is

Observa-

term

takes

This

contra-

zero the

vector zero

on

bevector

285

cannot

belong

to

Suppose Since

that

the

the

exterior

the

zero

function

f is

is

a compact

subset interval

Since

zero

the

both

Equation numbers

for

terms

in

takes

in

we

have

In conclusion,

The

of

equations tion set

of

in

the

is

to

to one

being the

C(~),

As

the

either

the

or

the

the

validity

in

negative

first

two

close

of of

is

c are

term

large

before,

C(~)

b and

third

x k sufficiently

boundary

set

endpoints.

all

the

the

I.

has

An

the

one-dimensional

[I,

p.

48]

travels

in

and

the

the

sought

to ~.

Equation

C(~)

(9).

and

our

intersection

M.

example

a straight

line

controlled

by

the

rocket

to

the

[2,

in

the

family

it

previous

is

equa-

terminal that

also

sections

space

of

the

the

knowledge

leave

state

section determines

Thus, with

the

than

As of

traa factor

the

where

the

suboptimal

derivative.

to

time-optimal

rocket

rather

this

The

sets.

Employing

points

antique

the

M.

terminal

set

in

condition

submanifold

of

infinite

Pontryagin

applied

an

has

studied.

the

is

necessary

that

submanifold

at

the

a family

an

rest

theorem

example

terminal

M designates function

Example

set

APPLICATIONS

of

for

the

example enter

determining

value

C(~).

the

the

numbers

~.

for

belong

each

corresponds

manifold

compact,

Thus

contradicted

In

equations

selected

jectories

to

bounded

condition

examples.

a family

of

proved.

necessary

three

of

positive

VIII.

to

is

Therefore,

exterior

close

must

exterior

c represent

negative.

large

are

again

zero

numbers.

the

only

(9)

the

U(~)

b and

both

x k sufficiently

Therefore,

is

is

to

and

real where

are

on

belongs

: v E U(~)}

the c]

they

Equation

theorem

of [b,

or

(9)

v)

vector

positive

C(~).

continuous

{ N(=).f(~, a compact

of

vector

illustrate regulator

p.

toward a thrust terminal

23].

the process

In this

a terminal program, set

theory of

is

set.

by

Leitmann

example,

the

given

a rocket

~ith

the

objective

is

and

render

the

mass

moving

horizontally

motion to

transfer

of

bring time

a

minimum. The a flat

equation earth

of

states

motion that

of the

a point acceleration

is

equal

to

the

thrust

above value.

286 In state equation form, we have

(16)

Xl = x2

where

xI

x2

the

is

£2

=

is

the

The

(Xl,

x 2)

go we

origin. in

the

to

an

condition

of

accomplished

component

theorem have

by

employing

obtain

equation

serves

The

is

state

the

function

time x 2)

=

set, the

and

the

the

two-dimensional

space

I]

rocket

X

is

the

criterion,

an

[-i,

terminal

v represents

fo

from

the

(~i~

the

is

for

two-

the

identically

initial

one

state

all

to

the

see

if

is

it

(Xl,

to

x 2)

at And,

the

con-

those

passing

contained

in

points is

Consider

we

in

where

the

optimal

point.

This

an

sought

the

Therefore,

infinite

orthogonal

vais

the

deriva-

determine

points

the

an

from that

Theorem.

to

the

select

coordinates

the

apply

necessary

Y2 ) such

sufficient

through X.

of (YI'

optimal to

the

for that

policies,

the is

check

change

direction

this

where

specially, and

st

coordinates

feedback

approach

possible

derivative

orthogonal

direction

submanifold

~ore space

normal

space Our

theorem.

new

control

state

state

an

x 2)

the

the

infinite

exist.

~2 ) be

of

optimal

derivative.

to

an

as

a normal possibly of

Let

(Xl,

Y2

of

to

coordinates

may

unity set

performance

for

points

(~i , ~2 ) in

the

function

tive

U(Xl,

infinite

condition

point

we

and

example

those

has

necessary

old

to

the

transfer

that

this

find

surface

arbitrary

lue

zero the

note

In

the

value

terminal

x 2 = 0.

E 2.

identically

solving

desire

value

the

to

control

normalized

x I = 0 and space

relative

the

X.

Without we

is

Here,

minimizing

Finally,

tained

rocket

and

thrust

with

is

are

the

rocket

Euclidean

function

of

the

reversible.

dimensional

since

of

maximum

are

point

position

speed

thrust. engines

-I ~ v ~ i

v

the for. transfor-

mation

[ cos(8) sin(e)]

where

8

submanifold

is

an M

angle having

yet at

unspecified. (GI,

N ( ~ I , ~2 ) = [ - s i n ( 8 )

G2 ) the

Suppose normal

that

(~i'

~2 ) lies

on

vector

cos(e)]

such that the optimal value function has an infinitive derivative at

a

287

all

points

of

M.

N(~I' where

that

I-~

If ~2

equals

is

sin(e)

N(~I

' ~2'

According

zero

(18)

product

~2)'f(~l

-l~v~l.

such

The

v)

to

in

+ v cos(e)

zero

then

(18)

a non-zero

this

if

~ satisfies

(19)

tan(e)

The are

angle

rotated

M at

the

can

be

the

equation M

e is

(~i'

(~i'

is

contained

M

through

new

coordinate

The

new

of

the

if

in

given

by

which

angle

e = e(~ I, ~2 )

set

and

the

the

Y2

is

coordinate M

a functional

is

only

if

boundary

of

the

Set

is

Yl

as

coordinates to

is,

a curve

relationship ~2

old

normal

in

(Xl, x~ submanifold

therefore, which

between

a function

the

tangent

locally,

~i

and

it

~2"

of

~I'

then

the

(20)

and

integrating,

If

slope

by d~ 2 d~ 1

Substituting the

an

O) = ± 4 / 2 .

submanifold

(20)

obtain

seek

is met

angle

~2 )" The

as

of

given

to

equation

the

if2 )"

expressed

is

the

that

point

M at

reduces

+ v cos(e)

boundary

condition

zero

the

we

v)

= -+ I/~ 2.

so

to

to

~2'

sin(e)

theorem the

~2

: -i -< v ~ I 1

e(~l, For

=-~2

the

contained

, ~2).f(~l,

Equation

tan(~).

(19)

into

Equation

we

solutions 2

(21)

~(~2 )

= -~i

(22)

½(c~2 ) 2

= c~1 + c 2

for

the

We

derived

equation

theorem. the

Thus,

optimal

sstisfy

of

these

these

if

value

M.

equations there function

equetions.

Here,

c I and

by are

c 2 are

constants

of

the

necessary

condition

utilizing submanifolds

has And,

+ Cl

an

of

infinite

indeed,

for

the

state

derivative terminel

integration.

space then

of on

they

conditions

the

which must of

288

el

= ~2

minal

=

0

set

the

together

rather

terminal

invoking

leave

conditions

trajectories equations

with

than

of

enter

the

the

sought

for

(24)

~(~2 ) The optimal Its

value

= e2

=

set

0

imply

(0,0)

enter

that

M.

cI =

rather

are

the

submanifold

than

reduced

c2 = leave

terThat

is,

O.

Since

it,

the

to

~2 ~ O.

function is

the

~2 > 0

= al'

derivative

trajectories

obtain

submanifold

½(~2 ) 2 = - ~ l '

[4].

~I

that we

terminal

(23)

2

it,

for

easily

this

example

obtained

since

is

given

therein

in

the

Stalford

optimal

value f u n c t i o n is g i v e n in a n a l y t i c a l form.

The d e r i v a t i v e is infi-

nite only at the points defined by E q u a t i o n s

(23) and

Example 2.

As a second example,

(24).

consider the r e c t i l i n e a r m o t i o n of a

rocket o p e r a t i n g at constant power.

In L e i t m a n n

[i, p. 29],

the equa-

tions of m o t i o n are given as

(25)

Xl

= v

-i ~ v ~ (26)

i2 =

The

terminal

the

rocket

transfer while low

set from

time fo

the

According

to

and the

the

(27) only

possible

Integrating

~2

c

submanifold

is

a

=

of

is

the

8 = (20)

function

state

space

orthogonal

component seek

an

is

set

to

and

go

is

identically

X

is

the

half

transformation be

angle

the

normal

~ such

that

transfer

render

the zero

space given

in

direction zero

is

be-

to

M.

con-

set

+ v 2 cos(e) if

objective

terminal

: -1 ~ v ~ 1 1

0 at

each

with

% =

point

of

0,

obtain

we

M.

c

constant M

the

The the

the The

Y2 we

Equation

(28)

where

the

theorem,

sin(9)

to

Thus,

Consider let

(0,0).

state

one.

boundary

t-v is

origin

identically

(17)

This

the initial

x I - axis.

in

v 2.

a minimum.

is

Equation

tained

is an

I

of

a straigh~

integration. line

parallel

Equation to

128) the

implies

x I - axis.

that

the

Applying

289

the

terminal

constant In (0,0)

be

this

any

the O.

it

is

M with

x 2 - coordinate

therefore on

M

function

is

Example

3.

pest

management

tors

are

[5]

particular,

to

introduced state

is

to

the

to

change

of

(30)

i 2 = x 2(x I - i)

x I represents

tolerable

actual

number

value to

of

of

minimize

the the

to state

cost

of

integral

secticide

models

before,

Equation

(17)

a point

zero

is

of

we and M,

contained

31> An

M.

the

of

crops an

no

value This

However, of

models

where

and by

preda-

insects.

insecticide

of

as

optimal

several

In

spray

biological

growth

be-

control

insecticides

where

is

control

the

is

predators.

are

the

o ~ v ~ 1

actual

The a desired

amount

of

system

number

state

x2

level

the

insect

the

of

insecticide to

of

is

pests

ratio

them.

of

The

used.

It

equilibrium

and

the

control is

point

desired (i,i)

and

I +

the

5v)

cost

d~.

associated

with

crop

loss

and

the

in-

used.

As

is

the

theorem.

to

set

This

criterion

~tf(5x

This

x I - axis.

terminal

derivative

done

and

integration the

contain

control

a program

of

the

not

where

- v Xl,

and

the

the

the

the

damage

pests.

predators

v corresponds

transfer

ratio

such

of

M,

population

- x2)

the

level

scope

used

model

Xl

a

for

does

agriculture

(29)

Here,

= Xl(l

only X

optimal

natural

the

(0,0).

unbounded.

is

the

of

the

given

of

reach

become

predators

equations

to

space

the

the with

exception

to

in

determine

coincides

normal

minimize

a model

nonharmful

we

non-increasing

presents

programs

utilized

0, M

the

outside

indeed

Vincent

=

possible

state

approached does

= ~2

not

the lies

is

~i

submanifold

on

Interestingly,

a point

The

The

point

example

value

of

zero.

example,

from

cause v =

conditions

to

inspection

consider let an

the

angle

in

the

- v of

Set

(31)

the Y2 ~ =

orthogonal

transformation

- component 0(~i,

boundary

sin(+ reveals

be

~2 ) is of

the

+ that

normal to

be

to

given M.

If

determined

in (~I'

such

~2 ) that

set

2+l-1) 8 must

cos + be

: o

a solution

v

l} of

one

of

290

the

equations ~2(e I - I)

(32)

(33)

tan(@)

=

~i(i

tan(e)

=

(I - ~ i ) /

Substituting

Equation

and

the

applying

(34)

"2 )

(32)

into

terminal

i.

Equation

conditions

(20),

of

~I

integrating

= ~2

=

i~

the

we

result

obtain

~ ( ~ 2 ) + ~ ( ~ I ) = ~ i - 1 + ~2 - I. The

(35)

inequality

~. x <

implies

that

(x-i)~

(~i'

x ~ <0,

~2 ) =

(i,

I) U (I~ ~)

I)

is

the

only

point

satisfying

Equation

(34). Substituting result the

and

e2

Equation

(33)

into

terminal

= ~n(~l ) - ~I resulted

Implementation state

Equation

(20)~

conditions

of

~i

integrating = ~2

the

= I,

we

have

terminal

this

of

set ~I

is

(i,i) e i is

indeed

infinite

value

function But

the

the

+ 2.

from

this

x I decreasing

straint

~.

(33)

the

equation

(36)

an

Equation

applying

in

set

than

leave

states

is,

addition,

value

into

function

out

the

(29)

(36)

optimal

value

Vincent

[5],

across

the

Assumption

in

enter

with

in

v = i.

results

submanifold

the

satisfy

value

trajectories

Equation

discontinuous does

control

Equation

of

where

pointed

the

since

it,

equation

As

optimal

for

Therefore,

derived of

(31)

value

time°

derivative. in

Set

control

rather the

the

the M.

con-

And,

function the

the the

optimal

submanifold 2.

has

291

REFERENCES [i].

Leitmann,

[2].

Pontryagin,

[3].

Stalford,

G.,

AN

INTRODUCTION

TO

OPTIMAL

CONTROL,

McGraw

Hill,

(1966). L.

PROCESSING,

of H.,

of Optimal

[5]

.

"An

and

Society,

October

Vincent,

T.

Theory,"

13th

Automatic California,

L.,

"Pest

August

1972,

Systems,

Management

Council, 16-18,

Nan

Washington,

Automatic

Stanford 1972.

and

Center,

1970.

1972 and

D.

Programs Control

the

OPTIMAL

Control

Research Theorem

of

OF

(1962).

California,

Proc.

IEEE

York,

in Optimal

Optimality

Processes,"

THEORY

Operations

Berkeley,

Society,

Control

70-13,

Equivalency

Joint

New

Conditions

ORC

9-12,

MATHEMATICAL

Publishers,

California,

Control

Cybernetics

THE

"Sufficiency Games,"

University Stalford,

et al.,

Interscience H.,

Differential [4].

S.,

Over

a Family

Int.

Conf.

on

Cybernetics

C.

Via

Conference University,

Optimal of

Control the

Stanford,

American

CONTROL OF AFFINE SYSTEMS WITH MEMORY

M.C. Delfour*
and

S.K. Mitter** (Massachusetts

i.

Institute of Technology)

INTRODUCTION

In this paper we present a number of results related to control and estimation problems

for affine systems with memory.

The systems we consider are

typically described by linear functional differential integro-differential

equations or Volterra

equations.

Our results may be divided into four categories: (i} (ii) (iii) (iv)

State-space

description

of systems with memory.

Feedback solution of the

finite-time quadratic

Feedback solution of the infinite-time Optimal

quadratic

cost problem. cost problem°

linear filtering.

The .main difficulty

in the study of the s y s t e ~

considered in this paper

The work of the first author was supported in part by National Research Council of Canada Grant A-8730 at the Centre de Recherehes Math6matiques, Universite de M~ntreal~ Montr~ai 101, Quebec, Canada. The work of the second author was supported by AFOSR Grant 72-2273, NSF Grant GK-25781 and NASA Grant NGL-22-009-124 all at the Electronic Systems Laboratory, M.I.T., Cambridge, Mass. 02139.

293

is that the state spaces involved are infinite dimensional

and that the equations

describing the evolution of the state involve unbounded operators. propriate

function space is chosen for the state space a fairly complete theory

for the control and estimation problems

2.

Once an ap-

for such systems can be given.

Affine Systems with Memory

In this p a p e r we shall consider two typical systems:

one with a fixed

memory and one with a time varying memory. Let X be the evolution space and U be the control space. X and U are finite-dimensional

2.1.

Euclidean

We assume that

spaces.

Constant Memory Given an integer N ~ 1 and real numbers - a = QN < "'" < 01 < 0

o

= 0 and

T > 0, let the system with constant memory be described by:

dx

N

(t) =

A

(t)x(t) oo

+

[ A.(t)x(t+G.) i= 1 l 1

f o +

Aol (t,~)x(t+0)dQ + f(t) -a

(1)

+

x(0)

=

h(0)

B(t)v(t)

,

in

[0,T]

-a _< Q < 0,

where Aoo , Ai, Ao! and B are strongly measurable and v C L2(0,T;

and bounded,

f £ L2(0,T;

X)

U).

We first need to choose an appropriate priate state space.

space of initial data and an appro-

It was shown in DELFOUR-MITTER

[i],

[2], that this can

294

indeed be done provided

dx d-~

=

that

(1) is rewritten

Aoo(t)x(t)

N I

+

in the following

I x(t+Oi)

{2)

I h I (t+Q i) , otherwise

o

h I (t+@)

"-a

+

x(O)

=

f(t)

+

B(t)v(t)

,

,

othezwise

in

[0,T],

h O"

We can pick initial where the solution

' t+0i>0

Ai(t)

i=l

!

form:

of

data h = (h°,h ~) in the product

space X x L2(-a,O;X),

(2) is x : [0,T] + X.

We can now define the state at time t as an element

x(t) of X x L2(-a;0;X)

as follows:

(3)

i

x(t} O

=

x(t)

}~{t)~{e)

=

.! x(t+0)

, t+e_>0

[

,

t

For additional

details

fixed duration

[-a~0].

2.2.

T_i,me Varying

Consider

(t+0)

see DELFOUR-MITTER

Memory

the system

otherwise.

[i],

[2].

System

(I) has a memory

of

295

t

~ (t) =

i

A (t)x(t) + o

A (t,r)x(r)dr I

o

+ f(t)

(4)

x(0)

=

+

B(t)v(t)

, in [0,T]

h O in X,

Where Ao, A I and B are strongly measurable

and bour,ded.

If we change the variable

r to G = r-t and define

Ii

O

(t)

:

A (t) o

(5)

I A (t,t+@) !

,

-t < @ < O,

,

-~<

\ oi (t,@) 0

equation

@<

t,

(4) can be rewritten in the form 0

d"t" : %oCt,xct,÷ I ~c,

% ct,o~ i xct÷01 , t÷e>0

-~

(6)

+ f(t) + B(t)v(t)

x(0)

with h I = 0.

=

h°

i h I (t+0)

in

1dO

, otherwise

[0,T]

in X, h I in L2(-~,0;X),

In this form equation

(6) is similar to equation

(2).

However

here we consider the system to have a memory of infinite duration in order to accomodate the growing memory duration to be the product X x L2(-~,0;X). X x L2(-~,0;X)

which is defined as

[-t,0].

The state space will be chosen

The state at time t is an element x(t) of

296

I

~(t)° =

(7)

x(t)

i

x(t+@) . -t < @ < 0

[ h l(t+C))

3.

, _oo < C) < - t

State Equation

It will be more convenient to work with an evolution equation for the state of the system rather than equations

(i) or (4).

state evolution equation corresponding to equation

(S)

I

H

=

X x L2(-a~0;X)

IV

=

{(h(0)~h)

In order to obtain the

{1) let

! h ~ HI (-a,0;X) }.

The injection of V into H is continuous and V is dense in H. its dual.

Then if V ~ denotes the dual space of V, we have

VF

H~V

~

This is the framework utilized by Lions tions.

We identify H with

Define the unbouxlded operator

(cf. J.L. LIONS) to study evolution equaA(t):

V "+ H by, o

I\(~(t)h)° = A°°

(t)h(0) +

(9) (A(t)h) i (Q)

=

and the bounded operator

B(t):

U

+ H by

~(@),

Z i=1

Ai{t)h(0 i) +

]-a

Aol(t,Q)h((9)dQ

297

(B (t)u) O (i0)

=

B(t)u

I

(B (t)u) I (0)

=

0

and f(t) S H by

(ii)

f(t) °

=

f(t),

f(t)l =

0.

Then for all h in V, it can be shown that x is the unique solution in

(12)

W(0,T) =

{z E L2(0,T;V)

I D z 6 L2(0,T;H)}

[D denotes the distributional derivative]

of

l d.~ t

=

A(t)x(t) + B(t)u(t)

~(o)

=

h.

+ f(t) in [0,T]

(13)

Similarly in the case of equation

(4) we let

X x L2(-~,0;X)

{(h(0),h)

I h ~ HI(-~,0;X)}.

We again have

VCHCV'. We now define A(t):

(A (t)h)

V ~ H as follows:

=

A

(A(t)h) I (0)

=

oo

(t)h (0)

(15)

a~-(O),

+

I OAol (t,0)h (0)dO

298

B(t) and f(t) be as defined in {6) and (7). For all h in V, x is the unique solution in

(16)

W(0,T)

=

{z £ L 2 (0,T;V)

I

D z ~ L 2 (0,T;H) }

I dx " t '

=

A(t)x(t) + B(t)v(t) + f(t), in

=

h.

of [0,T],

(17)

4.

Optimal Control Problem in [0,T]

We now consider a quadratic cost function,

~(vm)

=

<~ ~(T), ~(T))H - 2,,},x(T))H IT

(18)

[ (Q (t) x (t) ,

x(t) )H

-- 2(q(t)~ x(t))H]dt

]o [ T (N(t)v(t), v(t))udt,

I0 where L E

~(H)~

/ E H, q E L 2 (0,T;H) and Q:

are strongly measurable and bounded.

[0,T] ÷

~(H) and N:

[0,T] +

~(U)

Moreover L, Q(t) and Nit) are self adjoint

and positive and there exists c > 0 such that

(19)

Vt~ Vu,

(N(t)u,u) U > 0.

For this problem we know that given h in V, there exists a unique optimal control function u in L2(0,T;U) which minimizes J(v,h) over all v in L2(0,T;U). this optimal control can be synthesized via the feedback law

-i

(20)

u(t)

=

--N(t)

~

B(t)* [~(t)x(t) + r(t)],

Moreover

299

where ~ and r are characterized by the following equations:

(21)

(22)

i

d~ (t

=

A(t)*Z(t) + w(t)A(t) --~(t)R(t)~(t) + Q(t) = 0, in [0,T]

i~(T) R(t)

=

B (t)*N (t)B(t)

+

[A(t)

=

A

I

[~(t)

and ~dr( t,)

(23)

I

~r(T)

R(t)~(t)]*r(t) + [~(t)f(t) + q(t)] = 0, in [0,T]

Here a solution of (21) is a map 7:

[0,T] ÷

~(H) which is weakly continuous

such that for all h and k in V the map t ÷ (h,~(t)k) H is in HI(0,T;R); a solution of (23) is a map r:

[0,T] + H such that r E L2(0,T;H) and D r ~ L2(O,T;V').

For details see DELFOUR-MITTER [3] and BENSOUSSAN-DELFOUR-MITTER

5.

[i].

Optimal Control Problem in [0,~] We can also give a complete theory for cost functions of the form

(24)

J (v,h) =

[(Ox(t), x(t)) H + (Nv(t),v(t))u]dt O

with the following hypothesis:

i)

E,c/~(H), N E ~(U) are positive and self adjoint and there exists

c > 0 such that vu

2)

(Nu,u) U > c

lull;

X is the solution of I dx.t.

Ax(t)

+

BY(t)

in

[0,~]

(25) x(0)

3)

=

h

in

V;

(Stabilizability hypothesis) there exists a feedback operator

G E ~(V,U) of the form (26)

Gh

=

Gooh(0)

+

j=l~ Gih(0i) +

-

Gol (0)h(G)d@

300

such that the closed loop system

i

(27)

da(t)

=

!x(0) be LZ-stable,

=

[A + B G]x(t)

in

[0, ~]

h ~ V

that is [ co

(28)

I

%r h E H,

iI~t)il ~ dt <

~

)o For a study of the stabilizability When system

(25) is stabilizable,

w h i c h mini~nzizes J ~ optimal

u(t)

= -- N

via a constant

a unique u in L21oc(0,~;U)

for a given h.

feedback

[2].

Moreover this

law.

Riccati equation

~ * ~ - ~ + 5 = 0

A solution of

(30) is a positive

(30) is verified as an equation in decomposed

[i]

B* ~ x(t),

where 7[ is a solution of the algebraic

(30)

there exists

(v,h) over all v in L21oc(0,~;U)

u can be synthesized (29)

p r o b l e m see VANDEVENNE

self adjoint elemena of

~(V,V').

The operator 7[ in

ff~(H) such that ~(H)

into a matrix o f operators

ol 7[ (since H is either X x L2( - a,0;X) o r X x L2(--~,0;X))

i T oo

E $Z~(X)

,

~

o1

E/~(L2(-- a,0;X)

711o E ~(X,L~ (- a,0;X))

where

X)

~ 7[ii C ~(L~( - a~0;X)).

Moreover '

*~

~OO *

=

+ ~

(0)

+ ~

(0)*

+ Q -~

R~

,

t~ -~ ~[lO(O:): [-- a,O] +

=

0

7[ > 0 OO

~ l o h°) (0~) = T[lO(~)hO

c~(X)

can be

301 ~dn I----~O de (a) = ~

I0

(a) [A

O0

-- R~oo]

N- 1 2 Ai*Woo~ (~-@)

+

i= l

+ AOI(a)*WOO + W11(e'0)

lo (- a)

=

, a.e.

in [-- a,0]

~,~'*~oo

f o (w

h I Ca)

=

w

Ol

Ca)*h I (a)d~ 10

-a

I (W11hl)(a) = IiW*l(e'8)h1(~)d~ (a,8)~ ÷~Wl1(e,8): [--a.O]x,[-a.O],÷ /~(X) [~ ~] ~ (a,8)--Ao1(a)~o(B)+ ~1o(a)Aol C8)

+

+

N-1 2 Ai*W i= 1

IO 11 (- a,~)

=

~'*Zlo(~)*

(a,B)

=

~

~11

I1

io

N-I 2 ~ I0 (~)Aj~ (~-@i) (S) *~ (a-Q i ) + j=l

IO

, z II (a,-a)

=

~ Io (a)A's

(S,a)*.

Under additional hypothesis on A and Q we can also describe the asymptotic behaviour of the closed loop system

in [0, ~] (31) x(0) Definition

=

h

in

V.

Given a Hilbert space of observations Y and an observer M e~(H,Y),

System (25) is said to be observable by M if each initial datum h at time 0 can be determined from a knowledge of v in L21oc(0,~;U ) and the observation

(32)

z(t)

=

M x(t)

in

[0,~].

When System (15) is observable by Q I/2, for each initial datum h

302

(33)

x(t) + 0

as

t + ~,

where x is the solution of the closed loop system

(31)o

In the special case where

and Qoo is positive definite,

the closed loop system

(21) is L2-stable.

For

further details see DELFOUR-MCCALLA-MITTER. 6.

Optimal Linear Filterinq and Dualit Y Let E and F be two Hilbert spaces.

~(dx't') = (34)

~

% o (t)x(t) + +

I~

B(t)~(t)

We consider the system

~ Ai(tlx(t+0i) i=I + fit)

X(0)

=

h O + ~o

x(@)

=

hl(@) + ~I(0)

,

+

a Aol (t,0)x(t+0)d0

,

- a < @ < 0,

where ~ = (~o ~l) is the noise in the initial datum, and ~ is the input noise with values in F. (35)

We assume an observation of the form (with values in E) z(t)

=

C(t)x(t)

+ n(t),

where ~ represents the error in measurement BENSOUSSAN

and C(t) belongs to

[i] {~°,~l,~,n} will be modelled as a Gaussian

~.(X,E).

AS in

linear random functional

on the Hilbert space. (36)

~

=

X x L2( - a,0;X)

x L~(0,T;E)

x LZ(0,T;F)

with zero mean and covariance operator

P ~

(¢)

For each T we want to determine the best estimator of the linear random functional x(T) with respect to the linear random functional z(s), 0 _< s < T. to this problem see BENSOUSSAN

[2] and BENSOUSSAN-DELFOUR-MITTER

For the solution

[2].

303

References

A. BENSOUSSAN [1] FiltrageOptimal des Syst~mes Line'ires, Dunod, Paris 1971. [2], Filtrage Optimal des syst6mes Llnealres " ~" ......avec retard, I.R.I.A. report INF 7118/71027, Oct. 1971. A. BENSOUSSAN, M.C. DELFOUR and S.K. MITTER [1] Topics in S ~ e m Infinite Dimensional Spaces, forthcoming monograph.

Theory in

A. BENSOUSSAN, M.C. DELFOUR and S.K. MITTER [2] Optimal Filtering for Linear Stochastic Hereditary Differential Systems, Proc. 1972 IEEE Conference on Decision and Control, New Orleans, Louisiana, U.S.A., Dec. 13-15, 1972. M.C. DELFOUR and S.K. MITTER [1] Hereditary Differential Systems with Constant Delays, I - General Case, J. Differential Equations, 12 (1972), 213-235. [2], Hereditary Differential Systems with Constant Delays, II - A Class of Affine Systems and the Adjoint Problem. To appear in J. Differential Equations. [3], Controllability, Observability and optimal Feedback Control of Hereditary Differential Systems, SIAM J. Control, i0 (1972), 298-328. M.C. DELFOUR, C. McCALLA and S.K. MITTER, Stability and the Infinite-Time Quadratic Cost Problem for Linear Hereditary Differential Systems, C.R.M. Report 273, Centre de Recherches Mathematiques, Universit~ de Montreal, Montreal I01, Canada; submitted to SIAM J. on Control. J.L. LIONS, O~timal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. H.F. VANDEVENNE, [i] Qualitative Properties of a Class of Infinite Dimensional Systems, Doctoral Dissertation, Electrical Engineering Department, M.I.T. January 1972. [2], Controllability and Stabilizability Properties of Delay Systems~ Proc. of the 1972 IEEE Decision and Control Conference, New Orleans, December 1972.

Andrze j P. Wierzbicki~ Andrzej Hatko ~ COMPUTATIONAL METHODS IN HILBERT SPACE FOR OPTIMAL CONTROL PROBLEMS WITH DELAYS

Summary The paper consits of two parts. The first part is devoted to basic relations in the abstract theory of optimization and their relevance for computational methods. The concepts of the abstract theory (developed by Hurwicz, Uzawa, Dubovitski, Milyutin, Neustadt and others) linked together with the notion of a projection on a cone result in an unifying approach to computational methods of optimiza$1on. Several basic computational ccncepts~ such as penalty functional techniques, problems of normality of optimal solutions, gradient projection and gradient reduction techniques, can be investigated in terms of a projection on a cone. The second part of the paper presents an application of the gradient reduction technique in Hilbert space for optimal control problems with delays. Such an approach results in a family of computational methods~ parallel to the methods known for flnlte-dlmmensional and other problems: conjugate gradient methods, variable operator (variable metric) methods and generalized Newton's (second variation~ method can be formulated and applied for optimal control problemg with delays. The generalized Newton's method is, as usually, the most efficient; however, the computational difficulties in inverting the hessian operator are limiting strongly the applications of the method. Of other methods~ the variable operator technique seems to be the most promissing.

Technical University of Warsaw, Institute of Automatic Control, Faculty of Electronics, Nowowiejska ]5/19, Warsaw, Poland.

305

I. Basic relations in the abstract theory of optimization and computational methods I. Basic theory. Two basic rezults of the abstract theory of extremal solutions are needed in the sequel. Theorem I. (See e.g. [5] ). Let E,F be linear topological spaces, D be a nonempty convex cone (positive cone) in F. Let Q: E - ~ R I, P: E--~F, p e F and Yp = ( y 6 E: p-P(y)6 D 3 . and

(i) Suppose there exists ~ ¢ D ~ (called Lagrange multiplier) ~eY~ such that <~,P(y) - p> = 0 and

Q(~) + <~,P(~) - p> % Q(y) + The n ^) Q(y % Q(y) for all y GYp

for all

yeE

(I) (2)

(ii) Let Q,P be convex (P relative to the cone D). Let the cone D have an interior point and suppose there exists @

Yl ~ E such that P - P(Yl )~ D. Suppose there exists a point ~ satisfying (2). Then there exists ~E D ~ satisfying (I) and such that

- p>

=o

(3)

(iii) Given p l , P 2 & F suppose there exist in Ypl' Yp2 respectively. Suppose there exist (I) and (3) for ~I and ~2' Then <~I 'Pl - P2 ~ ~ Q(~2 ) " Q(91 ) ~

~2'

~i,~2 minimizing Q ~I' ~2 satisfying

Pl - P2 ~

(4)

Recall that the general form of a Lagrange functional is

L(%,?, with

y) =

oQ(y) ÷

~o ~ 0 , whereas the normal form, with

? o >

(5) O, is equivalent

to

L(?,X) = Q(y) +

(6)

Therefore, the theorem I (ii) gives a sufficient condition of normality of a convex optimization problem. The requirement of a nonempty @ D is fairly severe and by no means necessary (we shall give an @ example of the existence of a normal Lagrange multiplier when D is empty). However, weaker conditions of normality of convex problems have not been sufficiently investigated. The part (iii) of the theorem is basic for sensitivity analysis of optimization problems and results in the following corollary. Corollary 1. Suppose the space F is normed. Suppose there is an open set ~ = F such that the assumptions of theorem I, part (iii), hold for each

Pl,P2 e ~. Define the functional

~: ~ - P R I

306

by

~(~) -- Q(~) = mivn " Q(y). Suppose the normal S p Lagrange multipliers ? are determined uniquely for each the mapping #%

nal

Q

N: ~ --~ F,

~ = N(p),

is differentiable

and

gradient of the functional

~

is continuous.

[Q(p, ~p) = is

and

<~,

~p>

; hence the

-7 •

The properties of the mapping

N - uniqueness,

Lipschltz continuity - are quite important

continuity and

in sensitivity analysis

and X other computational aspects of optimization. not investigate

p ~

Then the functio-

However, we shall

these properties here.

Another theorem of basic importance, which snmmarlzes results proven in

[6]

,

[7]

Theorem 2. Let functional

, E

Q : E -~R i

in a given set

YpCE.

[8] , is the following.

be a linear topological space. Suppose the has a local constrained minimum at a point Suppose the set

has a nonempty internal cone Ki

Ki

at

Yq = ~

at

9 (,that is, a convex cone

each open cone

KO

containing

^) the set (K o + y ~ U ( 9 ) ~ Yp

qo / ~)

Ke

U(~)

such that (K i + ~)(~

has a nonempt~ external cone such that for each

k,

keKe,

m

qo £ K i

and

' ql ~ @

ql g Ke

Ke

for

and for each neighbourhood

contains more points shan only

there are nonzero functionals qo + ql = @ . The fact that

Yp

Q(y) < Q(9)}

(that is, a convex open cone

such that there is a nelghbourhood U(~) ~ Yq). Suppose the set

{y~E:

U(9)

~)

Then

such that

does not necessarily imply that

a corresponding lagrange functional has a normal form. Actually, Yp =

[ y a E: p - P(y)G D ~

, we need additionally

if

some assumptions

resulting in a form of the Farkas lemma in order to represent the elements of

Ke

by the elements of

Corollary 2. Suppose be differentiable

E,F

DW.

are normed spaces. Let

(with the gradient denoted by

K i" = [ ~ Q ~ (~), " ~ O } Let P: E - ~ F derivative denoted by P (y)).

Q: E - ~

RI

Qy" (y)); • hence

be differentiable

Cwith the

(i) Suppose Yp = { y e E: p - P(y) e D~ , where D is a nontrivial positive cone (inequality constraint). Suppose P..~ (9)D ~" is weakly ~* closed; hence, by Farkas lemma,

-- o ~

<~,P(~) <~,P(~)

- p>

= 0

Ke =

). Thus there exists 76 0 and

~y y + PF~ (~) ~ ^~(^) Lagrange functional has a normal form.

= ~

such that ; therefore,

the

(ii) Suppose Yp = ~ y g E : p - P(F) = @ 3 , hence D is a trivial cone (equality constraint). Suppose Py(y) is onto; hence, by Lyusternik theorem, Kme = P~(9) m ~. Thus there exists ~ y (obviously, <~,P(~) - p > : O) such that Q~(~) + P*(~)y ~ refore,

the Lagrange functional has a normal form.

F* : ~) ; the-

307

The sufficient condition of normality in (ii) - that is, the requirement that Py(y) is onto - is quite natural and useful. The sufficient condition of normality ~n (i) - the weak W closedness of O P~ (9) D*- is less restrictive than the nonemptiness of D, but ratY her cumbersome to check. 2. Projection on cones in Hilbert space

y a~,

Let D be a closed convex cone in Hilbert space ~ . denote by yD an element of D such that

Given

~yD _ yl~ = rain l~d- y~ (7) D 4eD The element y is called the projection of y on D. Lemma 1. (see e.g. [9] ). If E is a strictly normed space (such that ~Ix+yl~ : l~xll + l~Yll implies x : m y , ~ R I ; in particular, a Hilbert space) then the projection yD of y e E on a closed convex set D ~ E is determined uniquely. Lemma 2. The element yD& D is=the projectfon of 2 G ~ on a closed convex cone D " ~ if and only if (i) y D Y , D ~(8)

(ll) Proof. + {~d_yD~ 2 dGD such

--0

(9)

If (1) and (li), then ~Id-3,}l2 = ~lyD_y~12 + 2 < y D _ y , d > + > llyD_yl~ 2 for all d e D . If not (&), then there is that < 0 and there is a > 0 such that +&lldl}2< O. Hence llyD + ~ d y H2< ~yD-yI~ 2 ; since

D is convex cone, yD + 6 d e D, and yD cannot satisfy (7). If not (ii), then (yD , ND_ y~ ~ 0 (we have always (yD, YD -Y> >i 0 since y D - y G D * , yDa D). There is &l ~ 0 such that for all ~.~ (0, &~) the inequality - < y D , y D _ y > + 6~lyD~ 2 < 0 holds. Hence H( I- £)y~ - yll 2 < ~IyD-y~I 2 ; since D is a cone, (I- & ) y D e D for sufficiently small E.> 0 and yD cannot satisfy (7). Lemma 3. If ~IYD~ = min~Id{l , then YD = yD. Proof. Since

yD

satisfies (8) and ((9), we have

=l[d_yO, 2÷ llYOi{2÷ 2 >i [IyD[[ 2 for all d - y ~ D * . Lemma @. The projection following properties :

ildll 2

=

= l%~_yO i12 + llYOiI~ + 2 Since YD is unique, YD = yD yD on a cone in Hilbert space has the

(i) II Y~II ~ = 0 then yD=yl Proof. (1) Relation (9) implies [[yD [I 2 = ~< ~lyDl[ "ily[i" D D 2 D D _yD)+ = (ii) We have ~{yl-y 2 ~ = < y 1 - 7 2 , y1_Y2_(y ~ (y2_yD)>

308

=

- < h

. D D IlYI-:211 (iii)

llY1"~21i

- < Y2

as a consequence

llyD~i2 = < y D ,i.Y2 5

4
:-'2'h-Y2>

of (8) and (9). ~< I},D II II YII{ if

Y2eD~.

(iv) Since _<'D.y2~ ~ O, = 0 and , = ,l-y2, hence < , D - y , , 2 > ~. lly2112. Since < y - - y , y 1 > ~/ 0 and I~,D-,II 2 = = < y D ,,_y> , hence >~ |I YD''II 2 Therefore, II ,D-Y%|I2 = =

II~'D-'-'2 II 2 = ib'D'Y 112 +

11'211 2

. 2

g

O, and

D

='I"

Lemma 5. The functional ~ : ~ - ~ R I defined by ~(y) = ^,4 D IIyD~I 2 is differentiable and has the gradient Q,(F) = y • Proof. Consider the optimization problem: minimize Q(d) = = 0.5 Ildll 2 for de D~+ ,. The unique solution of the problem is

= 0.5

is " ::;f L : T f 3 "aSli ~ 0.5 ,Id + ~II 2 ~ 0 . 5 flyD+ ~,,2 yD 1nc:~ ~ e i ifn:qda ~Intl; sa o if ~ = _yD (otherwise d = - ~ yields a contradiction), hence ~ = _ D i s the unique element such th&t 0.5 ~d I~a + <~ , d-y~ 0.5 ~,ull~- + < ~ , , D - y > for all d~. Therefore, ~ = _yD is the unique, normal Lagrange multiplier for the problem (observe that the cone D may have an empty interior; consider the positive cone in ~ L2 [Dto,tl] ). Moreover, the mapping N: ~ - b ~ such that ~ = N(y) = -, is Lipschitzian- see Lemma 4, property (ii). Corollary I yields the conclusion of the lemma. ~. Application to penalty functional techniques. The penalty functional methods have bee~. applied to the cases when the constraining operator P is finlte-dlmmenslonal (see e.g. [5] , [I0] ) and in special cases of infinite-dimmenslonal operators (so called S-technlque, see [lq] ). The notion of projection on a cone makes it possible to generalize the penalty techniques to arbitrary infinlte-dimmensiona! operators. Consider the problem: minimize Q(y) on Yp= { 2 g E : p-P(,), _D} where E is a normed space, P : E - ~ , D is a self-conjugate cone (D~= D) in ~ . Observe that most of the typical positive cones are self-conjugate; however, the last assumption is made in order to simplify the analysis, and the results can be generalised for the case when D is not self-conjugate. Define the increased penalty functional ~ : E ~ R I-~ R I by

(Y,S)

= Q(x~ + o.5.~ 11 ('P(~,~ - P ) D I I 2

(10)

Theorem 3 . Let ~n~L tend monotonically towards Infinity. Suppose for each ~n there is a Yn minimizing ~ (y, 5 n ) for g E. Then

(i)

'

(yn'

309

(ii) inf Q(Y) ~ J (Yn' ~n ~ (iii) ~lim 9n I~(P(yn) - p)D~]2 = 0 A

(iv) (v) (vi)

If there exists y = lim Yn and Q, P are continuous, then Q(~) ~ Q(y) for all y e Y p . Denote Pn = (P(Yn)'p)D + p" Then lim Pn = p and each Yn minimizes Q(y) over Ypn = [ y m E : pn-P(y)e D ~ . Denote ~n = ~n (P(Yn I-p)D " If Q,P are differentiable, then ~n is the normal lagrange multiplier for the problem of minimizing Q(y) over Ypn =

= [ y e E : P n - P(Y)m D] with Pn = (P(Yn)'p)D + p° Proof. The points (i)...(iv) are an easy generalization of the theorems presented in [5] • To prove (v) observe that Pn - P(Yn )ED~=D' according to lemma 2; hence Yn ~ Ypn" Moreover, Q(yn ) + 0.5 an" "~{(P(Yn) - P)D~I2$ Q(Y) + 0 . 5 ~ n II (P(Y) - P)D~I2 for all y e E . But for all y e Y _ we have P2 = (P(Yn) - p)D _ (p(y) . p) e D = D~ Denote Pl = (P(Y~?- p)O~ D; hence P(y) P = P~ " P2 and, according ~o lemma 4 p. (iii), we have H(P(y)-p) DI~ ~ I~(P(Yn) - p)O~ for all YCYpn" Therefore, Q(yn ) ~ Q(y) for all y eYpnTo prove point (vl) observe that}Is differentlable according to _

~

+

p*

lemma 5 and ~y(Yn~ ~ n ) - Qy(Yn ) ~n y(Yn)(p(yn) . p)D = @ . Hence = ~n (P(Yn)-p) satisfies the Lagrange conditi o n. Moreover, ~ 6 D n and = E n < (P(Yn) - p ) D p(yn) . P _ (p(yn)_p~> =0 according to lemma 2. Therefore, ~ n has the properties of a Lagrange multiplier - see Corollary 2, point (1). The Lagrange functional corresponding to the penalty technique has a normal form - though no normality assumptions has been made. If the original problem is not normal, then the sequence {~ n~7 does not converge. But the sequence { Pn ~ 4 converges to p. Hence the penalty functional technique approximates optimization problems by normal optimization problems. Corollary 3. If the assumptions of Theorem 3, point (vi) are satisfied for all p & ~ , then the set of p e ~ such that the optimization problem is normal is dense in ~ . The corollary explains why the Balakrishnan's & -technique leads to a normal formulation of the maximum principle: singular optimal control problems are approximated by normal ones when applying penalty functionals. However, the increased penalty technique has one sincere drowbacM: the method becomes ineffectives computationally when $ increases. Thw computational effort necessary to sc~e the unconstrained problem of minimization of ~ increases rapidly with ~ ; this is due to "steep valley" effects, well known in computational optimization.

310 To overcome this difficu!$y, another technique has been proposed [12]. Define the shifted penalty functional ~ : E ~ x R I-~ R I by ~ C y , v , ~ ) = Q(y~ + 0.5 ~ ll(P(y) + v)Dll 2 (11) where v E ~ is not necessarity equal-p. The theorem 3, points (v), (vi), can be restated in this case: Theorem @. Given v E ~ and ~ > 0 suppose there exists Yv minimizing ~ over E. Then: (i) Denote Pv = (P(Yv) + v)D - v e ~ . The element Yv minimiQ(y) over v~pv y~ E: Pv _ P(y)~ D ] (il) Denote ~v = ~ (P(Yv) + v)D" If P,Q are dlfferentiable, then ~v is the normal Lagrange multiplier for the problem of minimization of Q over Ypv" The penalty shifting method consists of a suitable algorithm of changing v in order to a c h ~ e pv-~ p. An efficient algorithm was proposed first by Powell [13/ for equalit~ constraints in R n and later generalized in [ 1 2 ] for inequality constraints. Stated in terms of projection on cones, the algorithm has the form: zes

Vn+ j = (v n + P(Yvn)) D - p ; vI =-p (12) Theorem 5. Suppose there exists a solution Yo and a normal Lagrange multiplier ~o for the problem: minimize Q(y) over Yp = {y &E: p - P(y)e D } , where E is a normed space, ~ is Hilbert, D is a selfconjugate cone in ~ , Q: E -~R 1 and P : E--~75 are Frechet differentiable, p = pog~ is given. Suppose there is a neighbourhood U(p o) such that the optimization problem has solutions for all p a U ( P o ~ and the mapping N: U(p o)-~ ~, , ~ = N(p), is welldefined and Lipschitzlan, ~I N(P I) - N(P 2) H ~< R~ H Pl-P2 ~ for all p l , P 2 g U ( P o ). Suppose the shifted penalty functional ~ ( y , v , ~ ) has a minimum with respect to y m E for each S > ~ j and each v in a neighbourhood U(vl) , v I = -p~. Suppose v is changed iteratlvely: vq = -Po' Vn+q = (P(Yn) + Vn) - Po' where Yn are minimal points of q~(Y,Vn,~). Then: (i) There exists . ~ " ~ I s u c h that for all R>~N" the sequence {Vn] ~ teas

the

converges to sequenoe

[Pn~

Vo = +

~o

~ defined

- Po

and

[-Vn~~ ~

by Pn = (P(Yn)

+4

Vn~

D

U(v 1) whe- vn

con-

verges to Po and [Pn)4 ~ U(Po)" 1 + ~" , SUCh (li) Given any ~ > 0 there exists ~ > ~ ~", ~ >. ~ R~ that S > ~ Implies l~pn+1 - p o Jl ~( ~lJPn PoJ~ ' lJvn+1 " roll 6 ~ II Vn-VoH • Thus the convergence is at least geometrical and an arbitrary convergence rate can be achived. Proof. Since Vo = vl + + ~o' Voe U(vl ) for sufficiently large ~ and there is a neighbourhood U(v^) c U(v4) such that v I ~ U(v o). Since

Pl = Po + (I°(Yl) " Po )D'uhence -'by the Theorem 3 -

311 there is a sufficiently large ~

such that

Pl m U(Po)"

Suppose VnmU(Vo) and P n ~ U ( P o ). By the Theorem 4, ~n = S (P(Yn) + vn)D is the normal Lagrange multiplier for the problem of minimizing Q(y) over Thus we have: I I Ypn' v I n ....... S,~L ~ n - Pn ; Vn+1 = - ~ ~ n - Po : Vo ......~.. ~o - Po and

Vn - v° Jn (~n- ~o ) " (Pn'Po) ; Since ~n = N( ) is Lipschitzlan,

II Pn

- Po II ~

~S

Vn+l"V O =

II Pn " Po II ÷ II v n "

~

(~n " ?o )

Voll

and, if ~ > R~ ~lPn - Po II ~< ~ On the other hand

llvn+~, roll.< h hence

!1 Pn

~v n - v o~

- poll

IlVn+~..oll~ __h_ s- R7 ilvn " v°ll If E is sufficiently large, Moreower

Vn+IE U(v o)

IlPn+l poll -< ~

and

Yn+l' Pn+1

exist.

Poll

I!

Pn" and Pn+le U(Po). By induction, [v.~ ~ = U(v^) ~ U(v.) I,. iIJ 4 U / { P n ] l c U(Po). The convergence rate ~ = ~ can be and made arbitrarily small. ~- R~ -

The assumptions of Theorem 5 can be made more explicit by investigating the conditions of the Lipschitz-continuity of ~ = N(p), the existence of minimal Yn etc. However, these problems shall not be pursued in this paper. It should only be stressed that the assumption of Lipschitz-continuity of N is essential, what can be shown by simple examples even in R I - see [12] . Although an arbitrary convergence rate o@ can be achived, it is not practical to require too small 0@ and too large ~ , since the computational effort of solving the problem of unconstrained minimization of ~ ( y , v , 9 ) ~ecomes rather large in that case. 4. Application to gradient projection and reduction techniques. Consider once more the problem of minimizing Q(y) over Yp = {y e % : p - P(y)eD] where Q : ~(y-~R1, p : ~ y _ , ~ are dlfferentiable, ~y, ~ p are Hilbert, D is a positive cone in ~ p . Assume there is a Yl G Y p given and construct a cone K(y I) =

~y~

.3V

p(

)_ ~py(

Y%,~o£,(o~f- Yl n a l - see t h e o r e m 2 - to t h e explicit, several regularity direction d = -Qy (Yl) does

) ~y~

.

YS D~ Assume the cone is e x t e r set Yp a t Yl ( t o make t h e a s s u m p t i o n a s s u m p t i o n s c o u l d be m a d e ) . Assume the n o t b e l o n g to

K(y 1)

However, the p r o -

312

j ection d k of d on K~s)provides for a good approximation of a permissible direction of improvement. Assume now that {~y¢~y : Py(yl) ~ y e o ~ ~ = Py(Yl ) D*(again, to make the assumption explicit, we should use Farkas lemma -Py (Yl)D being weakly closed - or Lyusternik theorem - D = Py(Yl ) a surjection) o This is the basic normality assumption ch makes it possible to introduce a normal Lagrange multiplier ~ at a nonopt imal point y° -

Since K(yl), = ~yg~ I - P (yl)~D.Y;D1} , DI = D + ~(p-P(yl)) ~ R Ic and ~D? = Y { ~ g ~ , p-P(yl); = 0} , hence K~'(yl) = {- y(~)~: ~ D ~, ,P(yL) - p > = 0 } . When projecting d on K(y~), we obtain - d ~ K ~ ( y q ) and ~ d k - d , d k ~ = O. If d = -Qy ~yl ), we have: (1)

~

D*;

<~, p-P(yl)>

= 0

(iii) -- 0 (13) Hence we have a normal Lagrange multiplier ~ for a nonoptimal point Yl ~ Yp~ It coincides with the optimal ~ for an optimal YI' since d k = 0 if Yl is optimal. The multiplier satisfies the usual conditions (i) ; but there is also an additional condition (iii) - trivial at an optimal point - which helps to determine ~ for a nonoptimal point. In a general case it is not easy to make a constructive use of the set of conditions (q3) (1), (iii). However, these conditions generalise the known notions of the Rosen gradient projection [I#] or, in a special but ver~ important case, of the Wolfe reduced gradient

05] Actually, assume D = {e~ , Py(y~) being onto; then K(F 1) is the linear subspace tangent to Yp at Yl (the null,pace of Py(yl)). The conditions (i) are trivial, the condition (ii) amounts to an orthogonal gradient projection on the tangent subspace, and the condition (iii) results in an explicite value of the Lagrange multiplier and the Rosen gradient ~rojectlon

]

y(Yl

The notion of Wolfe's reduced gradient applies to several optimal control problems when ~ y = ~ x X ~ u snd ~ p = ~ x " The most important feature of the class of problems is that the state x 6 ~ x can be explicitly determined (analytically or numerically~ in terms of the control u ~ u. The statement of such a problem is; minimize Q(x,u) for (x,u)~ Yo = ~(x,u) g ~ x x ~ u : P~x,u) = 0 ~]~.x} where

313

Q :~x~u -~ R 3 and P : ~ x ~ u - * ~ x are Frechet differentiable and Px(X,U) is onto for all (x,u)*Yo. The last assumption corresponds to the requirement that x could be determined in terms of u. If we take y = (x,u), Py(y) = (Px(X,U), Pu(X,U)), Q~(y) = * = (Q~(x,u) , Qu~X,U)) and apply the Rosen's projection on the subspace obtain

:i o+xu, -

= - Q

(x,u) - P

However,

x(X,U)

(x,u) 7

this is not the most useful way of introducing a La-

grange multiplier in that case. Since Px1(X,U) exists for (x,u)+ Yo' we can change the original variables y = (x,y) to ~ = (y+l,y2) where

~y~ = ~ x

+ P'1(x'U)Px " u(X'U)YU'-

~

~Y2 : ~u. Now

P~(y) =

= (Px(X,U), O) and m~(y)* = (Qx(X+ , u ) , mu(X,U)* Pu(X,U)P~(x,u) Q~(x,u)). The tangent subspace becomes K = {@]W ~ u and the projection is particularly simple. The new Lagrange multiplier (we keep to the original denotation) is

= -

(x,u) Qx(X,U)

(~6)

and the new projection of the negative gradient is

(i)

_ bk = _ Q x ( x , u

) _ Px(X,U)? = O - P (x,u)pThe relation (i) has a simple interpretation: we choose the new Lagrange multiplier in such a way that the Lagrange functional Q(x,u) + ~ ,P(x,u)> doeas not depend in a linear approximation on ~x. The Lagrange multiplier determined by )(16) differs obviously from the multiplier (15); however, at an optimal solution the multipliers are identical. The projection (15), (il) is a generalization of the Wolfe's reduced gradient; in the original space it is a nonorthogonal projection (but it is a projection, since the variable transformation is one-to-one). The resulting technique has been in fact widely employed in many approaches of the calculus of variation and provides for many computational methods of optimization. 5. Reduced gradient and the basic variational equations If the condition P(x,u) = ~ determines implicitly a mapping S : ~u-~x, x = S(u), then instead of minimizing Q(x,u) over Yo we have to minimize J(u) = Q(S(u),u) in ~ u " However, the determination of S might be cumbersone, and it is often useful to express the derivatives of J in terms of the derivatives of an equivalent Lagrange functional. This technique amounts actually to the gradient reduction technique. Assume Q,S are twice differentiable ; so is J. Denote J(u+~u)=J(u)+ ~b(u),~u~ +0.5 ~ u , A ( u ) ~ u > +~(~l~ul12) (18)

314

where b(u) is the gradient and functional J. On the other hand

A(u)

the hessian operator of the

J(u~ = L( ? ,x,u) = Q(x,u) + ~ ,

~(x,u)>

(19)

if L_ (@,x,u) = P(x,u) = 0 (20] which can be interpreLted~as'~ the basic state equation of the optimal control problem. By choosing ~ according to (16) we obtain L ~ ( ~ , x , u ) = Qx(X,U) + Px(X,U) ? =~) (21) which can be interpreted as the basic adjoint equation of the problem. Hence the gradient of the functional J (the reduced gradient of Q)

is

Lug( ? ,x,u) -- Qu(X,U)

The linear approximation

+

Pu(X,U)~

= b(u)

(22)

of the state equation is

~x' = - p -Xt ( x , ~ ) P U ( x , u ) ~ u ; ~x -- ~x' + o( II ~ u l l ) (25) Expanding L( ~,x + 6"x~ u + ~u) into second-order terms of ~x, ~u and applying (23) results in the hessian operator of the functional J A(u)=L L'1 L L-IL -L L-IL -L L'1 L + L (2z~) u~x9 xx ~x ~u u~ x~ xu ux x~ 9u uu with the d e n ~ t a t i o n s E o r t e n e a ih an obviou~ @ay; observe that '

L~x = Px (x'u)" However, the explicit expression for the hessian is n~t She most useful computationally. Phere is an alternate way: to expand L( ~ + ~ ,x+ ~x, u+ ~u) into second-order terms and choose an appropriate ~ . Hence the hessian operator can be determined by the set of equations (i)

A(u) ~u = L ,~V + L

u:

0

~X'+ L

~u uu

(iii) E) = ~ - x ~ X ' + L u ~ u (25) The e q u a t i o n s ( i i ~ , (iii) - where (iii) is equivalent to (23) - are the linearisation of the adjoint and state equation and are called basic variational equations. In some computational approaches an important problem is the inversion of the hessian operator, in order to determine the Newton's direction of improvement d = -A -I (u)b(u). Setting ~u = d, A(u)~a= = -b(u) in (26) and assuming (i) d = - L -1 ( L . _ ~ + L UU

U ~

~

UX

L -I exists, we get UU ~ + b(u)) ~

~

~ - " b(u) = (L X ~ -L X U ~ LU- "U n U ~ )~"?- + (nX X -L X U LU- 'U L U X )~-~-L L Ut~ ~ XU (iii) 0 = - L ~ u ~ I u L u ~ ~ + [L__-L. u L u u L u x ) ~ - L uLu'u b(u) (26) where (ii), (iii) are called can?onic~al variationa~ equations. Their solution is more difficult than the solution of basic variational equations; usually, the canonical equations represent a nontrivia! two-point boundary problem, and the tNpical method of their solution is the reduction to a Riccati-type nonlinear equation. (ii)

e

315

II. Application to optimal control problems with delays. 6. Optimal control problem with delays: the gradient and the hessian. Consider the problem: mlnimise the performance functional

(27) Q(x,u) = Sj~fo(X(t),x(t-r ),u(t),u(t-~),t)dt + h(x(t I )) where x satisfies the process equation P(x,u) = ~ ~ x(t) = f(x(t),x(t-lz ),u(t),u(t-~ ),t); x(t) = El(t) for t ~[to-1~ ,to] u(t) = $2(t) for t 6 [to-W ,t o ) (28) The analysis of the relations between this particular problem and the general Hilbert space problem leads to the known, following results. Denote H = - fo + ~ f and ~(t+r):~(t); x[t+1:)=z(t); x((t-~']:y(t); u(1;+r):w(t); u(t-1~):v(t) (29) Define the shifted hamiltonian function H by =

where

~ ~

It +

t+~

denotes the function

'

~

:

o for t~(t1-~,t I]_ -

(30)

with arguments evaluated at

I t+l~

t +I: . Then the process equation (29) can be rewritten in the form the adjolnt equationtions-takes the form

under appropriate differentiabilit2 assump@

: - Hx ;

@

~ ( t I) : - ~ x ( X ( t ~ ) )

(52)

and the gradient equation is b = _ H u@ (33) These results apply for the case without any additional constraints save (29). If, for example, a final state-function x(t) = ~3(t) for t 6 GI-I; ,tl] is given, the penalt~ functional approach is most useful. The basic variational equations take the form

~=H x~X+H~ ~S + H.uYU + H.v~v; ~(t) = 0 for t ~[to-~,to] ~:

- Hx ~ ? - H x ~ - H

x gF " Hxx~X'Hxz~Z-Hxv~V - HxugU - Hxw~W ;

~ (tl) ~: - hxx~ ( h ~ u? '~

u~,

u:~

ux

uz

uv

uu

Huw

(3~')

The canonical variational equations have a complicated form

~:AI~+B~+

C~+A2Y

~ + B 2y~+ c2 ~ +

~x

~ : A3~ ~ + B3;~ + C?Y~'p ~. + A @ ~ * B@~'"+~ C#S'~* ~ (357 where ~ ( t ) = ~'~(t-~),~x(t)=0 for t ~[to-l~,to],~'~(tI)= - h x x ~ ( t I), Aj, Bj, Cj are matrices determined by second-order derivatives of the hamiltonian function H (assumed a matrix related to Huu , Huv , Huw is invertible) and ~x' ~ are determined by the gradient b. A numerical solution of the equations (3~) presents no particular difficulties, since we can solve the advanced-type adjoint variational equation backwards after solving the retarded-t~pe state va-

3i6

rlational

equation.

A numerical

sents a major computational with two-polnt

boundary

solution

problem,

of the equations

(35) pre-

since they are of neutral

type

conditions.

Several techniques have been proposed in order to solve the equar ) , tions k35 or equivalently, to invert the hessian operator of an optimal control of a Riccati

problem with delays.

setting of an integral

setting results Riccati

type,

Recently,

in an integral

problems.

equations

However,

is iterative reasonable

to ommit

Rn

and recently

on the concept

basis

milies

tational

iteratlve

technique

for quadratic

(or "second variation method") cases.

Therefore,

of the hessian

direction

it is rather

for non-quadratlc

and variable

of an optimal

generalised

for Hilbert

called

of conjugacy family,

operator methods.

control

problem

in Hilbert

of several universal

com-

programming

in

space problems.

problems There are

conjugate

or

directions

methods,

is based

or variable

matric

A-orthogonality.

called variable

operator

is based on the notion of the outer product in ~ . A variable -I V i is an aproximation of A constructed iteratively by in

~

where

. A discussion

of optimization

effective

a simple

for the application

Vi+ I = V i + ~ V i products

of

to solve computationallyo

well known for nonlinear

family,

The second operator

such a

equation

of such methods.

The first

methods~

However,

example.

formulation

methods,

method

consist

with delays.

conjugate

is a natural

two families

proposed

the inversion

A computational The general

~

in non-quadratic

problems

7. Universal

space

type.

(35), which is quite effective

the Newton's

itlself

optimal control

putational

operator

or partial differential

which is also rather difficult

Chan and Perkins

for solving

Most of the techniques

experience

~V i

methods

seems

is determined

of general is given

to indicate

than the Newton's

properties in

~8]

with help of outer of these

that these methods

if the inversion

two fa-

. The existing

compu-

are more

of the hessian

is diffi-

cult. One of the conjugate dure

in solving

gradient

the following

methods

computalonal

M. Jacobs

and T.J. Kao in an unpublished

Kurcyusz.

The example

tional

techniques.

complete

state

me delay

system:

~(t)

= -x(t-:)

mizing

illustrates

+ u(t),

the functional

of time)

t~[0,2] Q = ~ ~ o2

;

paper and computed

in achieving

x(t) = 0 x(t) dt

as a subproce-

example - constructed

the effectiveness

The problem consist

(a function

was applied

= I,

for

by

by St.

of penalty

func-

a given final t g [~,2]

t 6~I,0]

in a ti-

while mini-

The problem has an analytical

317

solution, hence provides for a goo~d test of computational methods. The results achived by a penalty shifting technique applied to the final complete state are followings: No of iterations Final state of penalty shift constraint violation

No of computed functional values I

Penalty Performance functional functional

0 (beginning)

I .0

51 .0

-

I 2

9.4x10 .3 I .4xI0 -3

I 07 88

0.168 0.171

0. 167 0.169

3

t+.0xl0-@

17

0.I 71

0.169

The number of computed functional values per iteration decreases (which is typical for penalty shifting methods) instead of increasing (typical for penalty increase methods). Bibliography ~i~ Dubovitski A.J., Milyutin A.A.: Extremal problems with comstraints. Journal of Computational Mathematics and Mathematical Physics (Russian), Vol V, No 3, P.595-453, 1965. 2] Neustadt L.W.: An abstract variational theory with applications to a broad class of optimization problems. SIAM Journal on Control, Vol. V, No 1, p. 90-137, 1967. [3] Goldshtein J.G.: Duality theory in mathematical programming (Russian~, Nauka, Moscow 1971. [4] Pshenitshny B.N.: Necessary conditions of optimality (Russian), Nauka, Moscow 1969. [5] Luenberger D.G.: Optimization by vector space methods. J. Wiley, N. York 1969. [6] Neustadt L.W. A general ter and System Science, [7] Girsanov I.W.: Lectures problems. University of

theory of extremals. Journal on CompuVol. III, P.57-91, I£69. on mathematical theory of extremal Moscow, 1970.

KS] Wierzblcki A.P.: Maximum principle for semiconvex performance functions ls. SIA~M Journal on Control, V.X No 3 P.@@#-@59, 1972. 9] Galperin A.M.: Towards the theory of permissible directions (Russian), Kibernetika No 2, p. 51-59 , 1972. 50] Fiacco, A.V., Mc Cormick G.P.: The s~quential unconstrained minimization technique for nonlinear programming. Management Science. Vo.X, No 2, p. 360-366, 196@. ~

Balakrishnan, A.V. A computational approach to the maximum principle. Journal of Computer and System Science, Vol.V, 1971.

318

~

Wierzbicki AoP.: A penalty function shifting method in constrained static optimization and its convergence properties. Archiwum Automatykl i Telemechaniki, Vol. XVI, No 4, p. 395416, ~971. [13] Powell, M.J.D.: A method for nonlinear constraints in minimisation problems°

In R. Fletcher:

Optimization, Academic Press,

~4S

N. York 1969. Rosen, J.B.: The gradient projection method for nonlinear programming. Part I, II. Journal of SIAM, Vol. VIII, p. 181-

~5S

217, 1960, Vol. IX, p. 514-532, 1962. Wolfe, P° Methods of nonlinear prograw~ing. In J. Abadle: Nonlinear Programming, Interscience, J. Wiley, N. York, 1967.

~

Horwitz LOB., Sarachlk P.E.: Davidon's method in Hilbert Space. ~

SIAM J. on Appl. Math., Vol. XVI, No 4, p. 676-695, 1968. Wierzbicki A.P.: Coordination and sensitivity analysis of a large scale problem with performance Congress of IFAC~ Paris 1972.

~

0~

iteration. Proc. of V-th

Wierzbicki A.P.: Methods of mathematical programming

in Hilbert

space. Polish-Italian Meeting on Control Theory and Applications, Cracow 1972. Chart H.C.~ Perkins W.R. Optimization of time delay systems using parameter imbedding. Proc. of V-th Congress of IFAC, Paris 1972.

SUFFICIENT CONDITIONS OF OPTINALITY FOR CONTINGENT EQUATIONS V.I. Blagodatskih ~athematical Institute

of USSR Academy of Sciences, Moscow, USSR

I. Statement o £ t h e proble m In this paper we prove sufficient conditions of optimality in the form of maximum principle for controllable processes Which behaviour is described by contingent equatiau. Let Eu be Euclidean n space of the state x = (~,...,x n) with norm IIx~ = ~ and ~ (~) De space o~ alA nonempty compact subsets of E ~ with Hausdorff metric (F,G)

:

{d:

F c S d (G),

G c

S d [f~) denotes # neighborhood of set M in space E n. Let's consider controllable processes which behaviour is described by contingent equation

where

(1)

~ ~ F(x),

or the differential inclusion as it is also called. Here F : En-~ ~ ( E n) is certain given mapping. The absolutely continuous function x(t) is the solution of the equation (I) on the interval [0, T ] i f the inclusion x(t) g F(x(t)) is valed almost every-where on this interval. Let Mo, M I be nonempty closed subsets of E n. These subsets may be non-convex and non-compact. The solution x(t) given on interval ~0, T~ does the transfer from the set M 0 into the set M I for the time T if the conditions x(O) g Mo, x(T) g M I are satisfied. The time-optimal control problem is to define the solution of the equation (I), doing the transfer from the set M o into the set M I for m ~ i m a l time. Let G be an arbitrary nonempty closed subset of E n. The f~uction (2)

C ( ~ ) : max (f,~) f~G

of the vector y E E n is called the support function of the set G.If the maximum in the expression (2) for the given vector ~o is reached at the vector fo E G, that is, C ( ~ ) = (fo' ~o), then the

320

hyperplane Qx,~) = C(~c) is called the support hyoerolane for the set G at the point fo and the vector ~o is called ~he support vector at the point re. In this case the inequality (f - fo, ~ )6 0 is valed for any vector f g G. As one can see from the condition (2) the support function C( ~ ) is single-valued function of the set G. 0n the contrary, if we know the support function C(~), we can incite only convex hull of set G, that is,

~ E En

En ~ - ~ (En) is an arbitrary mapping, we can consider the support function of the set F(x) for any x g En; we shall denote this function by C(x, ~) and shall call by the support function of the m ~ F: If

F :

C(x, ~) =

max (f,W). z e F(x)

Next !emma follows directly from the definitions of the support function C(x, ~ ) and Hausdorff metric. Lemma I. If the m ~ F : E n - ~ 3 ~ (En) satisfies Lipshitz's condition (is continuous~, then the support f traction C(x, ~ ) satisfies Lipshitz's condition Qis cont~uous) in x for any fixed vector E E n. On the contrary, if the suoport function C(x, ~ ) satis-fi__es Lipshitz's condition (is c q n t i n u o u s ) ~ x , then the respeq tire mapping conv F : x --~ cony F(x) satisfies Li~shitz's condit i0n (is continuous). Together with the inclusion (1) let's consider the differential inclusion (3)

~

~ cony F(x).

Lemma 2, I_~fabsolutely continuous fuact,ion x(t) is the solution of the equation q l ) o n the interval ~0, T~ , then the ineq~it.y

is valed almost ever.ywhere_ on this interval for any vector ~ E n. On the contrary, if the condition__~)is .valed for absolutely continuous fuu@tiem x(t) almost ever,Twhere on interval [O, T] for any vecto_~r ~ m En~ then this function is the solution of the equation (~) on the inte=val tO, T J.

321

Proof

of lemma 2 follows directly from the definitions of the

support function inclusions. and

C(x, ~ )

and of the solutions of differential

Let C o ( y ) and CI( ~ ) MI, respectively.

be support functions of the sets

Mo

Maximum principle° Assume that thesuppor t function C(x, ~ ) is cgntinuously differentiable in x and the solution x(t) does the transfer from the set M o into the set MI on interval [O,TJ. We shall sa~ that the solution x(t) satisfies the maximum principle on interval [O, T~ if there exists nontrivial solution ~ (t) of adjoint system

C(x(t), W ) (5)

~

= -

~x

such that following conditions are valed: A) the maxi'mumcqndition

(6)

(~(t), W (t)) = c(x(t), ~ (t))

i_ss vale~d almost ever.~where on interval ~0, T] ; B) trausversality condition on the set ~o: vector ~ (0) is th e suoport vecto r for the set M o at the point x(O), that is,

co( '/:" (o)) = (x(O), ~ (o)) ;

(2)

C) transversality condition on the set M 1 : vector - ~ (T) is the Support vector for the set M 1 at the point x(T), that is,

~8)

%(-

~ (~)) = (x(~),

- W (~)).

~ufficient conditions of optimality in the form of maximu~ principle can be received in the following section. 2. The ms~n result

The region of reacheh~lit ~ YT for the equation (1) is the set of all points x o E En from which we can do the transfer into the set M 1 for the time not exceeding T. The set M 1 is strongly stable if set M 1 lies interily in Y ~ for any ~ ~ O. In particular, if set

M1

consists of the single point, then the defini-

322

tion of the strong stability coincides with the definition of the local controllability in small of the equation (1) at this point (I). Theorem I. Assume that the se~ M I is s~rongly st__able and the ~ort function C(x, ~ ) ~ Lipshitz'~ condition in x fo___~ fixed vector ~ , the@- the inclusion Y ~ ~ int Y ~ is valed

fo._r~Z

z~,z'~

, o.<'z"z
Proqf. Let x o ~ Yr~ , sad x(~) be the solution of the equation (I), doing the transfer from the point x 0 into the set M I for the time i~ ~ r I , that is~ x ( O ) = xo, x ( ~ ) ~ M I. Since the set M 1 is strongly stable, M~ c iut Y ~ and x ( r ) @ int Yr~-g Thus, there exisSs & > 0 that S z ( ~ ~ ) ) c y ~z_~ . Since the support function C(x, ~ ) satisfies Lipshitz's condition in x, by lemma I mapping conv F(x) also satisfies Lipshitz's condition. Thus, the theorem of the contiuuous dependence of the so±utiom on the initial conditions is valed (2) for the iuclution (3). It follows that there exists such neighbourhood U(x o) that there exists the solution y(t) of the inclusion (3) for any poiut Yo ~ U(xo)' and this solution does the trausfer from the poia% Yo into the set S~ Qx[r)) for the time T , that is, ~ r ) ~ S ~ (x(r)). A~y solution of the inclusion (3) cam be approximated with auy accuraty by the solutions of the equation (1) (see theorem 2.2 in paper (3)), therefore there exists the solution x*(t) of the equation (1) with initial condition x~(O) = Yo that ~!x"(t) - ~t)~ .< ~ . Thus, x*(r) ~ S~(x(~)) and we cam do the transfer from the point x'(r) into the set ~ for ~he time .< ~ ~ 2- , that is, yo~Y[~. ~ Q.E.D. Therefore, U~x o) Y~z, that is, Xoe imt Y ~ . The support function C(x, ~) for the equation (q) is concave ~ E n, for any points Xl, x 2 a E n and ia x if for a~y vector O, ~ + ~ = I, condition for stay numbers ~ , ~

(9)

~c(~,Wj

+o/3 c(~2,~) ~< C(<x I +j3 x2, / )

is valed. This condition is equivalent to that of concavity of the multivaled mappiug F(x), that is equivalent to condition

~or any xl ~

~n.

323

in

If the support function Ctx, F ) is continuously differentiable x, then the condition (9) is equivalent (4_) to the condition

9 c(x. ,/.')

~)

9x

c(x2,~) - c(~, ~ )

£or any vector ~ E E n and for any points xl,x2~ h m Let's define a weaker condition on the support function C(x,~). Let's say that the support function O(x, ~) is concave in x at the point % in the direction ~o , if the condition

9c(%,~) (lo)

(

9 x

' x - xo) ~ C(x'~) " c(%'~)

is valed for any x ~ L ~. Let x(t) be solution of equation (I) om interval [0, T] , (t) be respective solution of adjoint system (5). Let's say that the solution x(t) satisfiesstrong trausversality condition on the set ~ , if the condition

%(-W(t)) < (x(t), -~(t)) is valed for any O~t(T. Note, that if the set M I is strongly stable, solution x(t) satisfies maximum principle on interval ~0, T~ and the support function C(x, ~) is concave in x at point x(t) in the directio~ ~(t), O ~ t @ T , then the condition (II) is valed. Indeed, as it will be shown in the proof of theorem 2 under the given conditions vector - ~/ (t) is the support vector for the region of teachability YT-t at the point x(t). And since M 1 c int YT-t for any O ~ t < T condition (II) is valed. The main result of this paper is Theorem 2. Assume that Mo, M 1 are nonempty close d subsets of E n, the solution x(t) qf the equation (I) does the trausfer fro m the set M o into the set M 1 on interval [0, T~ and it satisfies maximum principle on that interval and ~/(t) is the solution of the adjoint system. Assume that the support function C(x, ~) is concave in x at the pqint x(t) in the direction V~ (t) foz. any ta[O,T]

324

and the solution x(t) satisfies stron~ transversalit~ condition on the set M 1. Then the solution x(t) is optimal. Proof. Let ~ t ) be an arbitrary solution of the equation (1) defined on the interval [0, T] . Inequality d (12) ~ ( y ( t ) - x(t), ~ ( t ) ) ~ 0 dt is valed almost everywhere on this interval. Indeed, using lemma 2 and conditions (6) and (10), we get d ~(y(t)-x(t), dt

~ (t)) = (y(t), ~'(t)) - (x(t),~(t)) +

+ (~t)-x(t),~(t))$

/

9 C ( x ( t ) , ~ (~))

l

C(Y(t), ~(t)) - C(x(t), ~(t)) -

, y(t) -x(t)).< o.

9x

Let ,~r ~ 0 ~ ~ ~ T, be hyperplaue passing through point x(r ) and orthogonal to vector ~ ( T ) . It is impossible to do the transfer from the hyperplaue ~ into the set M 1 for the time 8 < T - Zo Imdeed, let y(t) be arbitrary solution of the equation (1) with the condition y ( ~ ) ~ ~r . I n t e ~ a t i a g an inequality (12) on the interval [~, ~ + e] we receive

( y ( ~ + e) ~ x ( ~ + e),

~ ( ~ + e)).< o.

From the strong transversality condition (11) it follows that

C l ( - , ~ ( z + e ) ) < ( x < z + e), - , ¢ ( ~ + e)) =

= (~+ ~(r+ that is,

e) - x ( ~ +

e), V ( t +

e))-

(~z+

e),

e)),< ( y ( ~ + e), - , / , ( r + e)),

C1(-~(~+

e))<(~+

e), - ~ ( r

+ ~)). It means that

point yi~ + 8) does not belong to set ~ . If point ~ ~ ) satisfies the condition (yi~)-x(~), ~(~))<0, then it is impossible to do the transfer from this point into the set M 1 for the time T - r . Iadeed, integrating an inequality (12) on the im~terval IT , T] we can receive (F(T)-x(T), ~(T))
325

It contradicts the traasversality condition (8) on the set the ayperplane IT_~

/~

at the poin~

~,

Thus,

is the support for the region teachability xkz)

with the support vector

- W (~)

From the transversality condition (7) on tae set M o

for all

it follows

that M o D Y T C ] ~ . Thus, it is impossible to do the transfer from the set M o into the set ~ for the time < T, that is the solution

x(t)

is optimal.

Q.E.D.

RemarK. Since the solution xkt) of the equation ('i) is also ~he solution o% the equation (3), then by the theorem 2 the solution x(t) is also optimal for the equation (3). Corrollary. Assume that the set M 1 is strongly stable I the support functi0n C(x, ~ ) is concave in x and the solution x(t) does the transfer from the set M o into the set M 1 on interval [0, T ~ a n d i t satisfies m a x i m ~ r ~ c i p l e on this interval. The~ the solution x(t) is optimal. Pr0of of this corrollary coincides with that of the theorem 2 but we have to use the result of theorem I instead of strong transversality condition (11). In case sets M o and M 1 are the points and F(x) is convex for all x E E n, this corrollary was proved in the author's paper (5). Thus, to solve the time-optimal control problem, in case set M fl is strongly stable and the support function C ( x , ~ ) is concave in x , it is sufficient to find at least one solution of the equation (1) which satisfies maximum principle. Note, the solution may not be the single one. In case, we have some solution

x(t), O ~ t ~T,

and want to know wether it is optimal or not, it is sufficiemt to verify all the conditions of theorem 2. Note, we have to verify the conditiom of concavity of the support function C(x, ~ ) only at the points

x(t)

in the directions

~ (t).

3. Examples Now we consider "classical control process". It behaviour is described by the system of differential equations x = f(x,u), u e U . Then the support function is

326 C(x, ~ )

= m a x (Z(x,u), ~ ). u~U

The sufficient condition of optimality similar to that in the above corrollary for the liner control processes X=

AX+

was proved in paper (6)

V,

V~V

taking into the account the additional as-

sumption of convexity of sets

Mo, M I

and

dition of strong stability of the set transversality condition on the set

MI

V. In paper (7)

the con-

was loosed up to strong

MI .

Example I. Assume that the behaviour of the control system is described by differential equation of order x (n) = f(x, x , . ..,

(13)

n

x (n-l) , u),

where vector u belongs to the set U(~) in space E , depending on y = (x, x,o°., ~n-lJ). The set ~I consists of one point x =0. Suppose, that following conditions are satisfied: 1) functions

f1(y) = min

f(y,u)

and

f2(y) = max

u eU(y)

f(y,u)

u ~ U(y)

are continuously differentiable; 2) function 3) point

f1(y)

x = 0

is convex and function

f2(y)

is the interior point of the set

is concave;

z(o,u(o)).

Then the support function

c(y, ~) = y 2 ~ +

is concave in

o..+ Yn ~ n-1 + ~I (y) - - r - - - + ~2 (y)

y ~ It was shown in paper

2

(1_) that the proccess (13)

is locally controllable in small at the point x = 0 in the assumption 3). Thus, maximum principle is sufficient condition of optimality for proccess (13) in the assumption 1) - 3). The oscillating objects which behaviour is described by differential equation of order 2 were considered in paper (8_). Some conditions were reoeived by the method of regular synthesis under which ms_~'mum principle is sufficient condition of optimality. It is easy to varify that oscillating objects satisfy the above assumption

327

1)

-

3).

Thus, o p t i m a l i t y of t r o j e c t o r i e s can be r e c e i v e d d i r e c t l y from the above c o r r o i l a r y w i t h o u t r e g u l a r s y n t h e s i s . We can r e c e i v e d more general r e s u l t s by the given method, For example, the c o n d i t i o n of = r a ~ e c t o r i e s reaching s w i t c h i n g l i n e s under non-zero angles can be omitted. The support function C(x, ~) was concave in x in the example I and we made use of corrollary to show that the solutions, satisfying m 8 ~ m u m

principle,

are optimal. The support function is

not concave in the following example and sets Mo, M I and F(x) are not convex but the given solutions satisfy all conditions of theorem 2, amd thus, are optimal. Example 2. Consider the control system

~=~ 2 3 (q#)

%

-~-e - - -

=

2 3 + x1

+

u

2

2

u=+l Set

MI

5"

-7

consists of two points (E , 3) and ( E ,

consists of two sets { ~

= -3, ~ 7. o } ~ d

{~, = o, ~

Sets Mo, M I and FLx) are pot convex and set ~'I stable for the given sys=em. The support funu=ion

C(x,~)

= x2Y'l

+3

~/~ * i % f 2

is continuously differentiable in

_Xl

x

3), and set

e_4

~ -3

Mo

~.

is not strongly

~",-i~.~1 2

, but is not concave in case

of definition (9). The adjoint system is

'~ = (I -

Two solutions

24) e-4

~'~ - /,~'.~i

2

x~(t) = ( ~ t 2 - 3t + ~

,

-

,

=

-

3) and the solution ~ (t)~ I of the adjoiat system 3 t + i , 3t satisfy all conditions of the theorem 2 and both of them are optimal.

328

References 1. Blagodatskih~ V.I., On local controllability of differential incluti~u (Russia=), Differenc. Uravn. 9, No. 2, (1973) 361-362. 2. Filippov, A.F. ~ Classical solutions of differential equations with multivalued right-baud sides (English trams.), SIAM Control, 5 (1967), 609-621. 3. Hermes, H®, The generalized differential equation xmR(t,x), Aav. in ~ath., ~, No. 2, (1970) 149-169. @. Ponstein, J. ~ Seven kinds of convexity, SIAM Review, --9,No. 1, (1967) 115-119. 5. Blagodatskih~ V.I.~ Sufficient condition of optimality (Russian), Differens. Uravn., 9, No. 3, (1973),~16-~22. 6. Boltyamskii, V.G., Linear problem of optimal control (Russian), Differenc. Uravm., 5, No. 3, (1969) 783-799. 7. Dajovich, S., On optimal control theory in linear systems (Russian), Differenc. Uravn., 8, No. 9, (1972) 1687-1690. 8. Boltyauskii~ V.G., Mathematical methods of optimal control,

Moscow, (19e9).

VARIATIONAL APPROXIMATIONS OF SOME OPTIMAL CONTROL PROBLEMS

TULLIO ZOLEZZI Centro di Matematica e di Fisica Teorica del C.N.R.-Genova

I t Necessary and sufficient conditions are investigated such that the optimal states and c o n t r o l s ~

and t h e v a l u e of a g e n e r a l optimal c o n t r o l problem depend in a c o n t i -

nuous way on the data. Let us consider the sequence of problems P n , n ~ O ,

2n

g i v e n by (u c o n t r o l ~ x s t a t e )

t

minl

t

fn(t,x, u)dt + hn[X(tln),X(t2n)]~, In

= gn(t,x,u) a.e. in (tln, t2n) , with constraints (tln, X(tln),t2n, X(t2n))E B n , (t,x(t)) ~ A n if tEEtln,t2n], u(t)~Vn(t,x(t))

aoe.in (tln,t2n), ~uI~

Po is given, and

P

,n~l

Lp

~<e •

n

,is to be considered as "variational perturbation"

n of Po" Assume that there exist optimal Un, X n for P ,n~/l. Variational convergence n ............. of {Pn} to Po means t h e f o l l o w i n g : t h e r e e x i s t optimal u o , x ° for Po such t h a t

(perhaps

for some subsequence)

(Un, X n) ---9 (uo~x o) min P

n

---~ min P

o

(in some sense), .

This means (a) e x i s t e n c e of optimal c o n t r o l s

(b) " v a r i a t i o n a l Pn

(depending

stability"

for Po ;

of Po i f P

~ Po v a r i a t i o n a l l y n o b v i o u s l y on convergence o f { (Un, X ) ~ )°

for "many" sequences

2. Sufficients conditions for variational convergence. In the above generality, we get min Pn ---~min Po ~

Xn --gxo

uniformly

(and~generally

speaking~ no "usual" convergence is obtained about u ) under general n conditions on An~Bn,Vn, en, and not very strong assumptions a~bout fn, g n. Moreover gn linear in

u

ir~lies

u

n

--~ u o in L p.

330 Such convergence can fail for variable end time problems (simple examples show min P

n

~min

P

o

for time - optimal problems with uniform convergence of data).The

general case is considered in Zolezzi ( $ ) o Assume now that tln~t2n are fixed (for every n),and gn(t,x,u) = an(t)x + bn(t) u + Cn(t) , fn = O ~ hn depending on x(t2n) only~ V (t,x) a compact polytope independent on (t,x) . n Moreover suppose that lim inf hn(Yn)2/ho(y o) if Yn ---~Yo' and for every z o there exists z

n

such that lim sup hn(Zn) ~ h o ( z o) •

Then

(an~C n) __i (ao,c o) ~ b n --~b ° in L 1 implies (for some optimal Un,n ~ O) min Pn --~min Po~ Xn---~x o uniformly~

Un--* u o in every L p, p < ~

•

The same conclusions hold with b --J b o in L 1 only, if either n (a) u is scalar ,b n is piecewise continuous and bn(t) = O for no more than

r

points (r independent on n) ; also if g is non-linear in u~ with a monotoniaity assumption on g(t~x,.), or (b)

u ~ R S ~ s ~i~

and the following regularity assumption holds: for every

p~q6 extr Vn ~ any orthonormal basis (Yl'°'''Ym)'~71(t)bn(t)(p-q)YJlfor no more than r points or intervals~

r in.dependent

on n (

O

n beeing the

principal matrix of ~ = a n x). Moreover u o is piecewise constant,

and u --~u o ~n~formly on continuity inte[ n

vals of u o, Same results hold if we minimize either

t2n f t

b~(t,u(t)~Tdt In

t2 or

i~ n (~#*rt ~t ~ ~ x ~~ + ~ l u l P ) dr, P 2 i, in

If Po has uniqueness~

(~ 2 0

same results hold when we minimize

~2n f

fn(t~x,u) t

dt

In

assuming that convergence of { fn} is "coercive"~that

is

331

lim inf fn(t'x'u)~ f0(t,x,u) + ~

(Ju -vl p) , p ~ i,

convex and strictly increasing. Clearly applications can be made to variational stability of classical problems of calculus of variations. About the above results, see Zolezzi ( 4 ) -

3£~ Nec@ssary conditions for variational con v@rsence. Take gn(t,x,u) = a(t)x + b (t)u + c(t) , n so perturbing now only b, and minimize minly - x(T) I ,t2n = T fixed , y given (minimum final distance problem).

Then

b n --~ b o in L I is a necessary condition for strong convergence of optimal controls for y in some restricted region, when either Po is (completely) controllable,

or

the regularity assumption holds and uniform convergence of optimal controls u ---~u n o does not destroy optimality of u o. Moreover b n --~ b o in L 1

is a necessary condition to strong convergence of optimal controls

minimizing ~y - x(T) I ÷ z x(T) ~

y and z given

About such results see Zolezzi ( ~ )

4.

•

.

Among the (few) result on such problems (applications of which can be found in

many fields connected with optimization~ for example perturbation, blems) see results of

sensitivity pro-

Cullum ( I ) , Kirillova ( 2 ) o

Such known results are generalized and substanti~lly extended in this work. All the above mentioned results can be shown to be a by~product of a general method~ called "variational conversence" by the present author~ generalizing the clas sical direct method of the calculus of variations, and useful to obtain, for general minimum problems, both existence and "stability" under perturbations (from variational point of view). See Zolezzi ( 6 ) subject.

about some abstract results on this

a

332

References (i)

CULLUM~J.

Perturbations and approximations of continuous optimal control problems. Math.Theory of Control~edited by Balakrishnqn-Neustadt. Academic Press, 1967.

(2)

KIRILLOVA~F,M. On the correctness of the formulation of optimal control problems. S,I.A.M.J. Control 1,36-58(1963).

(3)

ZOLEZZI~T.

Su aleuni problemi debolmente ben posti di controllo ottimo. RIC. di MAT.21~I84-203(1972).

(4)

ZOLEZZI, T.

Su alcuni problemi fortemente ben posti di controllo ottimo. To appear in ANN.MAT.PURAAPPL.

(5)

ZOLEZZI,T~

Condizioni necessarie di stabilit~ variazlonale per il problema lineare del minimo scarto finale. To appear in B.U.M.I.

(6)

ZOLEZZI~To

On convergence of minima. To appear in B.U.M.I.

NORM PERTURBATION OF SUPREMUMRROBLEMS ([)

J. BARANGER , Institut de Math6matiques Appliqu~es, 38041 GRENOBLE C~dex

B.P. 53

FRANCE.

ABSTRACT Let E be a normed linear space, S a closed bounded subset of E and J an u.s.c. (for the norm topology) and bounded above mapping of S into JR. It is well known that in general there exists no s E S ~sueh that J(s) : Sup J(s)

see (even if S is weakly compact). For J(s) = ilx-sll (with x given in E), Edelstein, Asplund and Zisler have shown, under various hypotheses on E and S, that the set (s) : { ~ E

E~

s E S-such that IIs-xll : Sup IIs-xII}

xEs is dense in E. Here we give analogous results for the problem

sup s(S These results generalize those of Asplund and Zisler and allow us to obtain existence theorems for perturbed problems in optimal control.

1. THE PROBLEM. Let E be a normed linear space, S a closed and bounded subset of E and J an u.s.c. (for the norm topology) and bounded above mapping of S into ]{. We are looking for an s E S ,such that (1)

J(s) : Sup J(s)

see A particular (and famous) case of problem 1 is the problem of farthest points (i.e. J(s) = fix-eli, where x is given in E).

(~)

This work is part of a thesis submitted at Universit6 de Grenoble in 1973.

334

1.!. P r o b l e m ( 1 ~ has no solution in general (even with S weakly compact and J(s) = [Is-x[l with x given in E). Here is a counter example : Let E be a separable Hilbert space with basis el, i E S : {ei, i 6 ~,} O {0} is weakly compact. For any x E E. we have : !Ix-eillz = 1 + llxil2 - 2(x,e i) Now suppose that (x~ei) > O, ¥i E ~ ; then, we have : Sup Iix-siI = ~ 2 s6S

~ and this supremum is never attained.

].2. Existence results for the problem of farthest points. As we have just seen in i.i., this problem has no solution in general { however, Asplund [2], generalizing a result of Edelstein [i] -- who himself generalized a result of Asplund [i] -- has obtained the following : Theorem , (Asplund) Let E beareflexive

locally uniformly rotund Banaeh space and S a

bounded and norm closed subset of E. Then the subset of the x E E having farthest points in S is fat (x) in E.

2. THE PERTURBED PROBLEM. As it is impossible to assert that problem (i) has a solution, we consider the perturbed problem. Does there exist a s E S such that

(2)

J(~), fix ~II = sup (J<s), IIx-slI) sES

where (3) S is a bounded and (norm) closed subset of the normed linear of space E, J is an u.s.c, and bounded above mapping S into ~ , and x is given in E. We shall call an s E S verifying (2) a J farthest point (in short a JFP). We have the following generalization of Asplund's result : Theorem ] Le~ E be a locally uniformly rotund and reflexive Banach space ; then under hypothesis (3) the subset of the x E E admitting a JFP in S is a fat subset of E. Proof : The function r(x) : Sup (J(s)+llx-s!l) is convex, lipschitzian with constant sES i and satisfies : (~)

A fat subset in E is a subset of E which contains the intersection of a

countable family of open and dense subsets of E. By the Balre category theorem such a set is itself dense in E.

335

Sup

(~)r(x) : sup Sup (J(s)+blx-s[[) : sup [J(s) + sup Lb-siJ]

xEB(~,b)

x

s

s

x

= r(y)+b Theh Corollary of lemma

3 of Asplund [2] asserts that there exists a fat subset

G of E such that for every y E G~all p E Dr(y) (~) have a norm equal to one. Take such a p E Dr(y), I[Pll = i. We have : r(x) ~ r(y) + < p,x-y > , Vx 6 E,. Therefore (4)

r(2y-x) ~ r(y) + < p,y-x > , Vx E E.

E being reflexive there exists an x E B(¥,r(y))(we can always suppose r(y) > 0 ) such that : < p,x-y > = -r(y). Then (4) implies r(2y-x) ~ 2r(y). The converseis trivial

; so we have

r(2y-x) = 2r(y). Hence, for every n E ~,, there exists s

(5)

n

6 S ,such that :

ll2y-X-Sn[ I + J(s n) > 2r(y) - @ ( i

~ )

where @(g,t) is the modulus of local uniform rotundity (8) Set

u

=

n

s +x-2y n

if IISn+X-2yll ~ 0

llsn+x_2yil = ~

elsewhere

and (6)

tn = Sn+Un J(Sn)

'

We have tn+X-2y = Sn+X-2y+u n J(Sn) : Un(llSn+X-2yll + J(Sn)) so first

[ltn+X-2y[[ = ]J(s n) + t[Sn+X-2yl] [ --

a(s n) + tlsn+x-2yJl

this quantity being positive for n

sufficiently large by (5), and second u

-

n

t +x-2y n

lltn_X_2yll

(*)

B ( y , b ) i s the ball { x , tlx-y]t < b}

(~)

Dr(y) is the sub-differential

( )

for lltl[ = i

of r it y

~(e,t) = Inf {i - ~l[t+~l

, I[~I = 1 , l[u-tll >_ e}

336 Hence (5) gives

t]2y-X-tni I

--> 2r(y)

-

~(i

, r~_Yy))V-X

and this implies ! £ [

!ltn-xil

r(y)~

Thus~ tn converges towards x and Un towards ~

= u .

Finally there exists a sequence s n

Taking the limiz in q in (6) we see that s towards s = x - u 8 ,

!ls-ylt

q

+ J(s) : : > >

3. APPLICATION

such that lim d(Sn) = lim J(s)=8. n q q

converge (for the norm topology)

n

Then

q

ttx-uO-y]l + IIIx-yll-el+ tlx-ytl IIx-yll

J(s) J(s) :

+ J(s)

-8

= r(y), because of the uos.c, of J.

IN OPTIMAL CONTROL THEORY . We shall limit ourselves to just one example.

The state equation is : [ -V(uVy) = f

,

f 6 L2,(~)

(7) . y 6 H~(n) where ~ is open in IRn . A z d being given in L2(~) (or Hol(n)), we are looking for a ] 6 b a d

lly(~)-ZdliLe(or NO1)= u6lnf~dilY(U)-Zdll

(s) where

%d

such that

is a closed subset of : {u E LP~(~) ; O < ~ <_ u(x) <_ B a.e}

We take O < p < 1 in order to ensure the local uniform rotundity of LP([~). Notice that neither

~ad no___rrthe mapping u + y(u) are convex.

It is impossible to apply the theorem of Asplund to problem (8), the hypothesis being too weak. But we can apply theorem 1 to obtain

:

Theorem 2 For every E > 0 the subset of the w E LP(~) such that there exists E

~ad satisfying IY(U) ZdllL2(or H O)

L

: Inf !Iy(u)-zJIL2(o r Hl)-eilu-wllLp

is fat in LP(~). Proof : We apply theorem 1 with a(u) : - E! IIY(U)-ZdlI It remains nnly to show that J is u.s.c. In fact we have :

337

Lena

I u ÷ y(u) is a continuous mapping from LP(~) into H~(~) (these two

spaces being endowed with their norm topology). Lemma 1 is a consequence of : Lemma 2 u + y(u) is a continuous mapping from LP(~) with its norm topology into H~(~) with the weak topology. Proof of lenm~a 2 : Let um E LP converging towards u 6 Lp in norm. Put Ym = y (Um)" The variational form of (7) gives fOUm(VYm ")2 = f~f Ym Hence, by Poincarr~'s inequality f(VYm)2 ! I]~]L21lYmlIH~ Therefore there exists a subsequence Ym. which converges weakly in H 0i towards a y. ] Let us-now consider the variational form of (7) 1 fUm.VYm Vz = ffz Vz E H~ ] ] There exists a subsequence u which converges towards u (9)

m.

a.e. and

]k

l u . <~)-u<x)llw<x)l ! 2slv~(x)l e L~(n> . ]k Then, Lebesgue's theorem implies that u

Vz converges (for the norm topology in L 2) m. ]k towards uVz , We can now take the limit in (9) and obtain : i fuVyVz = ] f z Vz ~ Hd , so y = y(u). It is then trivial to obtain that the whole sequence Ym converges to y. Proof of lemma 1

: Consider :

X m = fUm(VYm-Vy)2 _> ~f(VYm-VY)2 _> 0 . We shall show that X X

m

converges towards zero

= fUm(VYm)e-2fumVYmVZ + fUm(VY)2 .

As the canonical injection from

H 01 into L 2 is compact

fum(VYm)2 = ff Ym converges towards

ffy = fu(Vy) 2

-2fUmVYmVZ converges towards -2fuVyVz (as we have seen in the proof of iemma I). Another application of Lebesgue's theorem shows that fUm(VY)2 converges towards fu(Vy) 2 .

338

4. OTHER RESULTS. Using Asplund's techniques, Zisler [i] has obtained three theorems which can be generalized as follows. Theorem 5. Let E he a Banach space whose dual is a locally uniformly rotund and strongly differentiable space (SDS) (~), S a closed and bounded subset of E ~ and J an u.s.c, and bounded above mapping of S into ~

.

Then the subset of the x E E~ having a JFP in S is fat in E ~. Theorem 4. Let E be a weakly uniformly rotund (~) Banach space, S a weakly compact subset of E and J as in theorem 3. Then the subset of the x E E having a JFP in S is fat in E. Theorem 5. Let E he a reflexive, Frechet differentiable Banach space, S a weakly compact subset of E' and J as in theorem 3. The same conclusion as in theorem 3 is valid. Proof : there is no difficulty to adapt the proofs given by Zisler using the same device as in theorem 2. We give here a proof of theorem 5 based on a different method than Zisler's. X being reflexive there exists an x such that ilx-yll = r(y) and r(2y-x) = 2m(y). Hence there exists s

E S ,such that : n ll2y-X-SnlI ~ 2r(y) - £n

(gn ÷ 0 when n + ~)

There also exists fn E X ~ such that ...llfnll= 1 and fn(2y-x-s n) = ..2y-X-Snll then r(y) > fn(Y-X) = fn(2y-X-Sn) - fn(Y-S n) > 2r(y) - e n

--

(X)

-lly-Snl

I ->- r(y) -

n

An SDS is a Banach space in which every convex continuous function is strongly differentiable on a G~. dense in is domain. (See [3] for more details).

(~)

A Banach space is weakly uniformly rotund if llXn[I ÷ i, tends imply Xn-YnV'weakly to zero.

ilyniI + i, IlXn+YniI ÷ 2

339 v

A theorem of

Smulian [i] now states that fn converges (for the

norm topology in X z) towards an f with II~I = 1 . Moreover r(y) ~ f(y-s n) = f(2y-x-s n) - f(y-x) so lim f(y-s n) = r(y). u->co S being weakly compact there exists a subsequence of s converging towards an n s 6 S. Such an s satisfies IIs-yll ~ f(y-s) = lim f(Y-Sn) = r(y) n the theorem is then proved for J = O. The device used in theorem 2 gives now the general case.

Remark. One may look for other perturbations llx-sIl2 hase been solved by Asplund [ 4 ]

than llx-sll ; the case

when E is a Hilbert space. We have obtained,

in collaboration with Tames [i]2 results for perturbation of the form ~(IIx-sIl) where is a positive, convex, increasing function such that lim e(u) = ~ . (E is supposed u->~o to be a reflexive Banaeh space having the property

:

(H) If a sequence x n converges weakly towards x and IIXnlI converges towards Ilxll then

ttXn-~l

+ 0

).

340

BIBLIOGRAPHY ASPLLBD~ E. [114

The potential of projections in Hilbert space(quoted in Edelstein Eli) •

ASPLUND~ E. [ 2].

Farthest point of sets in reflexive locally uniformly rotund Banach space. Israel J. of Maths 4 (1966) p 213-216.

ASPLUND, E. [ 3]-

Frechet differentiability of convex functions. Acta Math 421 (1968) p 31-47.

ASPLUND, E. [4].

Topics in the theory of convex functions. Proceedings of NATO, Venice, june 1968. Aldo Ghizetti editor, Edizioni Oderisi.

BAR~NGER, J. [ 1].

Existence de solution pour des problSmes d'optimisation non

convexe. C.R.A.S. t 274 p 307. BARANGER, J. [2].

Quelques r~sultats en optimisation non convexe.

DeuxiSme pattie : Th~or~mes d'existence en densit~ et application au contr$1e.

Th~se Grenoble 1973.

BAR~NGER, J.~ and T~I~M~ R. [ 1]. Non convex optimization problems depending on a parameter. A para~tre~ ED~TEIN,

M. [1].

Farthest points of sets in uniformly convex Banach spaces.

IsraeiJ. of Math 4 (1966) p 171-176. Sr@JLiAN, V.L. [1].

Sur la d~rivabilit6 de la norme dans l'espace de Banach.

Dokl. Akad. Nauk SSSR (N.S) 27 (1940), p 643-648. ZISLER, V. [1].

On some ex~remal problems in Banach spaces.

ON TWO CONJECTURES ABOUT THE CLOSED-LOOp T I ~ - O P T I M A L CONTROL Pavo i Brunovs
Mathematical Institute Slovak Academy o f Sciences Bratislava,

Czechoslovakia

Consider the linear control system (L)

~ = Ax + u

( x E ~ n, A constant), with control constraints u ~ U , vex compact poiytope of dimension

men

where U is a con-

imbedded in R n, containing

the origin in its relative interior, and the problem of steering the system from a given initial position to the origin in minimum time. While the theory o f the time-optimal control problem for the systems (L) has been satisfactorily developed as far as the structure of the open-loop optimal controls is concerned, this is not the case of their synthesis - the closed-loop time-optimal control. To synthesize the open-loop controls into a closed-loop controller is formally allways possible as soon as the optimal controls are unique. There are various reasons which make a synthesis desirable - the most important perheaps being that if a system which is under the action of a closed-loop optimal controller is deviated from Its optimal trajectory by an instantaneous perturbation, the rest of its trajectory will again be optimal for the new initial position. If the system (L) is normal, it is well known from the basic optimal control theory that the set ~ of initial points x 6 R n for which the optimal control Ux(t) , t E [ O ,

T(x)]

(T(x) being the optimal time

of steering x to O) exists is open in R n, the optlmal controls are unique, an~ they are plecewise constant with values at the vertices o f U. Its synthesis v is obtained by v(x) = Ux(O). It is ~enerally believed that the system under the action of the closed-loop controller v, (CL)

~ = Ax + v(x),

exhibits the following properties : (i) its behavior is indeed optimal

342

(ii) its behavior will not be severely affected by small perturbaDions. Formulating conjecture

(i) more precisely, it means that the so-

lutions o f (CL) coincide with the optimal trajectories of (L)~ the sense in which (ii) can be understood,

follows from Theorems 2 and 3 be-

low. Due to discontinuity o f

v, care has to be taken with the defini-

tion of solution o f (CL). Numerous studies of discontinuous differential equations in the fifties lead to the conclusion that the classical (Carath4odory) solution may not represent well the behavior of the system modeled by such an equation. In particular, it does not characterize the so called sliding (chattering) Which may occur along the surfaces of discontinuity. The necessity to modify the definition o f solution is clearly seen if one tries to investigate the conjecture (ii). The best and most universal definition of solution o f a discontinuous differential equation is due to Filippov. This, applied to (CL), defines a solution o f (CL) as a solution o f the multivalued differential equation (CLo)

~ E

Ax + V(x)

in the usual sense, where

: fl J-> o is the Lebesgue measure in and radius

tl

co el v

,

~(N) :0 R n and B ( x , ~ )

is the ball with center x

~.

Thus, in order to justify (i), we have to prove that the Filippov solutions of (CL), i.e. the solutions of (CLo) , are optimal trajectories o f (L). Unfortunately, i± turns out that this is not true in general and the following theorem~ which settles completely the case n = 2, shows that the systems for which (i) is not true, are not exceptional. We shall say that the closed-loop control v is "good", ifever~ solution of (CLo) is an optim~l trajectory of (L) sad, vice versa, every optimal trajectory of (L) is a solution of (~Lo). Theorem 1. Let

n = 2 and let (L) be normal. The____n, v i._ssgoo d i__f

and only i~ the following conditions are not met : _A has two distinct real eigenvalues and there exists e verbex w

suo

that

w>:

u>}

contains the oi envector

o__f -A correspondi_~n~ t__qoits larger eigenvalue but not the other

343

e igenvector o f -A . In particular, w is allways good if m = 1 . A typical example of bad synthesis is given at the end of the paper : v(x) = w I on the negative x I - semiaxis, which has measure zero, an@ therefore this value is supressed in the Filippov definition: the solution of (CLo) from a point x = (xi, O) instead of being a solution of ~ = Ax + Wl, will slide along the line of discontinuity x2 = O with epees ~I-- - x l =

co(Ax

+ w2~ Ax + w4}/%(x2 = O}

.

Let us now turn to the stability conjecture (ii). We shall model the pertur-bstions as measurable functions p(t) satisfying the estimate

Ip(t)i <

6

and we shall be interested in statements, valid for any

perturbation satisfying the given bound. Those can be conveniently expressed in terms of solutions of the multivalued differential equation (CLg)

E Ax + V(x) + B(O,~ ) Namely, (# (t) is a solution of (CL6) if and only if there exists

a measurable function ~(t), Ip(t)l~ 6 the equation

such that

~(t) is s solution o f

E Ax + V(x) + p(t)

Of course, one cannot expect positive results in case the synthesis is not good for the unperturbed system. Therefore, we restrict ourselves to n = 2 and we shall assume that the system is n o r ~ l

and no

vertex of U satisfies the conditions of Theorem I. Under these assumptions the following is true : Theorem 2. Let m = I. Then, for s~y compact K C ~ there exists an 6 > O po%nts x £ K

such that all solutions of (CL$)

and an~/ ~ > 0 st@rtin~ at

reach the origin in a time not exceedin~ T(x) ÷ 7 "

Theorem !. Le__~tm = I. Then, given sr~v ~

K ( ~ and ~

there exists an ~.>O such that any solution of ( C L % )

9>

O,

starting st _s

point x E K reaches B(O, ~ ) at tim___~enot exceedin~ T(x) and stays in S(O, ~ ) afterwards ~s far es higher dimensions are concerned, no definite results have yet be obteine~. However, there are some reasons to believe that no synthesis is goo8 if n > 2 is unclear.

and m > l, while the case n >

2, m = 1

344

F o r the

4etsi!ed

v e r s i o n o f the proofs o f T h e o r e m s

i - 3

cf. B r u n o v s k ~ .

xI = _ xI + uI

~

=

x2 + u2 ,

Wl=(l,O) , w 2 = (0, i), w 3 = (-1, 0), w 4 = (0, -1)

,

vcx~ - ~v4

¢cx ~ - "2

References

"

Brunovsk~,

P.~ The

publicstion Filippov,

closed-loop

time-optimal

control,

submitted

to S I A M J o u r n a l o n Control

A.F.

M stemati~eski_~

sbornik

51, 99-128

(1960)

for

COUPLING

OF

STATE

VARIABLES

IN

TRANSFER

THE

OPTIMAL

LOW

THRUST

ORBITAL

PROBL~I Romain HENRION Aspirant

F.N.R.S.

Aerospace Laboratory University

of Liege

BELGIUM

I. INTRODUCTION The high specific cost of orbiting a satellite explains the importance of the low thrust orbital transfer problem,

Indeed, mass and size of such an engi-

ne are much smaller than for an impulsive one, but its fuel expenditure can be as close as wanted,

if final time is left open. However, the nemerous studies

undertaken have stressed two main difficulties

:

- the possible existence of singular arcs (intermediate

thrust arcs, which do

not proceed directly from the PONTRYAGIN maximum p r ~ c i p l e ) ,

the opt m a l i t y

of which is not ensured; - the high s e n s i t ~ i t y tial conditions,

of o p t ~ a l

trajectories

with respect to the unknown

a consequence of the two-point-boundary-value-problem

ini-

intro-

duced by the maximum p r ~ c i p l e . We shall show here how these difficulties

are reduced to a great extent by de-

coupling the state variables with repect to the thrust amplitude 2. PROBLEM

FORMULATION

Us~g

non-dimens~nal

polar variables,

engine in a central inverse-square-force equations

control•

the plane t r ~ e c t o r y

field may be d e s c r i e d

of a low thrust by the following

:

r

=

u r

ue

•

r

2 • u r

ue

1

~----~+

(1)

sin ~

r

•

u6

U r U@ .

r

+ i

COS

C

where r , O , u r , u 8 are position and velocity variables, mass and c the ejection velocity•

g the instantaneous

Denoting by a the acceleration

factor, we

346

have the constraint

0<E~
The aim of o p t i m i z a t i o n minimize

is to choose the control variables

some cost function,

3. PONTRYAGIN

Yu%XIMUM

Denoting

~ and ~ so as to

e.g. fuel consumption.

PRINC IPLE

by i the adjoint

state vector,

the H a m i l t o n i a n

takes the form

H = H° ÷ $ H I

where

H = k ~ , q (q , ~ , ~)

As H is autonomous~

H=Ht

which

we have the w e l l - k n o w n

=O

shows that the numerical

The adjoint

equation

state variables

k=-H

value of H is a trajectory

are governed

constant.

by the equation

q

or explicitly 2 uo

u0 r

=

~e

2

ur ue

÷ ('-@ - ~ ) r

Xur - "-'2-r

r

~u~

XO = O

-

ur

r

.

X8

~u e

r

~

u8 r

u8 uo

Ur

2 r

[ sin

~

~u r + T

.

u Maximizing - for

H according

Zu

c o s ~ . au

+

sin ~ = X

] o

r to

~ue

the m a x i m u m principle, /%

m r

cos ~ = XUe/~

we get

: 9

}

where

+ ~2

X = (X~ r

u6

)

1/z

347

- for ~

However,

~ = 0

if

H1 < 0

= a

if

HI > 0

if H 1 vanishes

for some finite period, the principle no longer

fixes ~ for H is no longer an explicit

function of ~ . This case corresponds

to a so-called singular or intermediate thrust arc. Now, by deriving the condition H I = 0 several times,

it is possible to determine precisely that special

value { which keeps H I = 0 , But, its optimality

is by no means guaranteed.

We shall show that by decoupling the present state variables with respect to , we shall be led to a reduced system, governed non-linearly Applying then the LEGENDRE-CLEBSCH known optimality

4. PRACTICAL

conditions

LMPORTANCE

The practical,

for singular arcs.

OF

THE

DECOUPLING

i.e. numerical,

shown on figure I. We represented polar variables.

by a new control.

condition, we easily establish all the

OPERATION

importance of the decoupling operation

the flow-diagram

is

starting with ~ , of the

One easily sees that the coupling of this system is extremely

strong m as all the variables

influence one another directly.

Indeed, there are

only two stages. It is clear that this is very prejudicial bility

to numerical

precision and sta-

: some error in any variable has an immediate consequence

on all the

other ones. But, the larger the number of stages between two given variables, the lower the speed and magnitude of the influence of an error on the first upon the value of the second. Therefore,

our aim is to increase this number of

stages. We shall see that by making use of canonical transformations,

it can be

brought up to four. Results show also a serious decrease in sensitivity to errors in the unknown

5. CANONICAL

initial values.

TRANSFORMATIONS

Because of the hamiltonian

formulation of the problem,

best achieved by making use of canonical transformations.

the decoupling

is

They are described

by

the equation

k' d q - H dt = A' d Q - K d T + d F

(2)

which tranforms the set (q , t , % , H) into the set (Q , T , A , K).F stands for the corresponding transformations

generating

function. Except from the last one, all the

we shall use are of the MATIIIEU type where F = O.

348

5.1. Ist T r a n s f o r m a t i o n Initial

set : r ~ 0 ~, u t ~ u 0 , ~

r Associated

B

form : %r dr + X o d8 + k u r du r + k u8 du 8 + k u d ~ -

differential

H dt

(3) l The first Sransformation and is described

%

U

we shall do. has been given by FRAEIJS de V E U B E K E

by the relations

optimizing

,

$ . @ is n o w a new state v a r i a b l e

= k sin ,~ r

(4)

ue

New differential Substituting

form ~ Ar dr + k8 de + ld ul. + %~ d~ + %

(4) into

(5)

(5), we get

X

o u~

u=

(3) and identifying to

dB - H dt

sin ~ -

÷

cos (6)

u o = uk cos ~ + ~

M a k i n g use of derived

(4) and

sin

(6), the n e w expression of the H a m i l t o n i a n

is readily

:

=

HO

SH I

+

= u%

(sin ~ ~ %

r

+ cos ~ - - - - ~

+

~

sin~ r H I =~A

The n e w equations

q=H

r

sin ~ - cos ~ %r)

% 2

-~A

of state q and costate k can then be computed

k

q

from

:

349

(7) now shows that at present, governed

by ~ , namely

Their equations

there are left only three variables

uk , U and

have the follo~ing

uA = f(q,t,X)

+ i

~

directly

.

forms

.

:

. &

;

X

= u

5.2, 2nd Transformation The aim of this transformation Applying

the method

and KELLEY,

inspired

is to eliminate

from hydrodynamics

ul's

dependan,~y on ~ .

and used by FRAEIJS

de VEUBEKE 2

KOPP and MOYER 3, we solve the equation

du I

d~

1

1 c

which produces

u X = - c £n ~ + w

The integration The associated

constant

(8)

w wikl be used instead of ul as a new state variable.

differential

form is :

Ir dr + A0 dO + Xw dw + ~@ d~ + o d~ - H dt

Substituting

(8) into

(5) and identifying

(9)

to (9), we get

W

(io) U The expression

~

w

of the switching

function H I is now :

I H1 -- - --c ~

(ll)

It shows that a present, It may therefore

B is the only variable

be considered

as a new control

trying to apply the corresponding trivially decoupling

satisfied.

still directly variable

LEGENDRE-CLEBSCH

We shall therefore

now the system with respect

continue to ~ .

controlled

replacing

condition,

~ . However~

we see that

our decoupling

by ~ .

operations

it is by

350

5.3. 3rd Transformation As for the second ~ransformation~ the equations dr sin~

rdO = rd~ cos~ cos~

lead to the change of variables g=

r cos ~

h = r sin~

(12)

which implies g

= cos ~ , %

r

.......sin~ ~ , , (x~ +

kh = sin ~ ~ Xr

+ ze)

COS~

r

(~

Figure 2 gives the corresponding flow-diagram. We can draw now the following clues : I) ~ now governs only u ~ whereas h is the only non-ignorasle variable governed by

~ •

Moreover h appears non-linearly in all the equations. The application of the LEGENDRE-CLEBSCH condition to h, considered as a new controlj will produce a useful condition° This point will be developed in the next paragraph; 2) ~he coupling of the present system is much weaker than for the polar variables. Indeed, it seems that there are now four stages. However, for numerical integrations, the first stage is eliminated, as B has an analytical solution. In paragraph 8, we show how the number of four stages can be restored. 6. SINGULAR

ARCS

A singular arc being characterized by the vanishing of the switching function H I p we get by multiple derivation o

HI : 0

~

: 0

(t3)

~iI = 0

~h = 0

(14)

HI = 0

k2 _& %w

k

w

(g2 _ 2 h 2)

(h2 + g2) 5/2

= o

(15)

351

, this implies

As FRAEIJS de VEUBEKE I has shown that

W

g2 ~ 2 h 2

(16)

which is equal to the well-known relation I 1 - - - ~ sin @ .< - -

If

(17)

/f

(13) and (14) now reduce the Hamiltonian to

H = H

=

g

o

• X ~ ~w

3 h3 X w (h2 + g2) 5/2

(18)

Figure 2 shows that ~ influences the variables appearing in H only through h. So, as already indicated above, h may be considered as a new control. Indeed, (15) expresses the stationarity of H with respect to h. O = ~H

Xg 2

Xw (g2 _ 2 h 2)

Applying the LEGENDRE-CLEBSCH condition :

O > 32H -

-

,

3 Xw . h ,

-

~h 2

.......

[3

g2

-2

h2

]

(2o)

(h2 + g2) 7/2

By (16), this leads to

h ~ o

(2l)

and in conjunction with (17) :

-

1 --4

sin @ ~< O

.

(22)

If Now HI = O

- 6 g h (g2 + h 2) %g + n h (3 g2 _ 2 h 2) Xw

+ g (g2 _ 4 h 2) ~

HI--O

B

~ O

(23)

h (3 g2 - 2 h 2) X2 - 3 h (3 g2 + 2 h 2) w + 3 2

(g2 - 2 h 2 ) %~ + ~ g (6 h 2 Xg -2h

X o H w -

X

12 g2 ) X W

g

352

+ (3 g2 - 4 h 2) k2 + h ([3 g2 - 8 h 2) %

g

L0

÷ 3(4 g4 - 2 h 4 + 3 g2 h 2 ) %2 = 0 g

(24)

g X + h %h ~ E - C Zn ~ + w + - - - - ~ Iw

where

Equation (24) gives the value of ~ . The corresponding arc is physically feasable if Special

0 < ~ ~ a ® cases

H being autonomous and

%

= O

~ the numerical values of Ii and %

to

are

to

constant on the whole trajectory. a) H = ~

= 0 : this case corresponds to free final time and polar angle. to

I

One easily shows * that necessarily

~ & O. Sop in this case there are no

singular, intermediate thrust arcs. 5) H " O

~

~

# O

Eliminating %

(LAWDEN's spiral)

~ 7, and w from (24) by (15), (18) and (23), we get g to

= 3~ h (g2 _ 2 h2)(9 g4 + 4 h 4 - 7 g2 h 2) (g2 + h2)(3 g2 _ 2 h2) 3 (16) and (24) then imply ~ ~ 0 , which means that LAWDEN's intermediate thrust arc is not optimal. It therefore follows that optimal arcs can only exist for H#O. With this result~ one easily shows then, that (21) and (22) actually reduce to

h
7, NEW

sin@
CONSTANT

OF

MOTION

(SINGULAR

ARCS)

Making use of (12), (13) and (14)p it is straightforward to show that the following expression is constant on a singular arc : C=3Ht+wk

w

-2gk

g

Clearly, it may prove very useful for numerical integrationsof singular arcs.

353

8. TWO

ADDITIONAL

CANONICAL

TRANSFORMATIONS

In order to improve decoupling,

two additional

transformations

are useful.

8.1. 4th Transformation First, we shall substitute

to g, h and w the n e w variables

~ , 6 and v

according to the relations

k a = g kw I8 " h i w

v

=w-~g-Sh

The associated

k

differential

d~

form

+ ~8 dfl + Iv dv + I~ d~ + o dV - H dt

leads to =

6

tg/X w

-

" - Ih/I w

v

w

8.2. 5th Transformation With the differential

form

Aa da + 13 dfl + A n dq + It0 d0~ + < dB - H dt implies by (10) : I

II

q

v

v Putting n o w ~. q

~

We are led to

ep

- ~ eO = I

and the corresponding

P

differential

form with a non-vanishing

% d~ + A6 d8 + X~ d~ +%V dB + %P do - H d t - d~p Hamiltonian

:

generating

function

354 H = H° + ~ H

o

I

o~

p

- Xt3 exp (3n) (X2c~ +

(25)

2.-3/2

as)

H 1 = exp (0)IP - X / c

State equations -

,

a = - 2 ~g + 3 exp (30) I

= 2

_

~2

+ exp

(3n)(2

X B (l

5/2

+ Xg )

2 . x2)(x2 + X2)- 512

X8

a

a

o = 6~

(26) e

c.

XB = -

Xp + 2 ~ X

o~ + 2 B X

B

o

-

XO = 3 exp (3p) 18 (~2 + X2)

312 -

~

exp ( p ) / ~

o~

X

= ~ exp (0)/~ 2

Figure 3 displays the corresponding there are now four stages.

Even if ~ is eliminated,

So t t h e decoupling of this system is very strong and

it should lead to a serious Especially

flow-diagram.

the last variable

improvement

in numerical

precision

p should be rather insensitive

and sensitivity.

to a sudden change,

i.e. a switch of ~ , as it is separated by four stages from ~ . The following application

confirms this conjecture.

355

9. APPLICATION In order to cheek the quality of the system (26), we have used it to describe a fuel-optimal transfer trajectory, with open final time and angle. We have

H i k

- O

and consequently, there are no singular arcs m but only

thrusting ( ~ a a) and coasting (~ m O) arcs• Now, there are only five equations which must necessarily be integrated. Indeed : - the "orbital transfer of variables", introduced by FRAEIJS de VEUBEKE l allows to jump the coasting arcs; -k

=0;

- m is

igaorahle;

- k

is ignorable, provided the switching function H l is replaced by - H ; p o - adopting to = - ~c Po ' ~ may be replaced by - ~c t .

At present, with ~

= e -p Ie

;~ = e-P l B

~=e

-O I P

it is easily seen that p is also ignorable, and that only the five following equations are left : •

-

~--

~=

2 ~B+

2

5/2

3 ~I k2 (k21 + ~ )

- ~2 ÷ (2 X 2 2_

k21)(~ ÷ k~)- 5/2

X2 = 8 X2 + 2 a X I - X3 •

2l

X3 = - ~ X3 + 3 ~2 (X

2 -3/2

+ X2 )

+ ct

As the two following equations are ignorable, their integration has no effect on precision : P = 8 i

356

The corresponding

flow-diagram

is similar to figure 3, if B and %

suppressed and if %1 ' %2 and %3 are substituted to %

' 18 and %

are p

: there

are four stages, Numerical

results

We adopted

a = 0.03 c = 0.3228

which corresponds

to rather a low thrust level. The numerical

precision difficulties -

Figure 4 displays According

to the theoretical

degree of decoupling

of variables

and

the behaviour of p on one coasting and two thrusting forecast,

tion points where ~ switches,

- By integrating

stability

of this problem are well known : arcs.

the join of the curves at the junc-

is very smooth and neat. This confirms the high

announced by the flow-diagram;

several trajectories

forward and then backward,

the new system

exhibited on increase of precision of at least 2 significant

digits with respect to the polar variables; - The sensitivity

with respect to unknown initial values was reduced by I and

mostly 2 significant

digits. As a consequence,

points where the polar variables Moreover~

these improvements

did

convergence was achieved

at

diverge.

increased with the length of the trajectory

and the number of switchings.

I0. CONCLUSION The decoupling operation presented results

in this paper leads to two important

:

- an easy, direct theoretical - a considerable

examination

increase in numerical

with respect to initial conditions.

of singular arcs;

precision

and decrease

in sensitivity

357

REFERENCES

I. FRAEIJS de VEUBEKE, B. "Canonical Transformations and the Thrust-Coast-Thrust Optimal Transfer Problem" AstronauticaActa, 4, 12 , 323-328, (1966) 2. FRAEIJS de VEUBEKE, B. "Une g~n~ralisation du principe du maximum pour les syst~mes bang-bang avec limitation du nombre de commutations" Centre Beige de Recherches Math~matiqueso Colloque sur la Th~orie Math~matique du Contr~le Optimal. Vender, 55-67, Louvain (1970) 3. KELLEY, H.J.~ KOPP, R.E. and MOYER, H.Go "Singular Extremal{'in"Toplc8 in Optimization" (ed, G. LEITMANN), Ac. Press, chap. 3, 63-101, New York (1967)

x

r . . . . . . .

~

i

i

~

i

!

=

=S ~ 3 V I S

. .

i ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

J

L 3~IACJl3 •( " ' ) ~ = ~ uo!~enbe aq~ u! ~l~lo!ldxa s J ~ a d d e x

: su~gw

X

S39VIS

I

L ....................................

.;

359

r ........................

)

+

,"N,

,~, STAGES :

...........

+..............

I

,

~.-+

II

................

!

+.......

X

+

II

FIGURE 3

I,+ ~ 12

,!

;

\',

/

',

,

I

,

2

/:/ ".,",

Y "-

•

,,i •~

!

X,,,

:

../.

.... ,.Rus,.NoPN.sE

•

\

,..,

X

,

/, /

,/,

I

,

",, ,

6

,

7

,

8

>

t

-8

FIGURE 4

OPTIMIZATION OF THE AMMONIA OXIDATION PROCESS USED IN THE MANUFACTURE OF NITRIC ACID P. Uronen and E. Kiukaanniemi University of Oulu Finland

I. Introduction This work deals with a quite straightforward

engineering application

of modelling and optimization methods and therefore we will concentrate on the practical points of view, because similar methods may be used also in optimizing other processes using ageing type of catalysts, To be able to analyse and optimize the operation of a plant~ one must first have a mathematical model describing the real behaviour of the process.

The modelling of a plant can be done in many different ways;

one can use the physical and chemical relationships

the process is

based on~ or one can try to find the model experimentally.

So we can

have different models for the same process. The models based on physical and chemical relationships have a meaningful general structure~

ie. the right control and state variables

will be automatically taken into account.

However, the model will

include generally so many nonlinear differential and partial differential equations that the use of model without many approximations is impossible.

The empirical models normally can be quite simple in

form, but they do not have the same technically meaningful properties as models derived using the physical and chemical laws. Therefore a suitable combination of these both methods may be a good compromise and this semi-empirical method has been used in this work.

2.

Process descriptioq

The plant studied is the ammonia oxidation process used in the manufacture of nitric acid, Figure I shows schematically the whole pro-

361

Tail gases 1 7 Air

I

N H 3 ~

f---

,_L.1H 20

] I

i

! !

I

8

sll

8I I

T

I

f

Jq 6 ~II I' t___]

I

I J:1

!

i

I: 2: 3: 4: 5: 6: 7:

Compressor Preheater of air ~ixer Burner Waste heat boiler Cooler Oxidation tower 8: Acid coolers 9: Resulting acid Fig. !. Schematic representation of a nitric acid plant. cess. The raw materials are air and gaseous ammonia which are fed after mixing (about 10-11 Vol % ammonia in the mixture) to a reactor, where ammonia is catalytically burned to nitric oxide and water. The next steps are cooling of the reaction products, oxidation of nitric oxide to nitric dioxide and then absorbing NO 2 into water in several countercurrent towers to form nitric acid (55-60%). The first step, the oxidation of ammonia, is the most critical for the whole process. Various investigators /3/ have shown that the conversion in the oxidation of NO and in absorbtion is very high (9799%) and stable. Therefore the optimization of the ammonia oxidation process will mainly determine the optimum of the whole unit. The details of the chemical and kinetic phenomena included in the catalytic oxidation of ammonia are not yet fully known. Probably the

362

reaction takes place stepwise as fast bimo!ecular reactions

on the

catalyst surface and thus we can assume according to Oele /2/ that the rate controlling factor here will be the diffusion of ammonia molecules to the surface of catalyst. Platinum with 10% rhodium will be used as catalyst and it is conveniently provided in the form of wowen sieves

(4-6 sieves) of fine wi-

res. The service period of the catalyst varies from 3 to 5 months and its activity will decrease towards the end of the period principally according to Figure 2. A part of platinum will be lost in the use and this is an important factor to be taken into account in optimization of the process.

I conve rsio~i/~

98

94

9o

0.0

0.3

0.6

0.9

time normalized

_~ig. 2. A decreasing type activity curve of the olatinum c~talyst.

3. Mathematical model Based on the physical and chemical phenomena involved in the above process we can qualitatively conclude that the most important

factors

affecting on the conversion of the oxidation process are: ammonia to air ratio in feed, feed temperature,

total gas charge on sieves~

plant pressure, number of sieves and sieve dimensions. In following we will assume that the pressure will remain constant. As mentioned above we can assume that the rate of reaction will be controlled by the chemisorbtion the catalyst.

of ammonia moles to the surface of

363

Oele /2/ has derived the following transfer

(1)

in turbulent

:

0.924

.~

•

empirical

formula for the mass

flow Re 0 ' 3 3

, where

P Cp d a

is the mass transfer

l

is the thermal conductivity

Re is the Reynolds P

coefficient of the gas mixture

number

is the overall pressure

Cp is the molar spesific heat of the mixture d

is the diameter of the catalyst wire.

Using now w e l l - k n o w n balance

and the definition

equation (2)

formula for driving of conversion

= Inf(11.82 m o- 28.6)

number we get

, where

Cp~0.33d0.67G0.67

f

and Reynolds

(2):

In(l-x)

x

force in mass transfer, mass

i s t h e c o n v e r s i o n o f NH3 t o NO is characteristic

factor for sieves r e p r e s e n t i n g

wire

area - sieve area ratio n

is number of sieves

U

is the dynamic viscosity

G

is the total gas charge per sieve area

m o is the ammonia content Curievici and Ungureanu temperature

of the mixture

in feed

/I/ have shown that if we know m ° and To, the

of feed~ we can with a reasonable

mean temperature conductivity,

calculate

the

at the sieves and thus also the values of I, thermal

~, dynamic

near functions

accuracy

viscosity

and cp, molar spesific heat as li-

of m O and T O in the normal operation range.

So the

following model can be derived T .10-3(ggmo-228.7)+320m (3)

In(l-x)

=

o

.....

2-748.1mo-70.67 o

x

To'10-3(1.43+3.84mo)+13.06mo2+5.03mo+6.51 n

.

f

(27.45.10-3.To+93.44mo+13.61)0"33(d.106)0.67G0.67

This model can be adapted to a real plant using equation plied by a correction

factor k which can be determined

minal operating point values of the plant.

(3) multi-

using the no-

364 When this is done for example at three eonversion

levels representing

three points on the ageing curve of the catalyst, we can get the time dependence

for the correction

factor k. So the ageing of the catalyst

can be expressed as a polynomial of the normalized time t (normalized so that 1.0 represents An other possibility

100 days).

for matching the model to plant data would be to

keep all numerical coefficients

in equation

necessary amount of conversion measurements

(3) as unknown and if a could be made we could

formulate a quadratic error function and the values for the coefficients could then be found by minimizing this error function. The difficulty here is the reliable measurement of the conversion,

so

that the ageing curve of the catalyst could be estimated.

4. Simulation To test the model an extensive examplewise digital simulation was done. ditions

study

In this study the effect of small changes in process con-

(mo~ G, To, n~ f) on the conversion was investigated.

was carried out by a program written in FORTRAN IV. Figures and Tables

I and II represent the results.

out at one conversion

This

3, 4, 5

The simulation was carried

level which corresponds

to one age of the cata-

lyst i.e. one point on the ageing curve of the catalyst. The results of the simulation experiences

show good conformity with operational

and with previously published results concerning the

conversion/% m fo D G

95

94

i

=0. I05 =I. 39 =0. 0000?6 =0.253

N =5 T o n =443

93

.....

340 Fig°

360

380

3. Effect

400

47o

of the temperature

4- o on the conversion.

365

conversion/% I00 95

T o= 1 . 3 9

90

D G

=0. 0 0 0 0 7 6 =0. 253

N

=5

=443

85

- ~

~

~

"~-"

--. " - - ..

"-.. "--

mon=O. I05

~-. ,%

80

.~8o

.0

Fig.

.uto .d95 .I0o

.~85

4. E f f e c t of the m i x t u r e conversion.

conversion/%

IO0

.±o~

.±IO .II5

.I20

s t r e n g t h in feed on the

N

=

T

=443

5

m ° =0.105 fo = 1 . 3 9 D =0.0O0076

% =o.253 95

9O • 150

.I75

Fig.

.200

5, E f f e c t

.2'25

.~50

.275

.300

. ~ 2 5 "~ '

o f t h e gas charge on t h e c o n v e r s i o n . TABLE ii:

TABLE i: The effect of number of sieves on the conversion

The effect of sieve dimensions on the conversion

Constants:

Constants :

T m fo D G

T m G° N

='443 = 0.105

1.39 = 0.000076 = 0.253

= = = =

443 0.105 0.253 5

Conversion

number of sieves

Conversion

0.000076/1.39

0.000076/1.5 0.000060/1.2 0.000040/t.5

0.815120

3

0.9a0000

0.894678

4

0.940000 0.965819 0.980528 0,988907 0.993680 0.996400 0.997949 0.998831

5

0.961976 0.941904 0,990603

6

7 8 9 10 11 12

D/f

366

behaviour of nitric acid plants.

For example we can see, that in-

creasing total gas charge will decrease conversion,

Fig. 4. Physical-

ly~ increase in gas charge means shorter contact time and thus the result can be explained. 5. Optimization The model presented can be used to optimize the operation of the plant during one period of service. part in the process

As mentioned earlier the critical

is the oxidation of ammonia.

Thus the conversion

of the other parts of the process can be approximated as constants. The conversion of the oxidation process can be calculated with aid of the model as a function of the air feed around the nominal operating point values of the plant. three conversion

This was done in the example solution at

levels corresponding to three values for the factor

k i.e. three ages of the catalyst. I00

Figure 6 shows the result.

~ conversion/%

b

9O

/

8O

7O ~

O.7 Fig.

6.

These results

.

I

0.8 Effeo~

,,

t

t

0.9

i

li

1.0

1

i

I.i

IE

F3/F3N

1.2

of air flow on conversion.

can be combined by matching time polynomials

to these

curves and thus the final result for conversion will be function of the normalized time t and the air feed. (4)

x = x(tiF 3) = ao(t)+al(t)F3+a2(t)F3 2

The objective

function for optimization purposes can now be formed.

The total variable costs minus the price of produced nitric acid and export steam integrated over the service period will be used for mi-

367

nimization:

(5)

if ~

J

=

F2 F 3 _ K _ T__ 4 NT -- N KIF2N + ( K2+K3)F3 -

-

F2

KS£---~N - x ( t l F 3 )

J~

dt

%)

In this equation be individually

KI,..., 5 are weighted evaluated

ammonia and T is the temperature notes for the nominal values. mean the cost of ammonia,

cost coefficients

for each plant.

which must

F 2 means the flow rate of

rise in the burner.

Subscript

N de-

The first three terms in equation

(5)

the cost of energy needed for compression

of air and the cost of platinum losses which as a first approximation depend on the flow rate. The fourth term means the value of export steam and the temperature

rise can be shown to be easily calculated

when the values of F3, the air feed and mo, the ammonia concentration in feed are known /4/. The last term includes the conversion presses

and ex-

the value of produced nitric acid.

The linear relationships relative

changes

variations

have been assumed here between the costs and

from nominal values and this is reasonable,

if the

are not very large.

The exact values

for coefficients

K i should be determined

from real

cost curves of the plant. As constraints

for this optimization

I) mo~0.13 ~ the ammonia contents the lower explosion

problem will be: in feed is less than 13% which is

limit of ammonia-air mixture

2) 0.7 F3N e F 3 ~ 1.1 F3N , the capacity of air compressor. The solution of this optimization regressive

dynamic programming.

lues the solution was computed. graphically.

task can be evalueated

Using some preliminary Figures

va-

the results

From these curves we can conclude that the optimal straof ammonia at the

of the service period and air feed F 3 will increase

ammonia concentration period.

numerical

7 and 8 represent

tegy means a small air flow and high concentration beginning

by using

will decrease

Normal strategy

towards

and

the end of the service

is that ammonia concentration

will be held

constant over the whole period. The preliminary

calculations

also show that the optimal control stra-

tegy derived here will give about 2% better value for the objective function

in comparison with the conventional

method.

The realization

of the proposed method would be quite easy also with conventional

368

mo~ 13.0 12.0 II.0 ~

I0.0

___~~_

5__---u

9.0

0.0 Fig.

' 0.2

I

_

,

I0.I

, ~

0.4

t

i

m

0.6

e

0.8

normalized

7. Optimal control curve for NH3-concentration in feed. dynamic programming . . . . constant feed

I

t-/% 1.08

I

. . . .

.oo I O. 9O

i

O. 80

L_~_~ 0.0

•

1.037

, 0.2

0.4

0.6

,

,

,

0.8

~ time normalized

Fig. 8. Optimal control curve for the plant capacity. dynamic programming ..... constant feed

instruments providing that reliability and repeatability of the conversion measurement methods are remarkably improved. 6. Summar Z A semiempirical mathematical model for the industrial ammonia oxidation process has been derived. The model takes also the ageing of the catalyst into account. The model is used to simulate the behaviour of the plant and The results show a good conformity with the

369 operational experiences and published qualitative results. Based on this model an examplewise optimization study of the whole nitric acid plant has been carried out. The objective function used is the total variable costs minus total revenue value of production integrated over one service period of the catalyst. This minimization problem can be solved by using dynamic programming and preliminary numerical calculations give about 2% better result than the conventional strategy in plant operation.

7. References /I/

Curievici, I., Ungureanu, S.T., An analogical model of the reactor for ammonia catalytic oxidation, Buletinul Institutului Politehnic Din Iasi, Serie Noua, Tomul, XIV (XVIII), Fasc. I-2, 227, 1968.

/2/

0ele, A.P., Technological aspects of the catalytic combustion of ammonia with platinum gauze elements, Chem. Eng. Sci. 3, I-2, 146, 1958.

/3/

Roudier, Houssier and Tracez, An Examination of nitric acid process yields, Information Chimie, Vol. 12, 6, 27.

/4/

Uronen, P., Kiukaanniemi, E., Optimization of a nitric acid plant, Process Technology International, December 1972, Vol. 17, No. 12,

STOCHASTIC CONTROL WITH AT MOST DENU~RABL~

NUMBER

0F

C 0RRECTIONS J. Za b c zyk Institute of Fmthematics PAN Warsaw,

POLAND

Let us consider Markov processes space Eo By Markov ~orocess ~'~t,

pi)

where ~

mappings from

is a sample space with elements

pi

Xl

A strateg~_~ ~ ~2

d 1,d2,~o®

being

~-algebras

and probability Pix describe the motion

x ~E.

is a sequence

~°°"

~

= ((1~i,d±))

where

are stopping times with respect to { ~ t } and

are functions mapping

measurable with respect stopping times

; ~t' ~ are

o The measures

starting from

~

xt, ~t'

are random variables such that

--~--*--~, eta(s) = ~(t+s) are defined on ~

of the process

0 = ~1

xt

for all t >i 0, ~ 6 ~

operators 0t

defined on a state

we shall mean an object ( ~ ,

~0,+ oo) into E,

xt <~) = ~(t)

measures

~

X I, o. o , ~

to ~ I '

~ I ~ ~2' °'"

corrections and the functions chosen at the moments

~ ~2'

into

LI,2,°°",k~

,

~°°

respectively,

( the

may be interpreted as moments of d I ,d2, ..o

~I,~2'°'°

indicate the processes

)"

To simplify the next definition we shall suppose that the controlet chooses the moment ginning from =

e~( di+1~

~i+I

T io That means where

knowing the past of the process be~i+I

= e ~ . ( v J i+1 ~ l

~ii@,is any stopping time and

and

'

di@,any

di+ I =

~"--

measurable function° Every strategy

~

defines new measures

PN,Y~ , Px~ , N=I...,

371

x@E

which satisfy the conditions

1,~' ~x

d~(x) (~) = ~x (~)

,

N,~I~

Px

N-1

(A

~

~(B))

Px(A):Px (A), Let

,~"

= Ex

dN

(Px~N

Ae ,~'~ ~

(B) ; A)

B~"

J ~ ~

J~

, N:2,3,.-

be some non-negative Borel functions defined

C1,oo.,ck, g

on E° Let us put

"C~4 v N (x) :

N

I g(Xs)ds- ~d±(x~±))

(

O

~(x

[

vN(x) =sup~

~m v~ (x)

,

if the limit exists

-

,

otherwise

OQ

v~(x)

, v~(x) =s~

There holds the following theorem

(sea

(x) ,

~ eE.

C41 )

Theorem Io If the functions C1,o..,ck Gig,..., Gkg , where i ~ Gig(x) = Ex( ~ g(xs)dS), are bounded and (finely) continuous then O

I)

vN

are bounded, Borel functions also

2) v~+1(x ) = sup

~(

,

V~ ~ ~

,

g(xs)ds + v~(x~)) - ci(x)

~,i

O

~,i

Let us consider an example

.~xample. Let

E

Let

be stopping points for both processes, if the cost

-~, ~

be an interval

[-~

and

372

functions

c1~c 2

are equal to a number

c ~0

and the reward is

equal to the exit time from the open interval

(-~,~)

(see

0

[¢~ ) there exist two numbers

best strategy, in the case of if ~ ~ o process

N = + ~,

and the starting point X2

x

such that the

is the following:

is in

(-~,0~

and wait until it reaches the point

point choose the process and so~n~

~o' ~o ~

if

~<~o

choose the process

XI

choose the

~o

and at that

and wait until it reaches the point -~o

and the starting point X2

then

and

~2

= +~

x

is in

(-~,

" The number

~o

01

satisfies

the equation th2~

= 2~

-c

o

It is rather surprising that the correction points do not depend on ~

~o'- ~o

o

Profo Eo Dynkin posed the problem to find the best strategy for the model considered in the example if, at any moment

t ~ 0

controller knows the states of the process only at moments ~ i

the ~

to

As far as we know the exact solution is not found yet° In virtue of the solution of the example it seems reasonable to use the following strategy (for simplicity let us suppose that choose the process moment

-x +

X2

~ o

and

x e(-~,0~):

and as the moment of the next correction the

~o "

Now we give "excessive characterization" of the function A Borel function~is said to continuous for all

belong to

(see Ro Bl~aenthal, R° Getoor x @ E

~i [II

v~.

if it is finaly. ),

~ -

ci

and

and all stopping times i

~heorem 2. Under the same assumptions as in Theorem I~ the functio_n

v~

is the least function

v - Gig @ ~ i

for

v

i = 1,2,o.o,k.

which satisfies the conditions,.

373

Proof°

Since,

for every

i = 1,2,.o.,k

i @

and @

= ~z ~(x) - ~.~(~ ~(x~)) therefore Now,let

for

ci(x) + (v~- ~ig)(x)@ v - Qig 6 ~ i

N. = 1,2, . . . .

V~GIg

for

Indeed

~(v~-

i = 1,2,o..,k

v~

rio If

V~VN_

+

wN =

I

so

then

:

t,

.

the proof.

Analogous time problem°

then v ~ v ~

v ~ ~x( &g(xs)dS + v ( x r ) ) - c i ( x )

- c i , and therefore

This completes

Gig)(x~).

theorems

can be proved if we consider the stopping

in that case we define

max

%

vi)-

j4.,:--j

w@o= sup i~N N w N = sup w N where

~

and

w@~ = sup w o o

is a bounded,

function defined on The equations

2) and

W~.l(X) = sup

Borel measurable

and finely continuous

E o 3)

from Theorem

x(~,~N)(~ ) - oi(~

I have now the form

~uw~

= m~(~,w~)

~w~=

.~x(%~)

~,i

w~(x)

sup

-

~,i

An e~mple of that ~ n ~ An excessive

characterization

(N = 2) m s considered in [31 . of

w~

in the case o£

c i ~.

0

374

i = 1,2, o~o~k

was given by Griegelionis,

Shiryaev

L2~ .

References

~

RoMo Bl~menthal,

R~Ko Getoor, Narkov procasses and Potential

Theory, Academic Press, New Yor~ - London, 1968 [21

B.Io Griegelionis, A.No Shiryaev,

On controllad Narkov process6s

and the Stefan problem, Problemy P ~ d a c h i

luformaaii,

4(1968), ppo 60-72 3]

Jo Zabczyk,

A mathematical

correction problem, Kybernetika,

8(1972), pp.317-322 ~I

J. Zabczyk ,

Optimal control by means of switchings,

Y~thematica 45(1973), ppo161-171

Studia

DESIGN OF OPTIMAL INCOMPLETE STATE F E E D B A C K C O N T R O L L E R S FOR LARGE LINEAR C O N S T A N T SYSTEMS

W.J. Naeije,

P. Valk, O.Ho Bosgra

Delft U n i v e r s i t y of Technology, The Netherlands.

Delft,

SUMMARY In this paper the theory of linear optimal output feedback control is investigated in relation to its a p p l i c a b i l i t y in the design of highd i m e n s i o n a l linear m u l t i v a r i a b l e control systems. A m e t h o d is p r e s e n t e d w h i c h gives information about the relative importance of the inclusion of a state vector element in the output feedback.

The necessary condi-

tions of the o p t i m i z a t i o n p r o b l e m are shown to be a set of l i n e a r / q u a d ratic algebraic m a t r i x equations.

Numerical algorithms are p r e s e n t e d

w h i c h take account of this linear/quadratic character.

I.

INTRODUCTION

In the design of feedback controllers for linear, of high dimension,

t i m e - i n v a r i a n t systems

i m p l e m e n t a t i o n restrictions may result in the addi-

tional c o n s t r a i n t t h a t

the control is a function of only a limited set

of elements of the state vector.

O p t i m i z a t i o n theory can then still be

used if the structure of the controller,

as s p e c i f i e d in advance,

used as an additional constraint relation.

is

In this paper the controller

w i l l be assumed to be a t i m e - i n v a r i a n t m a t r i x of feedback gains. As many s e r v o m e c h a n i s m - and tracking problems can be reduced to regulator problems with a t i m e - i n v a r i a n t feedback matrix,

the optimal output regu-

lator p r o b l e m as d i s c u s s e d here can be viewed as the basis of a wide class of controller design problems. In existing literature on this subject,

the n e c e s s a r y conditions of the

m a t h e m a t i c a l o p t i m i z a t i o n problem are derived for a d e t e r m i n i s t i c problem setting. As questions regarding existence and uniqueness of solutions are easiest solved in the case of o p t i m i z a t i o n over a finite time interval,

this p r o b l e m has drawn a t t e n t i o n first [i] . E x t e n s i o n of the results

to the infinite time interval showed the d e p e n d e n c e of the t i m e - i n v a r i a n t feedback m a t r i c e s upon the initial conditions of the problem. Moreover, only n e c e s s a r y conditions could be given, and existence and uniqueness of the solutions could not be q u a r a n t e e d [2,3]. The c o r r e s p o n d i n g

stochastic output regulator p r o b l e m [ 4,5] leads to

e s s e n t i a l l y similar necessary conditions.

The n e c e s s i t y to choose noise

376

intensity ditions assume

matrices

in the d e t e r m i n i s t i c initial

n-dimensional intensity

conditions, unit

matrix.

be r e p l a c e d knowledge

In using

arise

At first,

portance back,

to be able

costs

and i m p r o v e d

the m a t r i x

paper

for w h i c h

to make

which

these

the n e c e s s a r y

two p r o b l e m s

y(t)

= C x(t)

+ B u(t)

state-vector

x(t)~

y(t)

noise,

characterized

input

of d i m e n s i o n s

Constraining

the

= F y(t)

J = E{xT(t) x(t),

F(S,F)

=

feed-

implementation

numerical

solution

conditions

of

of the

algorithm.

a short r e c a p i t u l a t i o n

This

of the

CONDITIONS

E{X(to) } = 0

(i)

feedback

u(t) ~ no£se v e c t o r

r, k r e s p e c t i v e l y . w ( t ) = 0, E{w(t)

positive

w(t)

and output

is G a u s s i a n

wT(~) } = ~ 6(t-T)

semidefinite

intensity

white

in w h i c h

matrix.

to

= F C x(t),

F C = L

problem:

(3) choose

F as to m i n i m i z e

the quadrat-

index Q x(t)

u(t)

If S = E{x(t) closed-loop

im-

system

vector

n, m,

by E{w(t))

to the o p t i m i z a t i o n

ic p e r f o r m a n c e

linear

+ H w(t)

> 0 is a t i m e - i n v a r i a n t

with

in litera-

(2)

vector

leads

NECESSARY

the t i m e - i n v a r i a n t

= A x(t)

u(t)

two

in the o u t p u t

a suitable

on

problem.

~(t)

with

after

using

design,

the r e l a t i v e

between

constitute

based

here.

exists

about

Secondly,

II. Consider

solution

can

of the system.

in c o n t r o l l e r

compromise

to of the

a unity noise

matrix,

setting

element

con-

such a trick

intensity

behaviour.

be p e r f o r m e d

technique

background

information

a suitable

problem must

treats

no s u f f i c i e n t needs

problem

theory

initial

on the surface

to a s s u m i n g

problem

of a state vector

control

equations

optimization

of a noise

regulator

the d e s i g n e r

of the i n c l u s i o n

optimization

is e q u i v a l e n t

on the p h y s i c a l

output

used

distributed

in the s t o c h a s t i c choice

to c h o o s i n g

So the w i d e l y

the use of a s t o c h a s t i c

the optimal

main problems

case.

sphere [I] However,

by the p r o p e r

is e q u i v a l e n t

uniformly

of or a s s u m p t i o n s

This m o t i v a t e s

ture.

in this case

+ uT(t)

R u(t)}

and F s a t i s f y i n g

xT(t)}

(i),

is the s o l u t i o n

(4) (2) and

(3).

of t h e v a r i a n c e

equation

for the

system: (A + BFC)S

+ S(A + BFC) T + H ~ H T =" 0

(5)

377

the p e r f o r m a n c e

index can be written:

J = tr{(Q + cTFTBTRBFC)S} Using the m a t r i x - m i n i m u m

(6)

principle [6] by introducing

a Hamiltonian

function H(S,F,P)

= J + tr{F(S,F)P T}

(7)

in which P is the adjoint matrix,

the necessary

conditions

follow

from: ~H(S~F~P)~s ~ = ~ = 0

_

~H(S,F,P)~p ~=

~H(S,F,P) ~F

F(S,F)

(8)

= 0 .~

The feedback matrix F, resulting

from the necessary

conditions,

is:

F = -R-IBTpscT(cscT) -I in which

(9)

the adjoint matrix P = pT > 0 is the solution of

P(A + BFC)

+ (A + BFc)Tp + Q + cTFTRFc = 0

(i0)

and the variance matrix S = S T > 0 is the solution of performance

criterion

(5). The resulting

is

J~ = tr{ PH~HT} A necessary

(ii)

and sufficient

which minimizes stabilizable.

for the existence

Necessary and sufficient

stabilizability

are presently

in [ 7]. Evidently, and

condition

(6) is that the system

a necessary

unknown;

(i),

conditions is that

feedback

for output feedback

a sufficient

condition

of a solution F

(2) be output

condition

(DT,A) be detectable with Q = DD T [ 8]. If an optimizing

exists,

uniqueness

cannnot be proved generally.

the necessary conditions the equations

(5),

solution,

e.g. by comparing

and investigating

the performance

feedback

the observation

forms a m i n i m u m variance y(t)

solution

= C x(t).

by the projection: [ 9, p. 88]

estimate

to that

the nature of this

index belonging

index obtained with optimal complete

From the structure of the feedback matrix F follows, output

use of

(I0) in which P > 0 and S > 0, implying

the closed loop system is stable, the performance

solution F

So the practical

consists of finding a numerical

(9) and

is given

(A,B) be stabilizable

to it with

state feedback. that the optimal

~(t) of x(t),

This linear estimate ~(t)

given

is determined

378

~(t)

: scT(cscT) -I C x(t)

(12)

and this v e c t o r is used in the same feedback structure as is e n c o u n t e r e d in o p t i m a l c o m p l e t e state feedback: u(t) = -R-IBTp i(t)

(13)

In the sequel, with no loss of g e n e r a l i t y the output m a t r i x C will be assumed to be C

=

[Cli 0 ] w i t h C 1 non-singular.

In that case,

(12) can

be w r i t t e n as

LI ~(t)

i0]

= - . . . . . ,-T -i~ S12SII I 0j

x(t),

(14)

S =

with p a r t i t i o n i n g c o n s i s t e n t with the p a r t i t i o n i n g of C.

III.

S E L E C T I O N OF O U T P U T V A R I A B L E S

For a p r a c t i c a l a p p l i c a t i o n of the theory of optimal output feedback, i n f o r m a t i o n is r e q u i r e d about the importance of i n c l u s i o n of each element of the state vector in the output feedback. choices by c o m p u t i n g the r e s u l t i n g p e r f o r m a n c e from a c o m p u t a t i o n a l point of view.

However,

C o m p a r i n g all p o s s i b l e

indices seems u n r e a l i s t i c

any c o m p u t a t i o n a l l y

involved m e t h o d of c o m p a r i s o n m u s t be of an a p p r o x i m a t i v e nature.

less Here

a m e t h o d is d e v e l o p e d b a s e d on a c o m p a r i s o n of t r a j e c t o r i e s of an ideal system

(e.g. using optimal

feedback of the c o m p l e t e state vector) with

t r a j e c t o r i e s of the same system under c o n d i t i o n s of o u t p u t feedback. Let x°(t)

be the t r a j e c t o r y of the system under c o n s i d e r a t i o n and L O

the o p t i m a l feedback m a t r i x using c o m p l e t e state feedback: ~°(t)

= A x°(t)

u°(t)

=

+ B u°(t)

+ H w(t)

(15) Let x(t) ~(t)

LO

xO(t)

be the t r a j e c t o r y of the system w i t h output feedback m a t r i x L:

= A x(t)

+ B u(t)

+ H w(t) (16)

u(t) = L x(t) = FC x(t) The d i f f e r e n c e b e t w e e n both trajectories,

e(t) = x(t)

- x°(t),

is

g o v e r n e d by the equation [ 2] 6(t)

=

(A + BL) e(t)

+ B (L-L °) xO(t).

Define the source term vector q(t) q(t)

= B

(L-L °) x°(t)

in

17)

(17) as 18)

379

Assuming

that A + BL is stable,

ries will be minimum source

in some sense if q(t)

term objective

I = E{qT(t)

q(t)}

the difference

between both trajecto-

is minimal.

Introduce

as a

function

= E{tr(q(t)

qT(t))}

Inserting

(18) in (19) and interchange

operators

leads to

(19) of the expectation

and trace

l I = tr I(L - L O) S°(L - L°)TJ

where

(20)

S ° is the variance matrix of the optimal

state feedback,

which

tion of the Lyapunov

is the symmetric

positive

system using complete (semi)definite

(A + BL°)S ° + S°(A + BL°) T + H~H T = 0 The necessary within

condition

the constraint

~-~I I,

for minimizing

(21) I over all possible

choices

(22)

Elaboration

of

(22) leads

to

L~ = L°s°cT(cs°cT)-Ic the inverse

minimizing operating

(23)

exists.

the source

[i0, Ch. XI]

$1

a projector

extreme value of the source

term objec-

(24)

of the output matrix C and the correspond-

of S ° and L °,

l

o,,

(24) and

o I" = tr[L2(S22

(23)

(25)

leads to

1 o - S~S~IS12)~2]

As a computational

alternative,

I" = tr[ L2S22IJ2 °~-l~I]

(S°) -1

by applying

state feedback matrix L °.

- L O)T]

structure

ing partitioning

S =

feedback matrix

is

Using the adopted

and combining

complete

. The corresponding

I ~ = tr[(L ~ - L °) S ° ( ~

=

Note that ~ , the output

term in (17), is obtained

upon the optimal

tive function

s°

L

L = FC is

= 0

assum!ng

solu-

equation

I S12 =-%--77C--1

(26) (26) can be converted

into[ ii] (27)

(28)

380

and p a r t i t i o n i n g Equation

(26)

estimating element state

and

the r e l a t i v e

eleme n t s

all

determine U I. This

(24)

as a f e e d b a c k selection

forms

matrix

procedure,

cases

in w h i c h

(17).

and further process.

U 1 are s e l e c t e d

elements

means

state vector

the optimal

Assume

complete

further

that a

on t e c h n o l o g i c a l

(l 4 n). F r o m the set U/,

to form the o u t p u t

of

vector.

This

grounds k ~ 1

selection

as follows.

i~ by o m i t t i n g

L ° and a p r o j e c t i o n

in eq.

output

of each

Assume

L O and S ° are known.

feasible

the i-th e l e m e n t

from U/,

i = 1,2.°./

and

I~ ±= m i n ±.. ~ ~erm . t u/_ 1 . Dy o m i t t i n g the j-th element from 3 l l process can be r e p e a t e d up to U k, It should be noted, that

Is in eq.

those

of i n c l u s i o n

structure.

elements

are to be s e l e c t e d

may be p e r f o r m e d Compute

importance

parameters

measured

(25).

a computationally

feedback

1 state v e c t o r

as p o s s i b l e

with

(27) provide

in the output

feedback

set of

consistent

the square of a m a t r i x

of L °, and that this projection, in the s e l e c t i o n based

engineering

can be r e p l a c e d

(24),

L~,

This

L ~ does selection

procedures

in

the s y s t e m

are p o s s i b l e

in the d e c i s i o n

feedback matrices

other m a t r i c e s

that a

be useful

not stabilize

can be i n c l u d e d state

of

is not used

implies

can still

other

complete

by c o r r e s p o n d i n g

(24),

matrix

constraints

the o p t i m a l

procedure.

on e q u a t i o n

a feedback

Based on eq.

Also

norm of the d i f f e r e n c e

which

L ° and S °

render

"ideal"

s y s t e m behaviour.

IV. For a n u m e r i c a l proaches a.

solution

OF N E C E S S A R Y

CONDITIONS

of the o p t i m i z a t i o n

problem

two basic

ap-

are:

Use of n u m e r i c a l gradient matrix

b.

SOLUTION

function minimisation

of the p e r f o r m a n c e

algorithms

index w i t h

respect

based on the

to the p a r a m e t e r

L;

Numerical

solution

of the m a t r i x

equations

e.g.

by iterative

pro-

cedures. The g r a d i e n t matrix

for a similar gives

~ =

~Ll

of the p e r f o r m a n c e

L can be a n a l y t i c a l l y

index w i t h

derived,

p r o b l e m [ 12, Ch.

5.7].

respect

using This

to the p a r a m e t e r

a suggestion

derivation

by K w a k e r n a a k

(see a p p e n d i x

I)

as a result:

2 RLISll

+ BTp

~s12J

(29)

381 in which the output feedback m a t r i x L has the structure L = [ LliI 0]

(30)

and L and S are p a r t i t i o n e d

consistent with the adopted structure of

the output matrix C. S and P follow from (A + BL)S + S(A + BL) T + H~H T = 0

(31)

P(A + BL) + (A + BL)Tp + Q + LTRL = 0

(32)

Numerical

algorithms

that use

(29) need the solution of

for each step where gradient evaluation gradient evaluations

generally

is required.

is large,

this class of numerical

applicable insight

for high-dimensional

into the analytical

As the number of

convergent

For high-dimensional

properties

systems,

approximation

search

t e c h n i q u e s does not seem to be easily systems.

Moreover,

no use is made of

of the equations

nor the ana-

lytically known solution of the feedback matrix L, equation

by successive

(32)

either in a pure gradient

method with a small step size or in quadratically methods,

(31) and

(9).

solution of the matrix equations

techniques

(5,9,10)

seems useful. At first, proper-

ties of the matrix equations must be investigated. Equations

(10,9) can be written as

ATp + PA + Q - PBR-1BTp + [ I - c T ( c s c T ) - I c ] •

PBR-IBTp[I

- scT(cscT)-Ic]

Using the adopted structure projector

(33)

: 0

for C and the partitioning

of S (14), the

in (33) can be written:

4 thus

.

:

$_

(34)

kol

(33) becomes: 0 I-S 1 S12

F(P,S)

= ATp + PA + Q - PBR-IBTp +

0: PBR-iBTp

If consistent

partitionings

f

0

T -i = 0 -S12SII : IJ

(35)

are made:

(36) FI2

F22~

382

then F22 turns out to be linear in P22 and i n d e p e n d e n t of S: T + T T + = 0 F22 = P22A22 + A22P22 PI2AI2 + A I 2 P I 2 Q22

(37)

while FII and FI2 are q u a d r a t i c in PII and PI2' So F(P,S)

is a m i x e d

l i n e a r / q u a d r a t i c m a t r i x e q u a t i o n in P. It can be noted that the a p p e a r a n c e of S-terms in

(35) can be eliminat-

KL °I

ed by a p p l y i n g a s i m i l a r i t y t r a n s f o r m a t i o n on the state vector:

x+

= i

x

T -i -S12SII

(38)

I

A l t h o u g h this t r a n s f o r m a t i o n can not be c o m p u t e d in advance and so has no p r a c t i c a l

significance,

c h a r a c t e r of

(35)

it does not affect the l i n e a r / q u a d r a t i c

and it shows that the role of S in e q u a t i o n

(35)

is

only limited to a t r a n s f o r m a t i o n of state space. As a conclusion~ requirements At present,

a c o m p u t a t i o n a l a l g o r i t h m p r i m a r i l y m u s t m e e t the

set by the l i n e a r / q u a d r a t i c c h a r a c t e r of F(P,S) two i m p o r t a n t a l g o r i t h m s

A x s ~ t e r a l g o r i t h m [ 13, p.

31~,

for e q u a t i o n s

s u g g e s t e d by A n d e r s o n and M o o r e

a.

are the

based on [5] and later adapted to the

t i m e - i n v a r i a n t case by Levine and A t h a n s

these a l g o r i t h m s

(5,9,10)

= 0.

[14], and a simpler a l g o r i t h m

[13, p. 314]. M a i n d i s a d v a n t a g e s of

are:

The a l g o r i t h m s do not take a d v a n t a g e of the p r o p e r t i e s of the m a t r i x e q u a t i o n s as m e n t i o n e d before;

b.

The a l g o r i t h m s require a s t a b i l i z i n g initial output feedback m a t r i x w h i c h can be d i f f i c u l t to determine;

c.

The m a i n d r a w b a c k of both a l g o r i t h m s is the n e c e s s a r y c o n d i t i o n that the c l o s e d - l o o p of the iteration. stability,

system m a t r i x remains stable in the course + The a l g o r i t h m s provide no g u a r a n t e e for this

and p r a c t i c a l a p p l i c a t i o n s

s i m p l e s t cases both algorithms that A x s i t e r ' s J = tr(PH~H T) system remains

show that in all but the

fail for this reason. A l s o the fact

a l g o r i t h m converges

in the p e r f o r m a n c e index

[5,14] is only valid as long as the c l o s e d - l o o p stable,

and so has no s i g n i f i c a n c e as a proof of

convergence. Better a l g o r i t h m s m i g h t be d e v e l o p e d if the m e n t i o n e d properties of the e q u a t i o n s a.

(5,9,10)

are taken into account:

The p r e s e n c e of the S-terms in eq. of the state space;

(i0) results in a t r a n s f o r m a t i o n

383 b.

The equation

(i0) has a mixed linear/quadratic character; as a

result of the adopted structure of the output matrix C, the linear and the quadratic parts appear in separate partitions of the matrix equation. If a Newton-Raphson algorithm could be analytically derived for the equations

(5,9,10), these properties would be incorporated in the

algorithm. However, the complexity of the equations prohibits such an approach. Thus, a linear converging algorithm is the only realizable proposition. As the role of S in eq. applied to eq.

(i0) is limited, Newton-Raphson

(10) separately, assuming P as the only variable, may

provide a basis for an algorithm. The result is:

(see appendix II).

Pk+I(A - BR-IBTPk) + (A - BR-IBTPk)TPk+ 1 + ~kPk+IBR-IBTPka ~ + ~kPkBR-IBTPk+Ie ~ + Q + PkBR-IBTp k - ~kPkBR-IBTPk~ ~ : 0

(39)

-SIIS I ~k = Eq.

I

K

(39) is a linear matrix equation in Pk+l" Numerical solution re-

quires the use of a Kronecker-product and this is inefficient or impossible for high-dimensional systems. As Lyapunov matrix equations can be efficiently solved, even for high-dimensional systems [15,16J, an adaptation of (39) to the Lyapunov structure is desired. Three algorithms that perform this step will be suggested. All are based on the replacement in eq. ~k~k+IBR-IBTPk

(39) of the term

+ PkBR-IBTPk+ 1 - PkBR-IBTPk~ ~

by a term AQk:

Pk+l (A - BR-IBTPk) + (A - BR-IBTPk)TPk+I + Q + PkBR-IBTp k + AQ k = 0 (40) (Algorithm !) AQk = ~kPkBR-IBTPk~ ~

(41)

(Algorithm I I ) A Q k

(42)

(Algorithm

III)

= ~ k < - T : - - 1 BR-IBT -. pR ~K~ kj

AQk = a

'

BR-1BTPk + Pk BR-1BT

PkBR-1BTPk~k ~ T

(431 PR in (42,43) is a predicted value for P22 and is determined by an equation representing (37), the linear partition of (35): T + Q22 = 0 PRkA22 + A2]PRk + (PI2)kAI2 + AI2(PI2)k

(44)

384

Due to the a p p e a r a n c e of the c l o s e d - l o o p system m a t r i x (40), the i t e r a t i o n in S~ based on

(A - BR-IBTp k) in

(5) can be chosen as

Sk(A - B R - I B T P k )T + (A - B R - I B T P k ) S k + HCH T

!%:

0

[0;

-1 T I-,. . . . . . . . . . + BR B Pki ~ ~ T~-I~-I

(45)

0

I. . . . . .T. . -i . + ~;i-$22-S12SIISI

So the a l g o r i t h m s consist of i t e r a t i v e l y solving by

(41),

(42) or

(43). In the latter two cases,

PkBR-IB T = 0 k 1

(40,45) w i t h AQ k given also

(44) must be solved

at each step. These algorithms have the following properties: i. If

(A - B R - I B T P o ) is stable,

k = 1,2...

then

(A - B R - I B T P k ) is stable and Pk > 0,

(algorithms I, II; see a p p e n d i x III).

2. The initial s t a b i l i z i n g feedback m a t r i x -BR-IBTp back matrix,

is a state feedo a l l o w i n g the a l g o r i t h m to start on the stable optimal

state feedback matrix.

Known algorithms require initial stabilizing

output feedback. 3. C o m p a r i n g with known algorithms,

the range of c o n v e r g e n c e is signifi-

cantly i n c r e a s e d due to the g u a r a n t e e of s t a b i l i t y of the closedloop system m a t r i x and because the linear and q u a d r a t i c p a r t i t i o n s of the m a t r i x e q u a t i o n However,

the c o n d i t i o n s

(i0) are treated s e p a r a t e l y in the algorithms. for c o n v e r g e n c e can not e x p l i c i t l y be given

due to lack of k n o w l e d g e about e x i s t e n c e and uniqueness of the solutions to eq. 4. In a l g o r i t h m s

(5,9,10) . II and III, A22 m u s t be a stable m a t r i x for

e f f i c i e n t l y solvable [ 1 5 , 1 6 }

Under this condition,

(44) to be

PR = P22 in a

s t a t i o n a r y solution of the a l g o r i t h m s as can easily be p r o v e n by reg a r d i n g the fact that P22 is a p o s i t i v e d e f i n i t e solution of a quadratic m a t r i x e q u a t i o n and hence is unique.

IV.

APPLICATIONS

In the a p p l l c a t i o n to t e c h n o l o g i c a l

systems,

the suggested output se-

lection p r o c e d u r e has p r o v i d e d s a t i s f a c t o r y results. No numerical problems were encountered. An a p p l i c a t i o n to a 1 4 - d i m e n s i o n a l o p e n - l o o p stable boiler system

~7~

is shown in fig.

r e s u l t i n g o r d e r i n g of state vector elements

I. The a p p l i c a t i o n of the in optimal output feedback

w i t h a s u b s e q u e n t d e c r e a s i n g number of output e l e m e n t s yields s u b s e q u e n t increasing values of the p e r f o r m a n c e

index,

fig.

2. The fact that this

sequence is flat over a c o n s i d e r a b l e range can be i n t e r p r e t e d as a satisfactory result of the s e l e c t i o n algorithm.

The results of fig.

2 were

o b t a i n e d using a l g o r i t h m II° C o m p a r i n g the c o n v e r g e n c e p r o p e r t i e s of the

385

proposed

algorithms with the algorithms

Moore as applied Anderson/Moore

to the same system,

algorithm

of Axs[ter

failed when using 7 or less output variables

and the Axs~ter algorithm when using 6 of less. to instability of the closed-loop I, II, III showed convergence spectively. in fig.

system matrix.

The proposed algorithms

for 8 output variables

is that convergence

number of output variables,

that the algorithms

Both failures were due

up to 3, 2 and 1 output variables

The speed of convergence

3. The general experience

decreasing

and of Anderson/

showed as a result that the

m e t h o d of R.A. Smith

equation computations,

[15 ]was L .

slows down with

fig. 4. It should be mentioned

II and III exhibited practically

iour. For all Lyapunov

identical behav-

the accelerated

used.

10-'

ISELECTION

| STEP

10-z

///

STATE-VECTOR

~

12 L 5 11 1 7 6 % 2 9 ] fig. 1

PERFOR-

Selection procedure

]35

WiANCE J~ INDEX

ELEMENT

8 13 10

applied to 14-dim.

boiler system,

BOILER-SYSTEM DIM.=14

30

25

20 #OUTPUT-FEEDBACK ELEM.

15

1L 13 12 11 10 9 S 7 6 5 ~ 3 2 I fig.2

Performance upon number

re-

is shown

of optimal output-feedback of.output elements.

dependent

series

386

Log i tlLk+l-Lkll 110 ~

1 ]0'

"•., DI . . . .

"*",

&

,\

":.

<

-%,.....NAXAT F_R ""A L G ]I ~ ITERATION

=

10'u__.

1

23

L 5

6

?

t5 9 10 11 12 13 ]& 1S

fzg.3 Convergence behaviour of numerical algorithms.

l0 ~

t

Log

i

~1Lk+{ L k tl

'

.....

"

.....

SOILER-S'Y SJEM [)1~1.=14 ....... A L G I

z ~" " ~ OUTPUTELEMENTS

- - - : A LG X

~'~6 #

! o ig.

4

2

3

& 5 6

7 S

ITERATION

g 10 11 12 13 l& 15 16 17 1~ 19 20

Convergence of proposed algorithms dependent upon number of output elements.

V.

CONCLUSIONS

The results of this paper involve an a l g o r i t h m for s e l e c t i o n of output variables

in linear optimal output feedback control and improved numer-

ical a l g o r i t h m s feedback;

for solving the n e c e s s a r y c o n d i t i o n s of optimal output

these a l g o r i t h m s take into account a n a l y t i c a l p r o p e r t i e s of

the r e l e v a n t l i n e a r / q u a d r a t i c m a t r i x equations.

With these new results,

387

the range of a p p l i c a b i l i t y of optimal output theory in linear output c o n t r o l l e r design certainly can be increased,

as only very few ~18,19~

numerical a p p l i c a t i o n s have appeared that use existing algorithms. However,

further improvements in the use of the s u g g e s t e d algorithms

may be e x p e c t e d if q u e s t i o n s regarding n e c e s s a r y and s u f f i c i e n t conditions for e x i s t e n c e and uniqueness of solutions to the r e l e v a n t m a t r i x equations are better understood.

REFERENCES I. D.L. Kleinman, M. Athans, The design of suboptimal linear timevarying systems. IEEE Trans. AUt. Contr. 13(1968), 150-160. 2. R.L. Kosut, S u b o p t i m a l control of linear t i m e - i n v a r i a n t m u l t i v a r i a b l e systems subject to control structure constraints. Ph.D. diss. Univ. Pennsylvania, 1969; also IEEE Trans. Aut. Contr. 15(1970), 557-563. 3. W.S. Levine, T.L. Johnson, M. Athans, Optimal limited state v a r i a b l e feedback controllers for linear systems. IEEE Trans. Aut. Contr. 16(1971), 785-793. 4. P.J. McLane, Linear optimal stochastic control using instantaneous output feedback. Int. J. Contr. 13(1971), 383-396. 5. S. Axs~ter, S u b - o p t i m a l time-variable feedback control of linear dynamic systems w i t h random inputs. Int. J. Contr. 4(1966), 549-566. 6. M° Athaus, The m a t r i x m i n i m u m principle, inf. Contr. 11(1968), 592-606. 7. M.T. Li, On output feedback s t a b i l i z a b i l i t y of linear system. IEEE Trans. Aut. Contr. 17(1972), 408-410. 8. A.K. Nandi, J.H. Herzog, Comments on "Design of s i n g l e - i n p u t system for specified roots using output feedback". IEEE Trans. Aut. Contr. 16(1971), 384-385. 9. D.G. Luenberger, O p t i m i z a t i o n by vector space methods. Wiley, N.Y. 1969 i0. M.C. Pease, Methods of m a t r i x algebra. A c a d e m i c Press, New York 1965. ii. T.E. Fortmann, A m a t r i x inversion identity. IEEE Trans. Aut. Contr. 15(1970), 599. 12. H. Kwakernaak, R. Sivan, Linear optimal control systems. WileyInterscience, New York 1972. 13. B.D.O. Anderson, J.B. Moore, Linear optimal control. Prentice-Hall, E n g l e w o o d Cliffs, N.J., 1971. 14. W.S. Levine, M. Athans, On the d e t e r m i n a t i o n of the optimal constant o u t p u t feedback gains for linear m u l t i v a r i a b l e systems. IEEE Trans. Aut. Contr. 15(1970), 44-48. 15. R.A. Smith, M a t r i x equation XA + BX = C. SIAM J. Applo Math. 16(1968) 4 198-201. 16. P.G. Smith, N u m e r i c a l solution of the m a t r i x e q u a t i o n A X + XAT + B = 0. IEEE Trans. Aut. Contr. 16(1971), 278-279. 17. O.H. Bosgra, A p p l i c a t i o n of optimal output control theory to a model of external power station boiler dynamic behaviour. Report N-95, Lab. Meas. Contr., Delft Univ. Techn., Stevinweg i, Delft, The Netherlands, 1973. 18. E.J. Davison, N.S. Rau, The optimal o u t p u t feedback control of a synchronous machine. IEEE Trans. Pow. App. Syst. 90(1971), 2123-2134. 19. M. Ramamoorty, ~. Arumugam, Design of optimal c o n s t a n t - o u t p u t feedback controllers for a synchronous machine, Proc. IEE 119(1972), 257-259.

388

APPENDIX

I

Let X be an element of L. Partial d i f f e r e n t i a t i o n Z-~(A+BL) ~-~

of eq.

(31) gives:

. ~ ~S ~S T ~ T S+,A+BL)~-+~]-(A+BL) +S.T~(A+BL) = 0

tr

(AI)

(Q+LTRL)S

: tr ~ Q + L

T

%S ~ RL)~+~(L

T

using

(6)

using

(32)

using

(AI)

RL).S]

~S+ 2~-.SL ~L T R ~i = tr _ {P(A+BL)+(A+BL) T P}7~ J

F ~ T , q%L~ L T 2tri~SPB+~-~SL R i This

last expression

is equivalent

to equation

APPENDIX

(29).

II

E q u a t i o n (i0)~ w r i t t e n in the form (35) as F(P) = 0, can be differentiated, using ~ as defined mn (39): dF=-dP. A - A T d P + d P B R - 1BTp+PBR- 1BTdp-~ dPBR- IBTp~-~PBR- 1BTdp~ = 0 (A2) Writing F and P as properly ordered vectors F and P, (A2) becomes: dE--=-(i~I) dP- (~ A T )dP+ (BR- 1BTp~I )dP+ (I®PBR- 1 B T )dP- (BR- 1BTp ~ )dP

-(~aPBR-IBT)dP

= 0

(A3)

In (A3), the derivative of F with respect to P is explicitly Inserting this derivative in the N e w t o n - R a p h s o n expression | d P l (Pk*l-Pk) L --2 K

= -F(Pk )

and making the conversion to (39).

(4)

from vectors

back to matrices

APPENDIX Proof of property

I for algorithms

totically

then by Lyapunov's

stablen

(40) can be written

directly

leads

III

I, If:

Q + PkBR-IBTp k + AQ k > 0 and the pair As

given.

If

(A - BR-IBTp k) is asymp-

theory Pk+l > 0, because

in (40)

(A,Q ½) is assumed to be detectable.

as:

Pk+!(A-BR-IBTPk+I)+(A-BR-IBTPk+I)TPk+I+Q+Pk+IBR-IBTPk+I+AQk+ +,p -P ~BR-IBT(Pk+I-Pk) t k+l k' and the solution

P .~ of

(A5) is positive

(A5)

= 0 definite,

by Lyapunov's

CONTROL OF A NON LINEAR STOCHASTIC BOUNDARY VALUE PROBLEM

J.P. K E R N E ~ Z

J.P. QUADRAT, M. VIOT

Facult~ des Sciences

I.R.I.A.

6, Boulevard Gabriel

78 - ROCQUENCOURT

21000 - DIJON

I - POSITION OF THE PROBLEM The aim of this paper is to describe an optimal feedback control of a biochemical system described by partial differential environment.

Such biochemical

systems have been described and studied in the deter-

ministic case in J.P. KERNEVEZ these membraneous sections.

systems,

equations and submitted to a random

(~

. In this section we give a short presentation of

leading us to the stochastic model studied in the following

In section 2 are given some indications

about existence and unicity of a so-

lution for the state equations.

In section 3 is given a way to approach an optimal

feedback control of the system.

In section 4 the particular

is considered partments

and numerical

I and 2. The membrane

In the compartments

case of a linear feedback

results are given. An artificial membrane

separates 2 com-

is made of inactive protein correticulated

are some substrate

S

and some inhibitor

I

with enzyme.

which are diffusing

in the membrane. S=I, I=0

COMPARTMENT

S=I, l=w

MEMBRANE

COMPARTMENT 2

I

x

S is reacting in the membrane because of enzyme which is a catalyst of a biological reaction.

In this paper we are interested only by the stationnary

case will be treated in a paper to be published

case. The evolution

(J.P. QUADRAT, M. VIOT

[7) ). Let us

call y(x) = substrate concentration i(x) = inhibitor concentration.

at point

x

in the membrane

(O < x <

I)

390

The stationnary

case equations i

~ y(x) = ]~i(x) +I y(x) [

i y" (x)

(1.1)

I

{ i(x)

are

=

wx + (l-x)

; y(O) = y(1) = I

e

I

i (x)

I

1

0

x

In ( 1 . 1 ) 2 parameters may be random : O depends upon how much activator not well-known,

w

~ . . F

and w .

O

is in the system and this quantity of activator

is the concentration

of inhibitor

To control the system we have at our disposal e , inhibitor concentration compartment.

Moreover

the control,

to be efficient,

of the system.

In the present case observation

is the flux of substrate entering

-y'(O).

Therefore

in the Irst

will have to work in a feedback

closed loop from an observation

that is

is

in the 2rid compartment.

the membrane

at x=O,

controls will be of the form

(1.2)

e = u(y'(O)) +

where

u

is some function from

~

into

IR . The cost function to minimize

average deviation between y'(O) and a fixed value (1.3)

rain

is the

zd :

Ely'(O) - Zd 12

uE ~ad where Uad

is some fixed subspaee of functions with values in

rently very simple unidimensionnal

~+,

For this appa-

problem we are however faced with 2 main difficul-

ties : using feedbacks (1.4) Existence

(1.2) leads to a boundary value problem of the type y~'(x) = F(y(x),y'(O),x),

y(0)

and

y(|) given

and unicity of a solution for (1.4) are not standard.

In section 2 we shall

391

see that we must, for instance, impose to the feedback law u(.) to be monotone decreasing. This condition has a physical meaning (see remark 2.3) and led us to the notion of resulatory feedback. Another difficulty is the stochastic aspect of the control problem. We can no more use a variationnal approach, as in BENSOUSSAN

(I) . Here we used an algorithm, called

of "independant simulations", which is an extension of the strong law of large numbers to stochastic control problems. This method was already tested in the framework of stochastic dynamic programming (J.P. QUADRAT

~)

, J.P. QUADRAT, M. VIOT

~ ) ) and

finds here a new field of applications (see section 3). In computations we looked for an optimum of (1.3) in a class of linear feedbacks : 'u(y'(O)) =

i

(1.5)

( ~ (Y'(0)-z d) + B) +

~

0

and bounded

B ~

O

and bounded

This paper gives only a presentation of the ideas and of the main results. A more detailed study including proofs can be found in (7) . II - EXISTENCE AND UNICITY OF A SOLUTION FOR THE STATE EQUATIONS Let us call (2. l)

u :~ 2

=

p

+ IR

÷

~+ ={

a continuous application

(~,w)[

~0

,

w

~0

)

= some measure of probability on

Let us assume that (2.2)

/~

( 2 + w2)d~<

We wish to solve the stochastic system (2.3)

y" = ~

;

y(0) = y(1) (2.4)

i(x,~)

(2.5)

0(~) = u(y'(O,~))

p.p.x

E )0,I (

; a. 8.60

(p.s.)

= w x + (:-x)

0(~)

(a.s.)

For a given ~ , one can prove existence (but not unicity) of a solution for problem (2.3), (2.4), (2.5) by compacity methods (J.L. LIONS (4)). Then one finds a solution to the stochastic problem using a "measurable sections" theorem. (For a similar situation see A. BENSOUSSAN, R. TEMAM ~ ) suit :

). Therefore we can state the following re-

392

Theorem 2.1 Under hypothesis

(2.i),(2.2), the stochastic system (2.3)-(2.5) admits a solu-

tion y such that (2.6)

Y E

L2(Q,D;H2(O,~))

;

O~

y<

]

(a.s°)

Remark 2.1 Without any feedback (2.5)~ the system would be (2 7)

y~, =

•

d

l+wx+(1-

;

y(O)

= y(1)

=

1

The second member being monotone increasing with respect to city of the solution of (2.7) for o,w,e

y, one gets easily uni-

given and positive.

Unfortunately, in our problem we lose this monotonicity because of (2.5). Therefore the unicity problem must be approached in a different way. For

O> O,

w ~ O

given and

@ varying in

~+, let us call

y@

B

the solution of

(2.7). Then we can prove the Len~na 2.1 The function for

@ = O

for

@ ÷ ~

y~ (O)

v :8

is continuous, strictly increasing, with

- ~ < Yo'(o) < o y~(O) ÷ 0

Remark 2.2 For

d > O

of the function

and v : e ÷

w>~O ' 0

Ye( )

given, let us draw in the plane (y~(O),@) and of a feedback

the graphs

u:y'(O) ÷ 0. e

~f~

U

> y' (o) Every point of their intersection is such that (2.8)

9 = u(y~(O))

i.e. the constraint (2.5) is satisfied. Therefore, in general, with any feedback law -! if u=v , it is

u, there is not a unique way for the system to work. For instance~ clear that every solution tone decreasing,

v

Ye ' @~ O, of (2.7) will verify (2.8). But if u is mono-

being strictly increasing, the 2 graphs have only one point of

intersection, and this for every Therefore we have shown the

~ = (o,w), ~ >0, w ~ O, the caseo =O being trivial.

393

V

-~

U

~' (o) Theorem 2.2 If

u

is continuous, monotone decreasing from ~ into ~+, the stochastic sys-

tem (2.3-2.5) admits a unique solution. Remark 2.3 The choice of a monotone decreasing feedback law

u

implies that when the flux

of substrate entering the membrane, -y'(O), is becoming less intense, the feedback regulates it by lessening the inhibitor concentration

in the Irst compartment,

u

so that

the transformation of substrate in product increases, y decreases in the membrane and -

y'(O) increases.

One can check the same regulatory effect of the feedback law

when -y'(O) is increasing.

u

Therefore one gets a stable steady state. This is what is

expressed by the preceding result of unicity. We shall call regulatory the monotone decreasing feedbacks. In the following sections we shall work mainly with linear resulatory feedbacks of the form (2.9)

u(y'(O)) = ( ~(y'(O)-z d) + B )+

Numerical integration for ~ For a given

~ =(O,w)

(~

O)

fixed.

and a given feedback law (2.9) the system (2.3), (2.4), (2.9)

is solved using the following under-relaxation method : let

Lg

be the operator from

H2(O,I) into H2(0, I) defined by

(2. 10)

L ( y ) = g(l-x

- s ) F ( y , y ' (O))ds +

( x - s ) F ( y , y ' (O))ds)+ ( l - a ) y

~0

(2.11)

F(y,y'(O))

=

0

u y l+wx+(1-x) (~(y f (O)-Zd)+~)

+

+y

Then the following sequence is defined (2.12)

Yo = l

;

Yn+l = Lg (yn)

For ~ small enoug there is convergence of (2.12). (The integrals are approximated by

394

a classical method : Newton-Cotes, Gauss,..~). III - THE CONTROL PROBLEM Let

Uad

be ~he set of linear regulatory feedbacks defined by

(3. i)

u(y~(O)) =(~ (y'(O)-Zd)+~) + ; -M]~<~
For u E al~d and

~0 = (~,w)E ~2

(3.2)

y~, = - -

mize on

Ua d

O~< 8.~< M 2.

let y(u,00) be the solution of equation

0 y I+wx+(1-x)u(y'(O))+y

Let ~ be a law of probability on

;

2 ~+

; y(0) = y(1) = I.

and

zd

a number ; the cost function to mini-

is given by

J(u) = /

(3.3)

ly'(u,~)(o) - Z d 12 dD(0J) ~2 + It can be shown, using hypothesis (2.2), that the cost function J(.) is continuous with respect to the parameters

~ and ~

defining the linear feedback u. So that the

control problem (3.1), (3.2), (3o3) admits an optimal solution, for every measure verifying (2.2). Remark 3.1 When the measure D is a discrete measure of the form (3.4)

'~ = ! '

r

r ~ 6 j2 ! ~ -J

2

Dirac measure at point

~j,

J the problem (3.2), (3.3) can be written

(3.5)

Y~ = |+w.x+(1-x)u(y'(O))+yj

(3.6)

min

;

yj(O) = yj(1) = I ; j=l,...,r r

J(u) = r-1

u EUad For any measure

j=!

~, the idea is to discretize it in a sequence I Dr = r

(3.7) (3.8)

IYj'(o)- df2 Dr

r

~

.1"= 1

6

co,j

(e1,..o ,~j,.. o) = sequence of independant simulations of co according to the law ~ .

Then we must solve the problem (3.5), (3.6); this is possible by purely deterministic methods (see section 4). This procedure is justified by the following result of convergence ; let

~r

(resp 4) the minimum cost associated to the measure (3.7) (resp

395

the initial measure

~) ; let

Ur = (~r

~ ~r)

an optimal linear feedback for D , then

Theorem 3.1 ~or almost e v e r y s e q u e n c e o f i n d e p e n d a n t s i m u l a t i o n s , (3.9)

lim r

~r

=

an d eyery convergent subsequence of the sequence

~r~r

) converges towards an ~ )

which is optimal for ~. Remark 3.2 It is clear that theorem 3.1 is an extension of the strong law of large numbers. it can be also expressed in the following abstract form :

Let (X~(~))~ E A

be a fa-

mily of integrable random variables and for every s E A , let (X~(~))j >I be a sequence of independant random variables following the same law that X~. Then under some hypothesis of continuity of X in ~ and of compacity of the set A, one proves that : (3.10)

min sEA

! r

r I j=l

Moreover, f o r ~ f i x e d ,

~(~)

let

converges almost surely towards

(3.11)

E(~).

~ (~) such t h a t r •

I ~

min SEA

r

f XJ% (~) = min ! j=l Sr SEA r

~ X j(~) j=l s "

Then every convergent subsequence of the sequence (~r(~)) converges towards an such that (3.12)

E(~)

= min ~6A

E(X )

In our case S= u , X (0~) = ]y'(u,~)(O)-Zd]2 . So that the linearity of feedbacks is s not essential in the conclusion of theorem 3.1. IV - OPTIMAL LINEAR FEEDBACK CONTROL Let

Ua d

be given by (3.1) and let (~l,...,~r) be

of the random parameters

r

independant simulations

~= (~,w). From (3.5) and (3.6) the cost function to minimize

becomes r Jr (s 'B) = -I r j~l]Y](O)-

(4.1)

Zdl2 ; - M E s ~ O

; O~

B~M 2 •

We use a gradient method with respect to s and B • Let us assume that, (4.2)

S(yj(O) - Zd) +~ > 0

then the partial derivatives

~jr ~s ' ~sjr T

V j =l...r,

can be obtained by the following

Theorem 4.1 The gradient of jr is given by the relations (4.3)

~jr = _ ! ~ r

j=1

1%J O

~

dx

396

(4.4)

X, ~

=--

3B

r j=1

J0

dx

J

~Y Fj (y,z,~,$) =

(4.5)

the

l+w.x+(1-x) I~(Z-Zd)+-~ 3

+y

Xj , j=],o.o,r being obtained by integration of the primal and dual systems : v~l "j = Fj(yj,yj(O),~,~)

(4.6)

~ !X0

~ ~Y

=

;

yj(O) = yj(!) =

T (yj,yj(O),~,~) %. J

/i

! (4.7)

~.j(1) = 0

Remark 4. ] The integration of (4.6) was made by under-relaxation using the operator M (X) = g [ ( 1 - x ) e +

-

(2(y '(O) -Zd) xas}

+ (l-s)

The sequence X o = O, ~ n+! = M if ¢

~z %

-

x

1

3F x (l-s) ~s as +

0

X

(%n) is then converging towards a solution of (4.6)

is taken small enough.

Remark 4.2 The optimal open loop control problem is included in the preceding one, by ta(O,M2). In the following we can compare the performances of

king ~ = O, B varying in

these 2 types of control and verify the improvement given by the feedback part. NUMERICAL RESULTS In the following 3 pictures we see respectively

:

. In the Irst one ~ is random on (30;40) with a uniform distribution, w is fixed at

w=6 and we have looked for an optimal open !00.~ control to approach

zd = -2.

All the curves are between the 2 extreme ones corresponding to ~=30 and ~ =40 . In the 2rid one ~ is still random and

w

fixed as in the Irst picture, but

now we have looked for an o_~timal linear feedback contro~ and the values of y'(O) for all the curves between

o = 30 and ~ =40

. In the 3rd picture buted between

fit very closely to

Zd=-2.

~ i s fixed at 36 and w is a random variable equally distri-

3.75 and 8°25. This time again an optimal linear feedback control gives

a good minimization of the deviation between y'(O) for all the curves and Zd=-2.

397

/ 0.5~

zd:-2 - - ~

++

0.4~

u"=40

/ /

0 3 // SOLUTION

0.2~ /

OPEN LOOP OPTIMALE cout=0.0173

~=

o.1// /

W'=

/ /

__

O"

Y(X ].0 / J

•~

jI

!

!

•i

I

01

0.2

Q3

0.4

0,5

Q6

"#

2.17 6.0

........

0.7

t

I

0,8

09

x"

l

/

0.9

0.8 o.7

.~ / /

/ /

o.5 0.4

S

zd=-2.0

"\

/

T=40

\

0.3 S / 0,2

SOLUTION

/

FEEDBACK OPTIMALE

/

o(='200 ~=2.15 W= 6.0

o.~ So

cout = 10-6

I

0.1

i

0.2

I

0.3

l

0.4

i

QS

I

0.6

I

0.7

I

0.8

i

0.9

F

X

398

Y(X) 1.0 0.9

/

% J /

O.8

0,7 0.6

0.5 0.4 O.3

SJ .d / S / .d

zd='2

%4",=3.75

/

/ /

..tK)LUTION

0.2 0.1

\

FEEDBACK ~ I M A L E

~,--2o

S/

/ /

cout = 4.10"~

(~=2.34

:36 --------+--

0

0.I

I

0.2

i

i

i

I

0.3

0.4

0.5

0.6

I

07

~,,

0.8

l

0.9

X

RE FERENCE S

(i)

A. Bensoussan,

[2)

A. Bensoussan,

Identification et filtrage, Cahier IRIA n°1Fgvrier

1969.

R. Temam, Equations stochastiques du type Navier-Stokes

(g pa-

raTtre).

[3]

J.P. Kernevez~ Evolution et contr$1e de syst~mes bio-mathgmatiques~

[4)

J.L. Lions, Quelques m6thodes de r6solutions des 6quations aux d~riv6es partiel-

i972, N°CNRS

1.0. 7246,

les non lingaires.

(5) (6)

th~se, Paris

Dunod Paris 1969.

J.P. Quadrat, Th~se Docteur-lng6nieur

ParisVl (1973).

J.P. Quadrat, M. Viot, M6thodes de simulations en programmation dynamique stochastique. Rev. Fr. d'Aut, et de Rech. Operat. R.I. 1973.

17]

Cahier IRIA. Syst~mes Bio-chimiques

(dirig~ par J,P. Kernevez)(~ para~tre),

AN ALGORITHM

TO ESTI~IATE SUB-OPTIMAL PRESENT

VALUES FOR UNICHAIN MARKOV PROCESSES WITH ALTERNATIVE

RE~JARD STRUCTURES

S. Das Gupta Electrical Engineering Department Jadavpur University Calcutta 700029, INDIA

I. INTRODUCTION

Howard's H-algorithm,

algorithm

determines

(3), henceforth to be mentioned

in a closed form, the optimal decision

for a class of discrete Markov processes and associated reasonably

alternative

general

as

in the infinite horizon

reward str~ctures.

and is applicable,

His approach is

among others,

to problems

for discount factor ~ between 0 and I and as well as, with some modification, uneconomic

for

~ = I . The algorithm however becomes

rather

for

(a) large-scale

systems

(b) discount factors close to unity

and or

Finkbelner henceforth

and Runggaldler

(2) proposed

to be mentioned as FR-algorlthm,

a sub-optlmal

algorithm that approached

an algorithm,

which is essentially

the optimal values by

changing from one pollcy decision to another better one and improving upon the present values,

often still further, by some

400

additional

iterations

to any pre-asslgned

since the process of optimization (4,2), the authors provided each iteration,

degree of accuracy.

exhibits

contraction properties

a formula to estimate,

the number of iterations necessary

at the end of to bring the

estimated preaent value vector within a pro-assigned neighbourhood of the optimal present value vector. When, in particular, number is reduced to zero or negative,

the last decision

present value vector are taken as the respective values sought.

The advantage

the computation

this and the

sub-optimal

of this method lles in the fact that

8tops when a desired accuracy is reached.

it also runs Into difficulty

However,

for cases when, in particular,

the

discount factor is close to unity.

2. A SUB-0PTIMAL ALGORITHM

For large-scale of certain

qu~utlties

conslderably~ initiate

systems,

the approximate

help the computational

Both H and F R algorithms

the computation.

aprlori

algorithm

require starting values to

If these are chosen on the basis of

the quick initial estimates

a large amount of Iteratlve

computation

This led to the derivation

estimates

may be reduced.

of steady-state

the notations

estimate

probability

of

and gain in ref.(1). Using

of Howard (3), we define the first order estimate

of the steady state probability

= '

and in general~

distribution

e'rP

.= ~ #

given by

(2.1)

the m th. order estimate

T'-

a s ~ (I)

~. ,~

of Yc will be

e

(2.2)

N

where P is the transition probability

matrix.

Consequently

the

401

m th. order estimate of gain, g(m), will be defined as

gCm) = Tg(m)q

(2.3)

where q is the immediate reward vector. Evidently, both ~(m) g(m) converge respectively to the steady-state probability

and

~

and

average gain g, as m approaches infinity.

For discounted Markov processes, the estimate of g plays an important role in the present algorithm. Here we start with an arbitrary policy to estimate vicorresponding to the present value vector v given by

v = (I

-

(2.4)

from the approximate expression

v i = (I

+

(2.5)

where I is an identity matrix of proper order. Then v i is fed into the policy improvement routine (3) to find a new policy. If the new policy does not match the

old

policy, a new set of v i is to be

determined by the value determination algorithm acaording to eqn. (2.5). A little consideration will show that this is already available from the results of the preoeding policy iteration algorithm.

Eventually, when there is a match between two consecutive policy decisions, the last policy is taken as the sub-optlmal policy and the last estimate of the present value, vi, is modified as follows to give the corrected estimate of the present value Ves t

402

Vest = vl +

~2

It is showl~ in the appendlx guarrantee

eg

(2.6)

that a sufficient

that the value Ves t is a better estimate

condition

to

of v than the

value v I will be when

_~ ( 2 )

>

1 -~ -~-

(2.7)

which is generally not too difficult

The estimate FR-algorithm value

to satisfy in actual processes.

Ves t may then be run for further iterations

with pre-set error level or use~

of

as the starting

of v in the H-algorlthm.

3. DISCUSSION

Severalproblems

OF THE RESULTS

were solved on the computer with various

values of discount factor. Howard's Instance~

one of them~ The results

with those corresponding

Taxi-Cab problem was, for on computing

time were compared

to the pure FR-algorlthm

run of the same

problems for the same desired accuracy levels. A saving of about lO to 20% was frequent.

When compared with the corresponding H-algorithm, sub-optimal

policies

almost always coincided with the optimal

policy*. Thus only one iteration

*One exception

the

cycle of H-algorithm was necessary

cited in ref. (I) for

~=I

403

in Such cases.

It is not necessary to use the exact value of g in eqn.(2.6). Estimates of g may be used. Use of g(1) in eqn. (2.6) and of g(4) has shown any hardly noticeable difference in the computer time saved. In all the cases considered, g(4) came within ½% of the value of g.

4. CONCLUSION

A new algorithm to obtain a sub-optlmal policy and estimate of present values for a class of discounted discrete Markov processes having alternative reward structure in an lnfinlte horizon have been discussed. Based on initial estimates of steady-state probability and gain, this algorithm determines ~ policy and estimates the present value vectors which could either be used as it is or in conjunction with FR-algorlthm or H-algorithm depending upon the accuracy requirement. In both the later cases it generally accelerates the process of computation.

5. REFERENCE

I. Das Gupta, S., Int, J, C0ntr0%, Vo! 14, N0.6,1031-40 (1971) 2. Finkbeiner, B., and Runggaldier,W., Computing methods in Optimization Problem-Vol 2, ed. by Zadeh, L. A., and Balakrishnan, A. V., (1965) 3. Howard, R. A., D ~ m m i c Programming and Markov Processes , Technology Press and Wiley, (1965) 4. Liusternik, L. A. s and Sobolev, V. J., Elements of Functional Analysis, Unga~Inehart and Winston, New York, [196!]

404 APPENDIX

I

It will be shown here that eqn, (2.6) will give a better estimate

of present value provided

satisfied.

that the inequality

(2.7) is

We assume that the transition probability matrix

,

P = L=VPiJ ~ has distinct elgenvalues with

N j:1 PiJ

where N is the order of the matrix P. Since the largest eigenvalue of P is I, we can write the matrix P in the following form (I) N-I [ ~ iT! i=I

P = s +

where S is the constituent eigenvalue

matrix of P corresponding

I p and T i are the other constituent

respect to the other (N-I) eigenvalue

!im

(!-2)

p~

to the

matrices with

Obviously S is given by,

S = e~

(I-3)

where e is a column vector each element of which is unity. of (I-3) and eqn® (2.6) we may express the correction

term in

eqn. (2.6) as

since

q

=

g

In view

(I-5)

405 The actual present value ~, according to eqn° (2.4) is

v

=

(I -~P)-Sq

= (I + ~ P ) q . [

Fnpnq

= vi + a

(1-6)

=

(1-7)

with a

~

~npnq

Similarly Ves t = V i + ~ ~msq

= Vi + b

(I-8)

with cO

b

=

[

~ msq

(I-9)

To find condition for which the distance llv - Vestll~ ~< ~lw- vi% ~

(I-I0)

we have only to find the condition when a-b - ½ ~l b ~ 2 where a d o t between two v e c t o r s s i g n i f y

0 i~mer p r o d u c t .

(I-11) This implies

t~at [

F m + n q T ( T c T ~ (n) - ½ ~cT~v )q>.O

(I-12)

in view of relations (1-7), (1-9) and (2.2).

Now a sufficient condition that the R.H.8. of eqn (1-12) be positive~ or zero will be when

7[ TTE(n) _ ½ ~T~i s positive semi-

definite for all values of n between 2 and ~ . This in turn requires that

~(n)_ ½~ > o

(I-13)

406

for all n between 2 a n d ~ , that each element

where,

in general, we mean by

x > y,

of x is greater than the corresponding

element

of y.

According

to eqn (2.2),

_ (n+l)

vL(n)p

(I-14)

~ P

(I-15)

w =

also as we have

(~(n+1) Now if it is known apriori

½~) = ( ~ ( n ) - ½ ~ ) P that

~(n)_ ~T~ > o and thatnone

(i-~6)

(I-!7)

of the columns of P can have all zero entries

Thus we have only to aheck if the inequality show that Ves t is a better estimate

than v i

(2.7) is satisfied

to

SOME RECENT DEVELOPMENTS IN NONLINEAR PROGRAMMING by

G. Zoutendijk, University of Leyden, Netherlands

I.

INTRODUCTION

The general nonlinear programming problem will be defined as max

{/~xjI)(

E I~ c_ ~;~ I ~

(1)

with

(2) in which _~! polyhedron;

and ~ £ ~

are finite index sets and

~

~

I~A~C}

a convex

is supposed to be connected and to satisfy some regularity

conditions (like being the closure of its interior);

the function / ~

continuous. Usually differentiability of the functions /

and/i

is

and existence

of second partial derivatives is also assumed. Three special cases can be distinguished : i.

All constraints are llnear~ to be subdivided into a.

linear programming if / / X J

is linear~

b.

quadratic programming

c.

(general) linearly constrained nonlinear programming.

if//xj is quadratic and

From a computational point of view two important subclasses of the last class may be considered~ the nearly linear problems (few nonlinearities in the objective function of a relatively simple nature and many linear constraints) and the highly nonlinear problems (few variables and constraints and a highly nonlinear objective function). 2.

There are no constraints: unconstrained optimization.

3.

The~e are also nonlinear constraints. Again it makes sense to consider the subclasses of nearly linear and highly nonlinear problems. Some methods will only work for convex programs ( / R

convex).

concave,

408

!I. LINEAR PROGRAMMING AND UNCONSTRAINED OPTIMIZATION In linear programming the problems are usually large and structured; there are relatively few non-zero elements in the coefficients matrix. The productform algorithm has been successfully applied to the solution of these large problems. Re-inversion techniques have gradually become more sophisticated in that they better succeed in representing the inverse of the basis by means of a minimum number of non-zero elements° For reasons of numerical stability special decomposition methods are being applied for the inverse. For this the reader is referred to Bartels and Golub (1969) as well as to Forrest and Tomlin (1972). Many special methods have been developed for special structures. Much success has been obtained with the so-called generalized upper bound technique (see Dantzig and Van Slyke, 1967). In unconstrained optimization (max / y X )

) most of the methods are hillclimbing

methods. Most widely used is the variable metric method. Writing ~=

74÷i 2~ o

~<

~=

~/~J

and

, the formulae are :

arbitrary

/-/o=~

(or any other ~

by ~

positive definite and symmetric matrix),

7"

/

determined by solving the one dimensional problem max / 6 e

S J.

This method like most unconstrained optimization methods has the quadratic termination property, i.e. it is finite f o r / / A j

quadratic, / ~ J r _ / ~ - ~ } < 7 " ~ X .

In

that

case it can be easily shown that the following relations hold : -/ 2.

/S~J~

s/=Oj ~/"

the directions

S~

or equivalently

/Z~jL

~-~ 0 //" > ~js

are mutually conjugate.

This variable metric method, suggested by Davidon and further developed by Fletcher and Powell (1963)~ is a member of a class of methods (see Broyden, 1967).

409

$~

~=~//,A~

~

VP2

for the matrix updated

according to the variable metric method the update formula for a general member of the family reads : /~'~,,

~

~

~

arbitrary.

Recently Dixon (1972) has shown that all members of the family generate identical points and directions (provided the llne search is carried out in a perfect way). This does not mean that all methods are the same in practice when line searches are not carried out in a perfect way and numerical stability of the H-matrix becomes of importance. In practice it has sometimes be worthwhile to reset the H-matrix to the unit matrix periodically. Research is going on to find methods in which no line searches are required. Alternatively for the unconstrained maximization problem one could use a method of conjugate (feasible) directions. These methods work along the following lines : i.

)
2.

for ~ = o,/, . . . ~ - / in X '~ require for

5 3.

after ~Z

$~

with

maximizing

A

steps :

a.

either start afresh with ×O(new) = K ~ (old) ;

b.

or use a moving tableau,

r,

~=-~-~./~

~-~,...~

i,e. require

$~-- O

for

~-/.

Depending on a d d i t i o n a l requirements to f i x the d i r e c t i o n s ~

i . e , depending on

the direction generator chosen another method will result. Variant

b.

in step

is usually better from the computational point of view. This family of methods of conjugate directions has been first proposed by Zoutendijk (1960, 1970 a, 1978). Recently the computational aspects and convergence properties of some of these methods have been worked out in detail. All these methods have the quadratic termination property.

3

410

A special method from this class is the following : I.

~ 0 arbitrary~ / ~ o = Z

2.

for

a.

~=0~_..

~-/

, calculate

~t

~=0

;

:

$~_~max

5 = O, f = O , / , . - . , ~ - / j

C. X4÷t~_ X ~ , A ~ $4[

, calculate

_~ V

3.

near-optimality test; if not passed :

4.

~*/

---- ~--O- i~ ~ J s

5 / - ¢ ,5 ~ /

:

) ~'

~ ,- Xe(new)= X & ( o l d ) ;

~/;= g * , , go 1;o 2, .

-/ To solve the direction problem

2 a.

we need ~

inversion is necessary after step 4. formula for

£~

rather than ~

, so that no

It is even possible to give an explicit

:

Since f o r a q u a d r a t i c f u n c t i o n ~

-.__ C

the matrix

f ~t

in the general case can

be considered to be an approximation to the inverse Hessian, so that the f i r s t of each cycle of ~

step

steps is a quasi Newton step. For a quadratic function the

method will terminate after at most chosen during the first ~z

~

steps. If the steplengths are arbitrarily

steps and in step ~ , /

A = / is chosen, then the method

will also terminate in the maximum of the quadratic function. For a general function we may therefore expect that this metricized norm method is less crucially dependent on the accuracy of line searches which might be an important advantage. The method has been developed by Hestenes (1969) and, independently, by Zoutendijk

(1970b).

III. LINEARLY CONSTRAINED NONLINEAR PROGRAMMING.

The linearly constrained nonlinear programming problem max

A X ~-

can be solved by applying one of the methods of conjugate feasible directions.

411

I,:

(here ~.

are the rows of A

"

2.

A~=min

3.

if ~ ~_ ~ if

)%,=

' max with A~.'=

, A~ add ~

)~j add

4,

~ S= O &

=

O

f

] and i~;(°J-~ Z (I< K°--- 41 ] aS

and A~" = m a x

}

); Z~ <) ~

A ~

]

to the direction problem; to the direction problem for the hyperplanes

just hit and omit the conjugacy relations. 4. S~

has to satisfy all the relations added to the direction problem during

previous steps as well as / ~ y 7 5 ~

~0.

If no ~

can be found either the

{A~}rS=O

oldest one of the relations has to be omitted or one of the 77relations ~, ~ = O has to be replaced by ~, $ <= O (the one with the most negative dual variable should be taken). If, for some Z ~ ~,~T3&
5. X#"""= X4'~ '~4[ 5 ~ Again any direction generator may be chosen, so that we have outlined a whole class of methods.

It is also possible to adapt a variable metric method to problems involving nonlinear constraints. One of the possibilities is : At A ° : find S o

by solving through complimentary pivoting (see Zoutendijk, 1973)

The solution can be written in,#the form ~/o = i - ~

~

A

~6~/X°Jforwhich

~. $ = O

constraints in

~ ~

~o= /~o~o

with

consisting of those

and the dual variable

>0

(essential

X o ).

At Xf I=0,/,~,,..: if if ~ =

~_

update ~ ~'

then

according to a variable metric formula;

H~+/ =~/~ - ~r %' ~ ~• g~ { ~ K.- - ~

being the hyperplane

;

412

just hit); If p

then start afresh after ~ - p

is the number of rows of

steps with

X ° (new) = X ~ - P (old).

iV. GENERAL NONLINEAR PROGRAMMING

In general nonlinear programming we can distinguish I.

direct methods;

2.

barrier function and penalty function methods;

3,

primal-dual methods;

4.

special methods fom special problems.

:

As far as the special methods are concerned we may mention : a.

Separable programming (objective and constraint functions separable, see Miller ( ! 9 7 3 ) ) ;

b.

Geometric programming for problems of the type

in which the <. [X~

are posynomials~ i.e. functions of the type

~,

p

=.,

i.,×i

" G >

"

>°"

It can be shown that the dual of a geometric program is a linearly constrained nonlinear programming problem (see Zangwill ( 1 9 6 9 ) ) . c.

Convex programming

(/~j

c o n c a v e , ~ convex);

for these problems we have the cutting plane method~ developed by Cheney and Goldstein (1959) and - independently - by Kelley (1960) as well as a decomposition method developed by Wolfe (1967). These methods are dual to each other.

The direct methods are especially suited for the nearly linear problems (large, structured, few nonlinearities). There are three different approaches a.

:

Direct extension of the methods of feasible directions; however, instead of requiring

~, 5 ~ =O

like in the case of linear constraints we require

413 V / ~ J ~ <0 when/6(~J= ~

(in practice whenAl~J~ Z,_~ ). Hence

we restrict the search to the interior of the cone of feasible directions. One way of doing this has been outlined by Zoutendijk (1960). b.

Interior point methods like the modified feasible directions method (Zoutendijk~ 1966).

Having an interior point

~

the nonlinear constraints) and a linearization ~ ] within L I X J

leading to a solution i

as a function of Xt

A

within ~

or in a boundary point ~

v/,I~dT(X-~J~O

with £

(interior with respect to ~ ~ we maximize ~ 7 / / ~ J ~

; we then m a x i m i z e / 2 ~

~(~- ~j

which may either result in an interior maximum . In the latter case the linear relation

denoting the constraint just hit will be added to

the linearized constraints set while a new interior point will be chosen on the line connecting

~

and ~

, If ~

is convex no feasible point will be cut

{~/(XJ-V/i~J IT(X-X~=0 set while X' can be taken as new

off. In the former case the conjugacy relation can be added to the linearlzed constraints

interior point. This procedure can be adapted to non-convex regions; nonlinear equalities are difficult to handle, however. Conjugacy relations have to be omitted if no progress can otherwise be made. c.

Hemstitching methods where in ~ ~ ~

we are allowed to make a small step in a

direction tangent to the cone of feasible directions~ so that in the case of a nonlinear constraint we will leave the feasible region. By projecting the point so obtained onto the intersection of the nonlinear surfaces concerned we obtain a new and better feasible point. This approach is being taken successfully in the Generalized Reduced Gradient Method (Abadie and Carpentier, 1969). Nonlinear equalities can also be handled by this method.

Barrier function and penalty function methods are well-known and widely used to solve nonlinear programs of not too large a size. Let the problem be defined by (i) and (2). Then an example of a mixed method is :

solve

max {7:'x p<JV

} forpo -'S<>,

7

z:

>""

°>

414 The s t a r t i n g point

×+ should s a t i s f y the r e l a t i o n s A " / X y < ¢ . ~ 6

The same will then hold for the subproblem solution

.'~,o

X

#

•

which will be starting

point for the next subproblem, etc. See further Fiacco and Mc Cormick (1968) and Lootsma ( 1970). In primal-dual methods we try to improve the primal and the dual variables more or less simultaneously. This could for instance be done by using a generalized Lagrangean function (Roode~ 1968), i.e. i.

max

rain ~ X

with ~

x~j :

rain

max

a function such that ~/X~ ~;

being a convex subset of a space of a certain dimension p

The two problems will be called the primal and the dual problem, respectively. 2.

The primal problem is equivalent to the original problem.

3.

The dual problem can be solved relatively easily.

In that case it makes sense to solve the dual problem instead. Writing

p/ 2 = m a x

~ { X , 1~y

we have to solve the problem

Each function evaluation ~ / ~ y maximization problem in X

entails the solution of a linearly constrained

,

From this it follows that for reasons of computational efficiency the number of function evaluations should be as small as possible. Buys (1972) has developed a method

of this type for the function

Recently Robinson (1973) has reported another primal-dual method which looks quite promising. Expanding

~;(j:/-IX/~j4.

/ ~ - / X 3 in a Taylor serious with respect to K ~ • V ~ ( 2 ( ~ J 7 " I X _ X O + . . , and w-~iting

415

assuming that at step

we have available vectors X ~, ~ ~ we solve the linearly

constrained problem :

which results in a new point X ~÷t

and a new dual solution

~ I

(dual variables

of the linearized constraints). Convergence to the original nonlinear programming problem can be proved.

APPLICATIONS Although nonlinear programming methods have been and are being applied to many different problems - makroeconomic planning, refinery scheduling/plant optimization, design and control engineering, economic growth models, pollution abatement models, approximation under constraints and resource conservation models - the number of applications is still limited. There are several reasons for this. First the

is highly nonlinear, non-convex, so that local optima cannot be

avoided. Secondly the word nonlinear is negative by definition. Often there is little theoretical knowledge about the process other than that the relation between certain variables is not linear; empirical relations without theoretical foundation might be dangerous to use since they might not hold anymore after some time. Then there is the problem of data organization and model updating which is already tremendous in the linear programming case. Finally theme are few computer codes available and those available are not very sophisticated; up to now there is little commercial motivation for computer manufacturers and software houses to supply these codes, This might change in the future, however, when the need increases to obtain better solutions to some of the nonlinear problems we have to face,

416

REFERENCES Abadie, J. and Carpentier, J.

1969

Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints~ pp. 37-47 of R. Fletcher (ed.), Optimization, Academic Press.

Bartels, R.H. and Golub~ G.H.

1969

The simplex method of linear programming using LU decomposition, Comm. ACM 12, 266-268.

Broyden, C.G.

1967

Quasi-Newton methods and their application to function minimization, Math. of Computation 21, 368-381.

Buys, J.D.

1972

Dual algorithms for unconstrained optimization problems, thesis, Univ. of Leyden.

Cheney, E.W. and Go!dstein A.A.

1959

Newton's method for convex programming and Tchebycheff approximation, Num. Math. ~, 253-268.

Dantzig, G.B. and Van Slyke, R0

1967

Generalized upper bounding techniques, J. Comp. Systems sol. ~

Dixon, L°C.W.

1972

213-226.

Quasi-Newton algorithms generate identical points, Math. Progr. ~, 383-387.

Fiaceo, A.V. and Mc Cormick,G.P. 1968

Nonlinear programming: sequential unconstrained minimization techniques, Wiley.

Fletcher, R. and Powel!~ M.J.D.

1963

A rapidly converging descent method for minimization~ Computer Journal i, 163-168.

Forrest, J.J.H. and Tomlin~ J.A~ 1972

Updated triangular factors of the basis to maintain sparsity in the product form simplex method, Math. Progr. ~, 268-278.

Hestenes, M.R.

1969

Multiplier and gradient methods~ Journal Optimization Theory and Appl. ~, 303-320.

417

Kelly, J.E.

1960

The cutting plane method for solving convex programs, J. Soc. Industr. Appl. Math. ~, 708-712.

Lootsma, F.A.

1970

Boundary pmoperties of penalty functions for unconstrained minimization, thesis, Eindhoven Techn. University.

Miller, C.

1963

The simplex method for local separable programming, pp. 89-100 of R.L. Graves and P. Wolfe (eds), Recent Advances in Math. Programming, Mc Graw-Hill.

Robinson, S.M.

1972

A quadratically-convergent algorithm for general nonlinear programming problems, Math. Progr. ~, 145-156.

Roode, J.D.

1968

Generalized Lagrangean Functions, thesis, Univ. of Leyden.

Wolfe, P.

1967

Methods of nonlinear programming, pp. 97-181 of J. Abadie (ed.), Nonlinear Programming, North-Holland.

Zangwill, W.I.

1969

Nonlinear programming, Prentice-Hall.

G. Zoutendijk

1960

Methods of Feasible Directions, Elsevier.

1966

Nonlinear programming: a numerical survey, J. Soo. Industr. and Appl.Math.Control ~, 194-210.

1970 a

Nonlinear programming, computational methods, pp. 37-85, in J. Abadie (ed.), Nonlinear and integer programming, North-Holland.

1970 b

Some algorithms based on the principle of feasible directions, pp. 93-122 of J.B. Rosen, O.L. Mangasarian and K. Ritter, (eds.), Nonlinear programming, Academic Press.

1973

On linearly constrained nonlinear programming, in: A.R. Goncalvez (ed.), Proceedings of the Figueira da Foz Nato Summerschool on Integer and Nonlinear Programming.

F_EENALTY METHODS AND AUGMENTED LAGRANGIANS IN NONLINEAR PROGRA~MING R. Tyrrell Rockafel!ar Dept. of Mathematics, University of Washington Seattle, Washington 98195 U.S.A. The usual penalty methods for solving nonlinear programming problems are subject to numerical instabilities, because the derivatives of the penalty functions increase without bound near the solution as computation proceeds.

In recent years, the idea has arisen that such

instabilities might be circumvented by an approach involving a Lagrangian function containing additional, penalty-like terms.

Most

of the work in this direction has been for problems with equality conStraints.

Here some new results of the author for the inequality case

are described, along with references to the current literature.

The

proofs of these results will appear elsewhere. E__quality C.onstraints Let

fo,fl,..~,fm

be real-valued functions on a subset

linear topological

space, and consider the problem

(i)

fo(X)

minimize

over

{x ~ Xlfi(x) = 0

for

X

of a

i=l .... ,m}.

The augmented Lagrangia__nn for this problem, as first introduced in 1958 by Arrow and Solow [2], is (2) where

n(x,y,r) = fo(X) + r ~ 0

fact, this

m ~ [rfi(x) 2 + Yifi(x)], i=l

is a penalty parameter and

y = (yl,...,ym) ~ Rm.

In

is Just the ordinary Lagrangian function for the altered

problem in which the objective function fo + rfl 2 + °'°+rfm 2'

fo

is replaced by

with which it agrees for all points satisfying

the constraints. The motivation behind the introduction of the quadratic terms is that they may lead to a representation of a local optimal solution in terms of a local unconstrained minimum.

If

~

is a local optimal

solution to (1) with corresponding Lagrange multipliers

Yi'

as fur-

nished by classical theory, the function m

Lo(X,Y) = fo (x) +

X Yifi (x) i=l

*This work was supported in part by the Air Force Office of Scientific Research under grant AF-AFOSR-72-2269.

419

has a stationary point at

x

which is a local minimum relative to the

manifold of feasible soltuions.

However, this stationary point need

not be a local minimum in the unconstrained sense, and have negative second derivatives at to the feasible manifold. rfi(x) 2,

x

L

may even

in certain directions normal

The hope is that by adding the terms

the latter possibility can be countered, at least for

large enough.

It is not difficult to show this is true if

x

r satis-

fies second-order sufficient conditions for optimality (cf. Ill). The augmented Lagrangian gives rise to a basic class of algorithms having the following form:

[

Given

(yk,rk),

minimize

L(x,yk,r k)

(partially ?) in

(3) ~x ~ X to get x k. Then, by some rule, modify Ito get (yk+l,rk+l).

(yk,rk)

Typical exterior penalty methods correspond to the case where yk+l = yk = 0 and r k+l = ar k (~ = some factor > 1). In 1968, Hestenes [10] and Powell [19] independently drew attention to potential advantages of the case (4)

yk+l = y k+2rk VyL(X k ,yk,rk),

rk+l

r k.

The same type of algorithm was subsequently proposed also by Haarhoff and Buys [9] and investigated by Buys in his thesis [4]. cussion may also be found in the book of Luenberger [13].

Some disRecently

Bertsekas [3] has obtained definitive results in the case where an e-bound on the gradient is used as the stopping criterion for the minimization at each stage.

These results confirm that the conver-

gence is essentially superllnear when

r k ÷ ~.

Various numerical

experiments involving modifications of the Hestenes-Powell algorithm still in the pattern of (3) have been carried out by Miele and his associates [15], [16], [17], [18]; see also Tripathi and Narendra [26]. Some infinite-dlmensional applications have been considered by Rupp

[24], [25]. An algorithm of Fletcher [6] (see also [7], [8]) may, in one form, be considered also as a "continuous" version of (3) in which certain functions of

x

are substituted for

y

and

then has a single function to be minimized.

r

in

L(x,y,r);

one

The original work of

Arrow and Solow [2] also concerned, in effect, a "continuous" version of (3) in which x and y values were modified simultaneously in locating a saddle point of L. Inequality Constraints. For the inequality-constralned

problem,

420 (P)

minimize

fo(X)

over

it is not immediately

{x c X I fi(x) ~ 0, i = l,...,m},

apparent

what form the augmented

Lagrangian

should have, but the natural generalization turns out to be m (5) L ( x , y , r ) = fo(X) + [ ~ ( f i ( x ) , Y i , r ) ,

i=l

where

(6)

]

l(fi(x),Yi~r)

rfi(x) 2 + Yifi(x)

<-yi2/4r In dealing with nonnegative, Lagrangian

(5), the multipliers

in contrast [21],

below.

Yi

are not constrained Kuhn-Tucker

[Ii], Lill

To relate the augmented

by Buys [3] and Arrow,

grangian

y = 0

function.

for problems

[27],

[28],

[29], Fletcher

[12], and Mangasarian

Lagrangian

should be noted that by taking penalty

in a

to the inequality-constrained

problem may be found in papers of Wierzblckl

"quadratic"

This

[23], the main results of which will be

Related approaches

[7], Kort and Bertsekas

to be

theory.

by the author in 1970 [20] and studied

[22]~

It has also been treated

Gould and Howe [i].

fi(x) ~ -Yi/2r,

fi(x) ~ -Yi/2r.

with the ordinary

was introduced

series of papers indicated

if

if

[14].

to penalty approaches,

it

one obtains the standard

Observe

also that the classical

with inequalities

Lag-

can be viewed as a limiting

case:

m

fo x) + (7)

lim L(x,y,r) r+0

= Lo(X,y)

The following properties L(x,y,r)

(once) in

it is convex in

x

x

(y,r),

if every

if

(X

fi

and)

referred to as the convex case. not inherited surfaces" Theorem

by

L

neighborhood tion to

~

and

slackness

~ > 0.

differ-

the latter is is

However,

as will be seen from with algorithms

x

of

L

in a

is a local optimal

solu-

vector in the classical

If the multipliers

~i

satisfy

as usually has to be assumed

it is clear that none of the

will pass through

be two or three times continuously

[23]:

along the "transition

multiplier

conditions,

of convergence,

surfaces"

[21],

differentiability

in connection

(x,y,r) such that

is a corresponding

in a close analysis "transition

(6),

fi

0,

y

Furthermore,

is convex;

centers on the local properties

sense of Kuhn and Tucker, the complementary

fi

Higher-order

to formula

of a point

(P),

if

if

and it is continuously

every

most of the interest

and their convergence

X Yifi (x) i=! y ~ 0.

is differentiable.

from the functions

corresponding

4 below,

~ \-=

of (5)-(6) can be verified

is always concave in

entiable

=

(x,y,~),

differentiable

and hence

L

will

in some neighborhood

421

of

(x,y,~),

if every

fi

has this order of differentiability.

(Certain related Lagrangians inherit higher-order concave

in

recently proposed by Mangasarian

dlfferentiability

everywhere,

but they are not

(y,r).)

The class of algorithms

(3) described

may also be studied in the inequality gives an immediate

generalization

above for the equality case

case.

In particular,

of the Hestenes-Powell

We have shown in [22] that in the finite-dimensional algorithm always c0nverges

globally

minimization

in obtaining

The multiplier ~,

vector.

vectors

xk

yk

y.

solution

solution sense.

Kuhn-Tucker

possess more than one such results on local rates of

in the equality case are applicable

the locally optimal

algorithm.

in a certain

converge to some particular

For convex and nonconvex problems,

convergence

(4)

This is true even if the

is only approximate

even though the problem may

rule

convex case, this

if, say, an optimal

exists along with a Kuhn-Tucker vector

vector

[14]

in question

if the multipliers

satisfy complementary

at

slack-

ness conditions.

Dual Problem. The main theoretical fundamental

properties

to all applications,

dual problem c o r r e s p o n d i n g To shorten the presentation assumption

that

X

to the global

L.

here, we henceforth make the simplifying

that this assumption

fi

no real restriction.

Mixtures

are continuous.

is not required,

setting is in fact the one treated

in [21],

It should also be clear that our focus on inequality involves

Lagrangian,

in terms of a certain

saddle point problem for

is compact and the functions

It must be emphasized the more general

of the augmented

can be described

and that [22],

[23].

constraints

of equations and inequalities

can be handled in much the same way. The dual problem which we associate with augmented Lagrangian (D)

maximize

L

g(y,r)

g(y,r)

over all

= min L(x,y,r) xEX

Note that constraint

y ~ 0

the condition

r > 0

represent

seen,

is nondecreasing

g(y,r)

Thus the dual problem g(y,r)

is concave

in

ously differentiable,

(P) in terms of the

is y ~ Rm

and

where

(finite).

is not present

in this problem.

a true constraint, as a function of

is one of unconstrained (y,r),

r > 0,

since, r

Nor does

as is easily

for every

maximization.

y. Further,

and in the convex case it is continu-

regardless

of the dlfferentlability

of

fi

[21].

422 c@

THEOREM

l[23].mln(P)

denotes a__nnarbitrary THEOREM

= sup(D)

s_~uence

e k ÷ 0.

optimal

to

If yk ~ 0, penalty functions mln(P)

it suggests but

with having

(yk,rk)

Solution

that a necessary that

to

L

even holds out the attracyk

and

here is whether (D).

rk

remain

a bounded maximi-

In other words,

under

has an optimal

and sufficient ~

of bounded

condition

to be a

be a (global)

1 and the definition

saddle point

therefore

maximizing

for

(globally)

(y,r)

optimal

of

L.

of the dual

to be an optimal solution

The following

show that our question

sequences

to (P) is

(yk,rk)

about

the

has an affirmative

for "most" problems.

THEOREM of

2

at all for

from Theorem

on saddle points

existence answer

a

instabilities

can it be said that the dual problem

(D) and

(~,y,~)

theorems

question

in

through

could yield improved

in which both

exists

of

(y,~)?

It is elementary Solution

Theorem

methods

Perhaps,

some of the numerical

r k + ~.

The fundamental

to vary.

such a method

of algorithms

what circumstances

are

fact in the theory

class of penalty-like

is allowed

reduce

zing sequence

x~

m~ maxa{0,fi(x))]i=l

a larger

yk yk,

and thereby

possibility

rk

of the sequence

the familiar

+ rk

good rule for choosing

bounded.

points

I asserts

convergence associated

and

den_~ote a_nny s_eequence wit___~h

+~

still r k ÷ ~

tive

bounded

(yk,rk)k= 1

bounded (but not necessarily with ~ ]~ L(x,y ,r ) over X to within e ,

= lim mln[fo(X) k xEX

More generally,

yk

where

(P).

Theorem that

r

which

y

Then all cluster

solutions

(8)

with

k k (y ,r )~=i

2123]~Let

sup(D) = lim g(yk,r-~) and k ~). k r ÷ Let x minimize where

= lim g(yk,rk),

3 [21].

In the convex

if and o n l ~ i_[f (~,~)

Lagrangian

L°

THEOREM

satisfies optimality

is a saddle

Point

point o_ff the classical

inn (7).

4 [23].

is differentiable (a)

cas__ee, (x,~,r)

is a saddle

Sup pose that

of class

I_~f (x,~,~)

near

is a ~lobal

the second-order in (P),

C2

an_d_ ~

x c int X c R n, ~.

saddle point

Decessar[

and that each

conditions

i_s_s~lobally

optimal.

o~f

L,

then

[5, p.25]

(x,y)

for local

f. 1

423

(b) [5,P.30]

If

(x,y)

satisfies th__~esecond-order sufficient conditions

for local optimality i__nn (P) and

optima! , then

(x,y,~)

x

i_~s uniquely globally

i_~s ~ global saddle point of

L

for all

sufficiently large. Part (b) of Theorem 4 stengthens a local result of A~row, Gould and Howe [i] involving assumptions of complementary slackness and the superfluous constraint

y > O.

A corresponding local result has also

been furnished by Mangasarian [14] for his different family of Lagrangians. solution

It is shown in [23] that the existence of a dual optimal

(y,~)

depends precisely on whether

(P)

has a second-order

stability property with respect to the ordinary class of perturbations. REFERENCES 1.

K. J. Arrow, F. J. Gould and S. M. Howe, "A general saddle point result for constrained optimization", Institute of Statistics Mimeo Series No. 774, Univ. of N. Carolina (Chapel Hill), 1971.

2.

K. J. Arrow and R. M. maxima, with weakened Nonlinear Programming, Univ. Press,

.

Solow, "Gradient methods for constrained assumptions", in Studies in Linear and K. Arrow, L. Hurwlcz and H. Uzawa ~ i t o r @ , 1958.

D. P.BertSekas, "Combined primal-dual and penalty methods for constrained minimization", SIAM J. Control, to appear.

4.

J. D. Buys, "Dual algorithms for constrained optimization", Thesis, Leiden, 1972.

5.

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: a~l Unconstrained Optimization T e c h ~ Wiley, i968.

6.

R. Fletcher, "A class of methods for nonlinear programming with termination and convergence properties", in Integer and Nonlinear Prosramming, J. Abadie (editor), North-Holland, 1970.

7.

R. Fletcher, "A class of methods for non-linear programming III: Rates of convergence", in Numerical Methods for Non-linear Optimization, F. A. Lootsma (editor), Academic Press, 1973.

8.

R. Fletcher and Z. Li!l, "A class of methods for nonlinear programming, II: computational experience", in Nonlinear Programming , J. B. Rosen, O. L. Mangasarian and K. Ritter (editors), Academic Press, 1971.

.

Se~uenti-

P. C. Haarhoff and J. D. Buys, "A new method for the optimization of a nonlinear function subject to nonlinear constraints", Computer J. 13 (1970), 178-184.

i0.

M R. Hestenes, "Multiplier and gradient methods" Appl. 4(1969), 303-320•

ll.

B. W. Kort and D. P. Bertsekas, "A new penalty function method for constrained minimization", Proc. of IEEE Decision and Control

J. Opt. Theory

424

Conference s New Orleans, Dec. 1972. 12.

S. A. Lil!~ "Generalization of an exact method for solving equality constrained problems to deal with inequality constraints", in Numerical Methods __f°r Nonlinear Optimization, F. A. Lootsma (editor), Academic Press, 1973.

13.

D. G~ Luenberger, Introduction __t°linear and nonlinear programmi__~, Addison-Wesley, 1973, 320-322.

14.

O. L. Mangasarian, "Unconstrained Lagrangians in nonlinear programming", Computer Sciences Tech. Report #174, Univ. of Wisconsin, Madison, 1973.

15.

A. Miele, E. E. Cragg, R. R. Iver and A. V. Levy, "Use of the augmented penalty function in mathematical programming, part I", J. Opt. Theory App!. 8(1971), 115-130.

16.

A. Miele, E. E. Cragg and A. V. Levy, "Use of the augmented penalty function in mathematical programming problems, part II", J. Opt. Theory Appl. 8(1971, 131-153.

17.

A. Mie!e, P. E° Moseley and E. E. Cragg, "A modification of the method of multipliers for mathematical programming problems", in Techniques of Optimization ~ A. V. Balakrishnan (editor), Academic Press, 1972.

18.

A. Miele, P. E. Moseiey, A. V. Levy and G. M. Coggins, "On the method of multipliers for mathematical programming problems", J. Opt. Theory Appl. 10(1972), 1-33.

19.

M. J. D. Powell, "A method for nonlinear optimiztion in minimization problems", in Optimization, R. Fletcher (editor), Academic Press, 1969~

20.

R. T. Rockafellar, "New applications of duality in convex programming", written version of talk at 7th International Symposium on Math. Programming (the Hague, 1970) and elsewhere~ published in the Proc. of the 4th Conference on Probability (Bra§ov, Romania~---~l~.

21o

R. T. Rockafellar, "A dual approach to solving nonlinear programming problems by unconstrained optimization", Math. Prog., to appear.

22.

R. T. Rockafeilar, "The multiplier method of Hestenes and Powell applied to convex programming", J. Opt. Theory Appl., to appear.

23.

R. T. Rockafellar, "Augmented Lagrange multiplier functions and duality in nonconvex programming", SIAM J. Control, to appear.

24.

R. D. Rupp, "A method for solving a quadratic optimal control problem", J. Opt. Theory Appl. 2(1972), 238-250.

25.

R. D. Rupp, "Approximation of the classical isoperimetric problem", J. Opt. Theory Appl. 3(1972), 251-264.

26.

S. S. Tripathi and K. S. Narendra, "Constrained optimization problems using multiplier methods", J. Opt. Theory Appl. 2(1972), 59-70.

425

27.

A. P. Wierzbicki, "Convergence properties of a penalty shifting algorithm for nonlinear programming porblems with inequality constraints", Archiwum Automatiki i Te!emechaniki (1970).

28.

A. P. Wierzbicki, "A penalty function shifting method in constrained static optimization and its convergence properties", Archiwum Automatyki i Telemechaniki 16(1971), 395-416.

29.

A. P. Wierzbicki and A. Hatko, "Computational methods in Hilbert space for optimal control problems with delays", these proceedings.

ON

INF-COMPACT

MATHEMATICAL

PROGRAMS

by Roger timum

is a t t a i n e d

~here

exist

aC a f e a s i b l e

scalars to

problem

an u n c o n s t r a i n e d

cal

by

to that

stability ties

the

m often

associated

of

coincides

problem.

If our

G(x)

~ where

~ 0

variational

with

point)

called

constraints

and

original

problem

P

of

of

the p r o b l e m

multipliers to r e p l a c e

optimal

For

is:

is a v e c t o r

(ii)

us

whose

solvability properties

G

and

Lagrange

problem).

to some

function

Wets

allow

problem

the o r i g i n a l

correspond

J.-B.

convex

the

valued

- that

solution

is

map;

identi-

mathematical

dual p r o g r a m .

inf

problems,

These

function

f(x)

then

can be

the m a t h e m a t i c a l

the v a r i a t i o n a l Find

is s t a b l e

subject

one w a y

properof

the

to

to d e f i n e

the

is as:

P(u)

= Inf{f(x)

I G(x)

~ u}

E

It

is e a s y

schitz ginal

at

to v e r i f y

that

O

effective

on

its

is

stable;

problem

problem

[6,7,12,13].

In this

that

ties w h e n

the

the

paper

objective

classes

in s e c t i o n

P

of

4 some

is

one

and

is

domain

stable

variational

compactness-type

certain

P(O)

if

is a s s y m p t o t i e a l l y

conclude

tisfy

if

lower also

finds

further

various

properties

is l o c a l l y

theorems

at

0 see

which

Lip-

the orithen e.

of the

allow

original

to e x t e n d

these

of

us

to

proper-

problem

sa-

results

programs,

composition

the

g.

the a p p r o p r i a t e

stochastic

for

then

dualizable,

possesses

and

P

< +=})

semicontinuous

calles

In o r d e r

problems

and

I P(u)

constraints

assumptions.

control

({u

function

the

finite

we

to

develop

inf-compact

functions. 2. N O T A T I O N S The domain are

fective

that are

domain

a function

of

f

dom

a function

. S~

f

space

a member

f = {x I f(x ) < +~}

is l o w e r

semicontinuous

i n f - c o m a~_a~.t if

was

introduced

the b a s i c

f

Banach

= {x I f(x ) ~ ~}

inf-compact

Supported

TERMINOLOGY

L~(f)

is

terminology

An

+=

AND

to be m i n i m i z e d

reflexive

is

definition

and V a l l a d i e r I lj.

functions

identically

closed,

This

of

a separable

not

is by

class

is

by N . S . F .

the Grant

always

subset No.

of

and of

. The

is

of t h e s e

and w h o

then

level

if all

its

set

its

for

developed

of

f

known

level

sets

all

~ g ~

Rockafella

see

that May

ef-

with

a minimum.

x X such

with

It is w e l l

functions~

possesses ]-~,+~[

]-~,+~]

(~-)

compact

who

to t h o s e

class,

~ ~ ~.

(l.s.c.)

L~(f)

GP-31551

range this

with

by M o r e a u

properties

function

is l i m i t e d

e.g.

The epi 1973

.

[4,5,8,

epigraph f =

427

{(~,x)

I ~ ~

by

~C

By

K

we

subsets X

f(x)}

and

is

shall

of

the

which

{x I (x,u) said

g D}

is

set.

on

for

is

schitz

where

~ ( u =)

We

to be d

= max(6(P,Q),

you

the

that

~(Q,P))

if

then

is

ven

one

defines

the

real

valued,

in w h a t

will f

jected

easy

~f(u)

the

~f

Proof:

Sup. ygQ

two

into

satisfies be

X x U

. The

map

in

X x U

between

is

. Such

is

K ( u I) X

pro-

~

d(<(ul),K(ua))

of

of

some

could

closed

the

a subset

K(u)

subsets

Lip-

and

then

d(P,Q)

llx-Yll

distance

follows

~f

is

reflexive

we

only

use

the

for

compact

extended

se~

and

d

definition

gi-

f

: X .

2.

is

can

the

: X

be

if

projection

of

and

f

. We

pro-

Inf

f(x,u).

can

we

view

where

about

u

~f

and

of

epigraph

the

=

. Sup-

~f

repre-

can

also

function.

f • +~

its

x U ÷ ]-~,+~],

spaces

= ~

consequence

only

~f(u)

U the

. In p a r t i c u l a r

said

x U ÷ ]-~,+=],

onto denote

problem

variational

immediate

f

Banach

of

and

Suppose

~f

optimization

whatever

x U we

that

cl

{ ( x , u ) Ix ~ X}

an

an

(linear)

X ~f

such

f = epi

if

the

of

. By

function

f

of

U

~ +=

~ epi

dom

convex

projection

the

Thus,

on

is

f ~ +~,

is w e a k l y

the is

convex

f

convex

definition convex

set X

is

epi and

and f

U

inf-compact.

both Then

inf-compact.

suffices ~

Given

not

if

THEOREMS

not

el

properties

separable

It

timization

, i.e.

Suppose

then

is w e a k l y

all

and

proposition

Proposition

Proof:

canonical

function

convex

a function

epi

is

is

distance

P,Q

PROJECTION

that

= +~

into I.

is

This

since

f

perturbation.

translated

then

the

of

variational

Proposition

for

denote

to v e r i f y

that

the

sent

~f

graph

function

Hausdorff

Hausdorff

: X x U + ]-~,+~] function

is

have

be

its

U

is

here.

pose

as

{x I x

denoted

otherwise.

a space

, ~(u)

of

is

= +~

where.

3.

It

=

subset

the

given

(P,Q) = I n f x~P Usually,

from u

C

and

specifically

if

Lipschitz

denotes

remind

K(u)

a subset

g C

given

a fixed

up pe K s e m i c o n t i n u o u s

said

x

function

each

, more

D

of

if

Typically

u}

where

= O

valued

, i.e.

empty

function

~C(X)

a set

X

depends

to be

a map

indicator by

denote

a space

possibly

perty

. The

defined

c ~

to But

show this

a mathematical problem

a "well-set"

in

that

the

sets

L~(~f)

follows

from

program

- by w h i c h

finite

problem,

or

the

infinite

i.e.

(i)

the

are

fact

that

we

mean

dimensional problem

is

weakly

compact

L~(~f)

= ~L~(f)

a constrained spaces solvable

-

it

,

opis

(the

or op-

428

i.e°

the

sets

and

L~(~f) thus

Proposition rable

are

are

3o

simply

weakly

Suppose

relexive

f

Banach

spaces

inf-compact

weakly

lower

semicontinuous.

Proof:

First

observe

ists

x of

x

such f

~ Le(g)

the

in

for

L~(g)

g Le(f)

=

x

u

. The

weak

Proposition

4.

dimensional

Euclidean

in

inf-compact,

convex L~o(~f)

-

Proof: is

P

is

to

know

s° k

Inf

the

from

there

of

exists

~f

x

f(uO+%v,x(l)) convex x ~ O x(X) with

is

,

u ~ O}

x(%)

~ O

of

it where it

,

the

lower

f(x,u)

= g(x) c P}

x U

it

of

follows

the P

is

and

in

ex-

now . By

. By w e a k

<

If

the

l.s.c,

and

to

the com-

a point

implies

For

for

each

%

there

with always and

that

is

case. From

of is

inf-compaet

2 that

Thus~

proposition

] and

This

implies

not

actually []0]

dom

zf u

x(1)

it

such ,

that

such

. On

the

other

hand

Ax(%) for

+ %

zf is

inf-

+

P

is

a

Bu

= p

,

(Weyl-Minkowski). ,

that

that

since

~ ~ O

attained

f(x,u)

I Ax

matrices

that

follows

. Now,

all

~f suffi-

((x,u)

are

or

it

x(%) E <(u°+%V)

for

g

a nonempty

that

fixed

as

.

finite

always

g

exists

written

f

unbounded.

have

in

of

where

is

be B

that

U

proposition

we

each

P

~f

functions

O}

and

and

infimum

convex % ~

X

this

inf-compactness

~ 0

us u°

+ ~K(u)(X)

unbounded.

l.s.c~

%v,

%v

to

semicontinuity

- is

from

proof

ray.

A

of

either

~f

this

follows

u° +

. Then

{u ° +

can

there

then

. Let

S L~(g)

is

inf-compact-

subsequenee

with

{x I (x,u)

on

= ~f(uO+Xv)

~ <(u°+%v)

follows

finite.

say

~ thus,

polyhedron,

.

=

by

properties a ray~

~ ~

when

g > -~

{x i}

semicontinuity

points

convex

the

g

~f

C L~(f)

< +~

sepa-

with

that

weakly

a converging

and

then

case

f(x,u)

U

Then

from (x,u)

f(x)

X

and

. Thus S

completes

decreasing

in

of

minimum

is

Inf

the

strictly

compact

f

~

infimum

then

the

compact

+ @K(u)(X)

converging

x U ÷ ~-~,+~

~(u)

bounded

that zf

of

which

consider

3 we

: X

spaces

subset

set

inf-compact

ces

is

the

If

f

convex,

polyhedral

if

upper

this

Suppose

follows

whenever

= 0

X

implies

, this

contains

~ From

= g(x)

a corresponding

~<(uo)(X)

u ° ~ L~(~f)

weakly

semicontinuous.

u c L~(~f)

s L~(~f)

~ f ( u °) i.e.

of

, f ~ +~,

. Moreover

g(x)

{u i}

, {x i}

, i.e.

f(x,u) upper

fixed

fixed

exists

L~(g)

, say

x E < ( u °)

for

f(x,u)

there

of

and

weakly

that

sequence

remarks

pactness

K

(x,u)

x

since

consider above

and

that

in

projections

: X x U ~ ]-~,+~]

weakly

hess

(linear)

compact.

Bu ° +

Since lBv

=p

sufficiently

429

large

x(l)EL~o+c(g)

x(l)

contains

From

the

ly

it

that

By

= O

decreasing

ceding

that

. Thus

sequence

polyhedral

~

corresponding The

x ~ u -~}

is

Inf

= ~o

L

~f(u)

(nf)

is

Suppose

f

spaces,

Lipschitz

on

and

{u

# ~}

It

l~f(u)

- zf(v) l ~ M

It

is

easy

to

~f(v)

nite

and

and

y

that

that

of

f

there

is

have

that

~f(u)

of

o(g)

%)

ray

, say

which and

not

simple

by

a strict-

{u ° + Iv,

in

x.

now

since

have

superfluous

~

~f

% ~ O}.

the

pre-

example:

. Let

= u -I

fact

for and

u ~

I

K(u)=

function. ~f

is

and

+~

{x I x ~ O,

In

convex

this

case

and

.

with

= g(x)

us

its

on

and

~f

then

the

M

such

hypotheses

~f(u)

and

also

assume

that

f(y,v) u

g

is

domain

that

f

that

for

separable

.

a constant

dom

U

where

effective

dom

exists

and

that

X

+ ~K(u)(X)

on

= -~

attained

]

valued

. In

assume

such

L

following

ux

u,v

~f(u) us

sequence

% ~ 0 can

inf-compact

there all

= ~f(u)

~ ~(v)

= zf(v)

and

]ly-x[I =

v

Inf

imply

~f(v) there

fi-

exist

, i.e.

. Moreover, {IIy-xH

are

x

that

the

assume

I y c ~(v)~

,

then I~f(u)

where

if

actually y

,

Lipschitz

that

~ ~f(v)

the

Lipschitz

for

let

~f(x,u)

is a

means

set

= #

f(x,u)

is

show

the

x U ÷ ~-~,+=~

is

IIu-vll

. Thus

that

: X

~f

to

verify

= -~

such

infimum

we

suffices

no

clearly

~ > eo

and

<(u)

. Then

Proof:

that

all

of

all

, we

on

function

Leo(~f)

for

for

~f

x ~ 0

is

the

= x + @ [ O , i ] (x)

semicontinuous

Banach

I <(u)

by

g(x)

to

g

reflexive

X

for

this inf

but

a point

g(x(%))

consider

see

aupper

5.

inf

To

unbounded

Proposition

lim

variational

= 0

to

thus

Bu ° - p = B v ( - l i m

is

function

an

+

restriction

subject the

Ax

on v a l u e s

Find

otherwise.

small,

x £ K(u°+lv)

g(x)

proposition.

arbitrarily subsequence

follows

semicontinuity

The

e

a convergent

above

implies lower

for

the

first

f(y,v)

, the

f

X

on

- zf(v)[

inequality second (with

~

-

f(y,v)[

B • IIx-F

II

B

llu-vll

follows

inequality

constant

[~f(u)

•

Kfrom

the

follows

B ) and

the

fact

from last

the

that

~f(u)

Lipschitz

inequality

~ zf(v) property

follows

from

430

the L i p s c h i t z ~(v))

property

N KIlu-vll ~ Now,

tually

attained

closest

point

points)

of

(or on

in

the

above

and

tions

6.

where

f(x,~)

are

for all

to

sis b u t

might

used

range

~

still

]-~,+~]

fo+(y)

It can b e

shown

a positively cription easy

Lemmao nuous.

need

is not

there

ac-

is no

e-optimal

to c o n t r o l

to e x t e n d

: X x~

so

÷ ]-~,+~]

in

space.

problems

a result

is

of

f + g . To

x

for

all

inf-compact

function.

concepts

somewhat

are

unfamiliar

concepts

its

which

recession

of

suppose

and u - m e a s u r a b l e on

~.

Then

standard

proposition

f ~ +~

is c o n v e x

function

fO +

the

in

F(x)

in C o n v e x

6,

func-

that

to the m a t h e m a t i c a l

to p r o v e SHy

is a~ f a m i l y

Moreover~

measure

=

Analyprogram-

they w i l l

function

is d e f i n e d

not

with

by

= L i m ~ - 1 1 f ( x + % y ) - f(x)l ~+~

that

this

limit convex

the p r o p e r t i e s

to show

results

a probability

homogeneous

of

<(u) (using

inf-compact

Euclidean

developments.

, then

we

are

convex

these

in f u r t h e r

in

IIy-xI[~ d ( < ( u ) , K(u)

the proof.

above

problems,

some be

a point

on

THEOREM

the

f(x,~)

a convex,

need

g

modification

yields

g

K ) since of

6 below:

be

introduce

We o n l y

or g i v e n

of

and

is a f i n i t e

is

We n e e d

be

f

inf-compact,

/ f(x,~)du

mer.

If

x . Let

infimum

standard

some

Suppose

X

constant

the

A COMPOSITION

the p r o p o s i t i o n

Proposition

(with

arguments

programming

[4] ~ viz.:

prove

the

to a p p l y

stochastic

Moreau

<

~(v))

~(v)

4. In o r d e r

of

if e i t h e r

of

is

independent

function.

recession

See

of

x

19]

and

for

functions.

that

fO +

a detailed

In p a r t i c u l a r

is

desit

is

that: Suppose

Then

f

f : X ÷ ~-~,+~]

is i n f - c o m p a c t

is

if and

convex

only

if

and

lower

fO+(y)

semiconti-

> O

for

all

y~O. Since valued sign. dard du

in p r o p o s i t i o n

functions In fact

all w h a t

Lebesgue-Stieltjes as

the

s u m of its

understanding that

in the

if

(i)

that

U{~If(x,~)

6 we

allow

integrand~ is

required

integral. positive either

= ±~I}

# O

part

possibility

to give

a slight usual,

and may

then

the

have

is As

part

for

we

we

of its possibly

of

modification define

the

negative be

the p o s i t i v e

infinite

a meaning

to

the

of the

/ stan-

integral

part

divergent, (negative)

with and part

f . the (ii) is

431

automatically that be

the

defined

integral

the v a l u e

where

of

finite,

integral

can

be

which > O

FO+(y)

y # O

of

If the

adopt part

integrand

the c o n v e n t i o n

is

the

same

properties

except

that

subaddivity

+~

whatever

is a l m o s t

to the L e b e s g u e - S t i e l t j e s

the

and

be

of c o n v e x i t y

quite

of c o n v e x

in v i e w for for

~

easily

every-

integral.

as

the

This

standard

replaces

lemma

since

the u s u a l

convergent)

readily

theorem

in

that

El4]

From

require

inf-

to s h o w i n g

fO+(y,~)

from a weakend

(we do not

but

F(x)

to e s t a b l i s h

is e q u i v a l e n t

is c o n v e x .

, it f o l l o w s

convergence

found

observes

Remains

above

F

can be

if one

functions.

of the

y # 0

dominated

integral

proof

reconstructed

that

all

6: The

combination

compactness,

the

integral

we

the p o s i t i v e

part.

essentially

of p r o p o s i t i o n

in fact

for

; in a d d i t i o n

property.

is a c o n v e x

sion

±~

whenever

the n e g a t i v e

Lebesgue-Stieltjes

Proof

+~

it c o r r e s p o n d s

possesses

additivity

to be

is

> O

ver-

here

that

that

FO+(y)

~ f fO+(y,~)d~

> 0 .

5. A P P L I C A T I O N S

A.

Nonlinear

G : ~n Gi ,

programs.

÷ ~m

,

i.e.

i=|,...,m

Let

G

f : ~n

. A nonlinear Find

nal

by

~

we

function

denote

F(u) It is o b v i o u s sufficient gram

(P

Moreover

l.s.c,

if

the

tions.

as

program:

Find

Gi for

Inf

or G

f ~ +~

and

with

let

components

then

N O ordering.

problem,

f(x)

The

~ u}

theorems

.

of s e c t i o n

dualizability (P

is c o n t i n u o u s

variatio-

is g i v e n by

I G(x)

stability

standard

of the n o n l i n e a r

is l o c a l l y and

3 provide

f

is

Lipschitz

inf-compact

proat 0). then

is s o l v a b l e .

somewhat

the

that

= {Inf

O)

the c o n t r a i n t

Since

it f o l l o w s

this

to e s t a b l i s h

at

program

G(x)

the p r o j e c t i o n

function

examine

as w e l l

all

conditions

is

the n o n l i n e a r We

that

to

with

is

,

function

f(x)

componentwise

associated

+~

valued

program

inf

subject where

÷ ]-~

is a v e c t o r

further

's are all

f(x)

the

functions

case w h e n

continuous

scalars subject

the o b j e c t i v e

G i , i=1,...,m (they

are

are

convex-finite

u i , the

feasible

to

~ u , is a c l o s e d

G(x)

function

convex

region

funcon

~

)

of the c o n v e x convex

set.

432

For

each

region

i , the

is

given

Proposition convex

{x I G(x) X ~ 0

is also

Proof:

Since

sets,

that

(see

any

e.g.

and

for

contained

can

prove 139])

function

and

be

~ 0 which

then

As

a corollary

of

{x

8o

Proof:

If

as

observe

that

the a b o v e

! O}

for

all

the only

is n o n e m p t y vectors

ray

u

ray

contained

in the c l o s e d

If

(x

~ O}

is the

only

follows apply

not

only

that

propositions

It is n o w

easy

3, 4 and

B.

Stochastic weak

an e q u i v a l e n t possible

terministic blish

class

Provided

deterministic an e x p l i c i t

problem

also

and

problem

,

we

of le-

implies

contained

of c o n v e x

above the

in

functions

hypotheses

of

the

GiO+(y)

conclusion.

have

map

then

of a c o n v e x {x

I G(x)

I G(x)

~ O}

set

{x

is

program

~ u}

restrict

and

if

f

one

the

the p e r t u b a t i o n s

of

the o r i g i n

is

inf-compact, but we

properties

fashion

trivial

~ u}

is s o l v a b l e

further

the

7 it is also

I G(x)

of

could

also

it

can

this

u

(which

then

problem.

apply

pro-

of p r o b l e m s .

the

original

problem

" c o n v e x ~' s t o c h a s t i c

convex

program.

simple

consequently is or

D =

{x + %y

the p r o p e r t i e s

neighborhood

in a s i m i l a r

to each

that

it is n o n e m p t y .

problem

2 to o b t a i n

5 to this

Programs~

if a g i v e n

the

and we

anyway)

the o r i g i n a l

to see h o w

to find

convex

is c o m p a c t

interest

assumptions,

Ix

to a c o m p a c t

I and

positions

very

of

% ~ O}

intersection

y = O , then by p r o p o s i t i o n

problem

region

such

Lui(Gi)

4 on

proposition

in

only

i G(x)

semicontinuous

the ray

the

is

yields

compact,

i.eo

the o r i g i n a l

under

for w h i c h

contained

direction

feasible

{x o + %y,

v

that

properties

the c o n s t r a i n t

ray,

of

with

is

all

as

in s e c t i o n

automatically

G

lower

% ~ O}

% ~ 0}

from basic

or rely

Suppose

I G(x)

is c o m p a c t

represendted {x o + %y,

{x + ky,

that

for

we h a v e

the

.

can be

to show

of

and

that.

u , the ray

~ Then

x e D

D

convex prove

a class

some

N u}

in

D

be

for

any

x g Lvi(Gi) ,

[;O,p.

Proposition

that

and

can also

, i=l,...,m

and

and

closed

. One

C = {x I G(x)

# @

C

~ One

recession must

Gi

it s u f f i c e s

for

Lvi(Gi)

is

Lui(Gi)

Suppose

in

~ v}

Lui(Gi)

~

7. Let

functions.

is c o n t a i n e d

vel

set

by

However,

analytic it m i g h t

is not w e l l

satisfies

program

it is not

representation be v e r y

set°

of

difficult

It is not

some

correspond usually this

de-

to e s t a -

possible

to

433

develop

here

We

intend

only

tions. see

Some

e.g.

the

implications

to i n d i c a t e

partial

[13]

for

where

a version

tions

are

chastic

full

here

results

have

of p r o p o s i t i o n The

inf x

functions

z = f(x)

f : ~n

functions

are also spect

already

of

3 appeared

in the

3 and

litterature,

and also

will

be

4.

applica-

of p r o p o s i t i o n

first

problem

section

elementary

appeared

+ E {Inf y

convex

in

G(x)

N 0

÷ ~ _~,+~

and

Gi

x ;

in

variable

q(y,~)

to

y

q : ~nx

and

to a p ~ o b a b i l i t y

random

of the m o r e

of a v e r s i o n

following

subject

convex

the r e s u l t s

5 and

some

called

Eli

applica-

here

a sto-

program: Find

The

some

an a p p l i c a t i o n

indicated.

of

[ H(x,y,~)

: ~n

÷~

÷ ~-~,+~

(x,y)

measure

~ O}

, i=l,...,m

and

Hj : ~ n x ~ .

are ÷~

respectively

and m e a s u r a b l e

that

the d i s t r i b u t i o n

defines

with

re-

of

the

~ . Let

Q(x,~)

= Inf

{q(y,~)

[ H(x,y,$)

< O}

Y and

~(x) where

E

= E{Q(x,~)}

(expectation)

f • d~

in s e c t i o n

written

as

is d e f i n e d

4. The

Find

equivalent

inf

subject We

shall

spect Then

assume

to

~

~(x)

0} ral.

that

(see e.g.

in

x

the

Since

recourse

shall

q(y,~)

Hyll . See

practical

of

problem

assumptions,

it

function

K

in

for e x a m p l e

<

of the is m o r e

that w o u l d

follows

~ O

x

for

Q(x,~)

a proof

defined . Both

of

the

integral

is

by

<(x,~)

then

recourse

to s a t i s f y

it

x

this

re-

case).

that

Q(x,~) in

y

= {yIH(x,y,~) are q u i t e

increases

model and

is h a r d

from proposition

with

natu-

for s e l e c t i n g

the c o s t

between but

linear

is i n f - c o m p a c t

assumptions

that

simple

technical fail

the

is m e a s u r a b l e

to be p a y e d

expect

difference

directly

program

| it f o l l o w s

q(x,~)

a penalty

the

than

.

that

that

and one m i g h t

is a f u n c t i o n

semicontinuity

assume

represents

y

[14]

manner

+ ~(x)

from proposition

simicontinuous

action

ly w i t h penalty

. We

shown

same

deterministic

G(x)

[2~ and

set v a l u e d

is u p p e r

to

since

the

z = f(x)

it can be

is c o n v e x

is c o n v e x and

that

in

[]5] w h e r e ~ . The

assumption.

With

Q(x,~)

the

upper

to i m a g i n e

3 that

a

strict-

a these is

434

lower

semicontinuous

if some w e a k l.s.c, tion

and

i G(x)

that

symptotically

can

use

Proposition tional

Let

us

just

to o b t a i n

vity

as used

compact,

the

in

one

can

only

it

problem

show

that

2 and

be p r o v e d If now

follows

then

is s o l v a b l e

again

the

governed here

that and

by

to o b t a i n

partial

the

for

~(x)

f

is also

from

proposi-

and

part

we

loose

is

the

is

K(x,~)

that ~

at least

as-

that

also

differential

the

conditions all

are b o u n d e d is

properties

then

inf-compactness

regularity

It is o b v i o u s

sets

6 to show

f + ~ { x I~G (0x )} i

existence [3].

can also

is s a t i s f i e d .

is compact,

equivalent

since

mention

used

condition

propositions

problems

property

(dualizable).

2 can be used

function

C. C o n t r o l

. This

~ O}

the

stable

If in a d d i t i o n then we

x

integrability {x

3 again

ix

of

the v a r i a -

inf-compact.

equations.

assumption rather

coercive

uniqueness

inf-compact.

of

functions the

can be

than

coerciare

solution.

inf-

435

References [l~

S. Gartska:

"Regularity

Manuscript [2]

P. Kall:

"Das

8 (1966), J.-L.

zwelteilige

Lions:

Problem

der stochastischen

Z. Wahrscheinlichkeitstheorie

Optimal

Control

Equations,

J.-J. Moreau: tions

of Convex

Programs",

linearen

verw.

Gebiete

IOl-ll2.

Differential ~]

for a Class

(1972).

Programmierung",

[3]

Conditions

of Systems

"Fonctionelles

aux deriv~es

Governed

Springer-Verlag,

Berlin

Convexes",

partielles,

by Partial (]971).

Seminaire

Coll~ge

sur les Equa-

des France,Paris

(]966-1967). ~5~

J.-J. Moreau: 258

~]

(1964),

Convex Functions",

R.I.,

[IO]

J. Stoer

N.J.

[1|]

graphes. ~]2~

E14~

in Nonlinear

Math.

Programming"

in Applied

Soc.,

Convex Analysis,

Problems

in Mathe-

Mathematics,

of Conjugate

123 (]966),

Princeton

Con-

46-63.

University

Press,

(1969). Convexity

D. Walkup

and R. Wets:

Nonlinear

Programs",

D. Walkup

and R. Wets:

W. Ziemba:

"Stochastic

Manuscript

(1972).

Ferm~s

notamment

R.I.R.O.,

"A Duality

Theory

to Optimal

22 (1968),

Programs

d'Epigra-

(1970),

for Abstract Control

Regularity

SIAM J. on Control,

15 (]967),

R-2

57-73.

Mathema-

Theory",

679-706.

"Some Practical

"Stochastic

in Finite

(1970).

Continue",

with Applications

Anal. Applic.

and Optimization

Berlin

de Convexes

and R. Wets:

SIAM J. AppI. Math., [151

in Extremum

Sets and Continuity

Amer.

"Integration

tical Programs

D3~

"Level

Trans.

Inf.-Convolution

R. Van Slyke J. Math.

Sci. Paris,

J. Math.,2](]967),167-187.

40]-422.

I, Springer-Verlag,

M. Valadier:

Pacific

Lectures

and C. Witzgall:

Dimension

and Stability

Sciences~

R. T. Rockafellar: Princeton,

C. R. Acad.

] 1 (]968),

R. T. Rockafellar: vex Functions",

~]

"Duality

of Decision

Providence, [8]

"Duality

R. T. Rockafellar: matics

inf-sup,

2720-2722.

R. T. Rockafellar: Involving

E7~

"Theoremes

7 (1969),

Programs

Conditions 430-436.

with Recourse",

1299-]314. with

Simple Recourse",

for

436

Mathematisches der U n i v e r s i t g t 5000 K~In Weyertal

Institut zu KSln

41 86-90

and

Department

of M a t h e m a t i c s

University

of K e n t u c k y

Lexington,

K e n t u c ky

NONCONVEX QUADRATIC P R O G R A M S r L I N E A R C O M P L E M E N T A R I T Y PROBLEMS, AND INTEGER L I N E A R PROGRAMS % F.Giannessi

¢% E.Tomasin

Abstract. The p r o b l e m of nonconvex q u a d r a t i c programs is considered, and an a l g o r i t h m is proposed to find the global minimum, solving the c o r r e s p o ~ ding linear c o m p l e m e n t a r i t y problem. An a p p l i c a t i o n to the general comp l e m e n t a r i t y problem and to 0-I integer programming problems, is shown. I - Introduction. The aim of this paper is to study the general q u a d r a t i c p r o g r a m m ~ g problem, i.e. the p r o b l e m of finding the m i n i m u m of a q u a d r a t i c function under linear constraints. Such a problem is often met in many fields of mathematics, mechanics, economics, and so on. If the objective function is convex, the p r o b l e m is well known, both t h e o r e t i c a l l y and comp u t a t i o n a l l y [2, 4, 6~, while, w h e n the objective function is n o n c o n v e ~ the problem of finding the global m i n i m u m is still open, even if there are many methods to find a local minimum. Among the methods proposed to solve the general case ~ , 5, 11, 13~, two kinds of approaches can be distinguished: a) enumerative methods [3, 111, and b) cutting plane methods [5, 13 3 . While the former approach seems to be not so efficient as expected, the latter till now, gave rise to methods which are not al ways finite ~ 6 ~ . In this paper a method to solve the general q u a d r a t i c p r o g r a m m i n g problem is proposed, where the quadratic problem is substituted by an equivalent linear c o m p l e m e n t a r i t y problem, and this is solved by a particular cutting plane method [8]. This way no method of p r a t i c a l l y efficiency is p r o d u c e d but after a successive investigation of the method and of the r e s p e c t i v e properties, by m o d i f y i n g somewhere the initial ba ses of the method, the algorithm was implemented E14~. Thus, the vertices of a convex p o l y e d r o n are found, which m i n i m i z e a linear function and satisfy a given condition, for example c o m p l e m e n t a r i t y (this way the p r o b l e m of nonconvex quadratic p r o g r a m m i n g is solved) or a 0-I integer condition. T h e n the method can be used also to solve a general linear c o m p l e m e n t a r i t y problem which can be met i n d e p e n d e n t l y from the quadratic (~)

Research supported by Netional Groups of Functional Analysis and its Applications of Mathematical Commitee of C.N.R.

t

Department of Operations Research and Statistical Sciences, Univ. o~ PISA, Via S.Giuseppe, 22, PISA, ITALY.

tt

Mathematical Institute, Ca' Foseari University, VENICE, ITALY.

438

programming 2-The

!ity,

problem,

or 0-I

general_quadratic

programs.

programming

The problem with which the f o l l o w i n g I: P

we

: min ~(x)=cTx

2.1

concerned

+ ½xTDx,

w h e r e A is a m a t r i x of o r d e r f o l l o w i n g t h e o r e m s hold: THEOREM

are

mxn

x6X

is, w i t h o u t

={x:Ax~br

and D is a m a t r i x

[1 ~ 1 0 ] )

(Kuhn-Tucker

problem. loss

of g e n e r a

x~0}

of o r d e r

nxn.

The

I f x i s a l o c a l minimum for P, t h e r e e x i t

v e c t o ~ y, u, vr ~uch t h a t : T

(2.1a)

c + Dx - A y - u = 0

(2.1b)

Ax - v = b

(2.1c)

X, y, u, V > 0

(2.1d)

T T x u = y v = 0

THEOREM //ty

([4],

2.2

p.146) o I f

~(x)

(2.2)

(x, y, u, v) £~ a s o l u t i o n of

= ~I ( c T x

(2.1), then t h e equa

+ bTy)

holds

THEO~M

2.3

([14])o i)

X

~ ~

ii)

~

~ {0}

iii)

X

is c o m p a c t

If

then •

{[I

±nf Define

(Jx

+

;~= A

(2.3)

---~.

T

i--

THEQREM

]y)}

The l i n e ~

2 .A min 2

-T c z'~

;z'=

;c= b

complemeng~y

b

(x) (>< Y

; z"=

\ ,,)

problem:

z ~ 6 { z : A-z ' - z" = b;

z' T

z .

= 0,

z > 0}

equivalent to P. I -

The "T" minimum

as s u p e r s c r i p t respectively.

ann

~'min" denote

transposition

and

global

439

T h e o r e m 2.4, w h i c h is a s t a i g h t f o r w a r d c o n s e g u e n c e of t h e o r e m s 2.1, 2.2 s h o w s that, t o s o l v e P, w e h a v e to s o l v e a l i n e a r c o m p l e m e n t a r i t y p r o b l e m [7, 9]. T h e n , i n t h e f o l l o w i n g s e c t i o n s , t h e l i n e a r c o m p l e m e n t a r i t y problem, will be considered. 3 - The For

linear sake

x ' T = (xi) ,

complementarity of s i m p l i c i t y

x "T= (Xm+ ~) ,

problem.

define

A = (aij) ,

now a T= (ai0) ,

c T=(cj) , i=1,..m; j=1,..n=2m,

xT=(x,T, x,,T) and consider the p r o b l e m : Q

: min

T c x,

x6X =

{x

:Ax = a;

x'Tx"=0;

x > 0}

N o w t h e p r o b l e m Q w i l l b e c o n s i d e r e d , i n s t e a d of (2.3), w h i c h is a p a t t i c u l a r c a s e of Q. T h e n , d e f i n e X ~ the c o n v e x c l o s u r e of t h e p o i n t s of X, w h i c h v e r i f y x'Tx~'=0; the f o l l o w i n g t h e o r e m s hold: THEOREM

3. I

The s~t of optAmal s o l u t i o ~ of Q is a face (in particular a v~ttex)

of X. PROOF. A v e c t o r x 6 R m u s t h a v e an o p t i m a l s o l u t i o n ( s h o r t l y to a f a c e of X, w h o s e p o i n t s The result follows. THEOREM

3.2

Q is equiv~ent

(3.1)

min

T

c x

at l e a s t n - m = m zero e l e m e n t s . T h e n e i t h e r o.s.) of Q is a v e r t e x of X, or it b e l o n g s are o.s. of Q.

3

to the linear problem.

;

xE X

w h i c h is a c o n s e g u e n c e of t h e o r e m 3.1. If X ~ w o u l d be k n o w n , t h e p r o b l e m (3.1) c o u l d b e s o l v e d in p l a c e of Q. As X ~ is u n k n o w n b u t it is s u f f i c i e n t to k n o w o n l y a s u b s e t of X ~ c o n t a i n i n g an o.s. of Q, s u c h a s u b s e t is d e t e r m i n e d to r e d u c e Q to a linear programming problem. 4 - A cutting

plane method

Consider

the p r o b l e m :

to s o l v e Q. T h e

Q0 The method previously o.s. of QO s a t i s f y i n g

outlined,

: min

T c x

consists

;

c a s e of n o n d e g e n e r a c y .

x 6X

firstly

in s o l v i n g

Q0'

as a n

x ' T x '' = 0 is a l s o

2 -

a n o.s.

of Q and t h e n of P.

In the sense that the @irst n elements o@ an optimal solution o@ the latter one are an optimal solution of the @ormer one.

3 - In the sense that they have the same o.s. and the same minimum.

440

But~ by the t h e o r e m 2.3 Q0 has no f i n i t e o . s . ; t h e n in s e c t i o n 6 a transformation is d e f i n e d , such t h a t a p r o b l e m e q u i v a l e n t to Q0' h a v i n g f i n i t e o . s . t i s o b t a i n e d . S u p p o s e then, t h a t Q0 has f i n i t e o.s. If an o°s. such that x ' T x " = 0 t e r m i n a t e the a l g o r i t h m . O t h e r w i s e j a l i n e a r ineq u a l i t y is d e t e r m i n e d , s u c h that it is not s a t i s f i e d by V^ (where V^ is the v e r t e x c o r r e s p o n d i n g to the c u r r e n t o.s.), but is s a t i s f i e d by ~very o t h e r v e r t e x of X. This a b v i o u s l y can be r e a l i z e d in m a n y ways; h e r e the s t r o n g e s t i n e q u a lity is g e n e r a t e d . T h i s h a p p e n s w h e n e v e r y v e r t e x in X a d j a c e n t to V 0 s t r i c t l y v e r i f i e the g e n e r a t e d i n e q u a l i t y . If there is no d e g e n e r a c y , this can be e a s i l y p e r f o r m e d ; o t h e r w i s e ; i t is m o r e c o m p l i c a t e d . In this s e c t i o n the case of non d e g e n e r a c y is c o n s i d e r e d . Let the c o o r d i n a t e s of V 0 be the b a s i c s o l u t i o n (briefly, b.s.) of the r e d u c e d form: (4.1) of the

xi system

+ ai,m+1

Xm+l

Ax = a; w h e r e

+ "'"

Xm+ I =

+ ~in Xn = ai0 ...

i=I,.oo;

m.

= Xn = 0.

If the n - v e c t o r (4.2)

(~IO~O~.,~m0

,

0, 0 . . . .

0)

s a t i s f i e s x'Tx" = 0~ then it is o.s. of Q too. Otherwise, a convex p~y~dron is d e t e r m i n e d , w h i c h ces of X, b u t V . 0 By the h y p o t h e s i s of non d e g e n e r a c y , it is: (4.3)

~i0

i = I,...i

> 0

has

only

the v e r t i -

m.

Define

x,=] Sup[x~]

: x.l = ~i0

- ~i]'x'3 --> 0,

i=I,...,

m},

j=m+1 .... ~n.

I

[0 m+1 ~j =

The

if

x. = + 3

i

if

x. 3

1

; j = m+1 .....

ll/x T a =

~

n;

< +

(~m+1 ~m+1 ~ .... ~ m + 1 ~n )

inequality T ~ x"

(4.4)

> !

is s a i d a cut for X. T h e i n t e r s e c t i o n , X ~between (4.4) is the r e q u i r e d p o l y h e d r o n , as it is s h o w n

X and the h a l f s p a c e by the f o l l o w i n g :

_4__-! t h e i n e q u a l i t y (4.4) ~ i) ~ not v e r i f i e d by Vn; ii) /S weakly v e r i f i e d oy a l l t h e v e r t i c e s of x which are a d l a c e ~ £o v 0, 1 1 1 ) ~ v e r r f r e d by every other ver r e x o~ x; i i i i ) e v ~ y v e r t e x of x i s a v e r t e x of ~ too.

THEOREM

Which is e a s i l y

prooved

[8].

The

cut

(4.4)

has

interesting

properties

441

L e t QI be the p r o b l e m (4.5)

obtained

by a d d i n g

~Tx" - X n + 1 = I ;

to Q0 the c o n s t r a i n t s

X n + 1 >_ 0

As (4.2) does not v e r i f y x ' T x '' = 0, the a d d i t i o n of (4.5) to Q does not c h a n g e its s o l u t i o n s . QA is then a s s o c i a t e d to Q in p l a c e of Q^. An o.s. of Q w e a k l y v e r i f i e s (~.4), so that it is an e l e m e n t of J ~ (V~) i.e. I A ' of the set of v e r t i c e s a d j a c e n t to V 0 in X. Let V I be o.s. of QI, and Xh, h > m ; t h e n o n b a s i c v a r i a b l e (shortly, n.y.) that m u s t b e c o m e b a s i c to go f r o m V 0 to V I. T h e c u r r e n t s y s t e m is then: x.+e. . .+e' x i z,m+lXm+1 +" i,h-1 h-1

~. +..4e. x =a. l,h+~h+1 l,n+1 n+1 z,0

1=I ,. ~m

(4.6) Xh+~m+1 ,m+IXm+1

+ . .+~w x +~l x +...+~i x =~ " m+1 ,h-1 h-1 m+1,h+1 h+1 m+1,n+1 n+1 m+1,0

IfV I at x j = 0 j = m + 1 , . . . n + 1 , j # h , v e r i f i e s x ' T x " = 0 ; t h e n it is o.s. of Q too. O t h e r w i s e , the p r o c e d u r e is i t e r a t e d and a n o t h e r i n e q u a l i t y is g e n e r a t e d w h i c h is not s a t i s f i e d by VI, but is s a t i s f i e d by e v e r y other v e r tex of the f e a s i b l e r e g i o n of QI" T h i s is r e a l i z e d in next section. 5 - The cut in the case of d e g e n e r a c y . A s s u m e that V I does not s a t i s f y x ' T x " = 0 . T h e n to d e t e r m i n e a cut, like in s e c t i o n 4, w h i c h cuts off VI, the d e f i n i t i o n of cut has now to be e n l a r g e d ; in fact o ~ ( V 1 ) may contain more than n+1-(m+1)=m elements, so that it w o u l d be i m p o s s i b l e to d e t e r m i n e an i n e q u a l i t y like (4.4), w e a k l y v e r i f i e d by all the e l e m e n t s of ~ ( V I ) . R e m a r k t h a t in (4.6) e x i s t s at least one [ £ { i = 1 . . . m } such that ~'~ =0, so that to d e f i n e a z0 g e n e r i c cut is to d e f i n e a cut w h e n d e g e n e r a c y occurs, i.e. w h e n (4.3) are n o t v e r i f i e d . T h e n c o n s i d e r the f o l l o w i n g linear p r o g r a m m i n g p r o b lem: (5. la) (5. Ib) (5.1c) and a s s u m e

min

(ClX1+... + C N X N )

xi+~ i ,M+IXM+1 +... + ~ i N X N = a i 0 ,

i=1 .... ~I ;

j=I,...N

x

> 0 3 -that the v e c t o r :

(5.2) (xi=~i0, i=I .... ,M; x.=0, i=M+I,...,N) 1 is an o.s. of ( 5 . 1 ) j w h i c h does not v e r i f y x'Tx"=0. If M=m, N = n (5.1b) c o i n c i d e w i t h (4.1); after the f i r s t cut (4.5) b e e n d e t e r m i n e d , if M=m+1 and N = n + 1 , ( 5 . 1 b ) c o i n c i d e w i t h (4.6). Assume, without loss of g e n e r a l i t y 4

(5.3)

~i0>0,

i=I ..... M;

~i0=0,

i=M+1 ..... M

,

has

(0<M<M).

T o d e f i n e a cut w h i c h does not v e r i f y o n l y (5.2) a m o n g the v e r t ~ e s of (5.1b,c) c o n s i d e r the l a t e s t M - M of (5.1b), w h i c h m a y be e q u i v a l e n t l y

4

-

In
i=I ..... Mnthen [5.1b,c] has only one vertex which does not

veri~y x'Tx"=O, a~then Q has no o.s.; i~ ~io>O, of section 4.

i=1 ..... M, we are in the case

442

written (5,4)

ai~M+IXM+I

+~ • °+ C~iNXN ! 0,

i = ~ + I , . . . ~N

Define C ={(x .~...~x ) : x . > 0; ... ; x > 0} and C , r = I , . . o , M - M 0 M+I N M+I --f as the i n t e r s e c t i o n a m o n g C o and the f i r s t r ~ a l f s p a c e s ([.4). To de ine a cut in this c a s e f i r s t l y the c o n v e x p o l y h e d r a l c o n e C=C ~ m u s t be d e t e r m i n e d . T h i s is r e a l i z e d in a g r a d u a l way; in fact s t a r t l n g f r o m C 0 w h o s e e d g e s are t r i v i a l l y k n o w n , the e d g e s of C r + I are o b t a i n e d k n o w i n g the e d g e s of C . T h i s w a y C M _ ~, and t h e n C, is d e t e r m i n e d [8]. A s s u m e t h e n to k n o w t~e p a r a m e t r i c e q u a t i o n s (5.5)

x] = Sijt

of the

edges

~

j=M+I,...,N

of C r e s p e c t i v e l y

; t>0;=

Hk •

T h e set O~ x (V 0) of the v e r t i c e s of the V 0 m u s t n o w be d e t e r m i n e d 5. T h e n put {s = Sup

convex

polyhedron

{t:ai,M+ I ~s,M+1+...+~iNBsN!~i0,

6 (5.6) can be a s s u m e d to be f i n i t e T h e n the e l e m e n t s of O ~ x ( V 0) are the p o i n t s (5.7)

V

=

(X

s

sj

=

6ij~0

indicated H i ...

(5.6)

i=1,...,k;

8

{s'

si

X, a d j a c e n t

i=1,.o,M}

to

s=1,..~k

7

j=M+I ..... N),

s=l .....

k .

N o w a cut can be d e f i n e d in this case. If k i N - M , u s i n g the m e t h o d d e s c r i b e d in s e c t i o n 4, an i n e q u a l i t y like (4.4) s a i d h e r e too cut, can be e a s i l y d e t e r m i n e d . If k > N - M an i n e q u a lity like (4.4) m a y not exist, then m o r e t h e n one i n e q u a l i t y is g e n e r a ted. Define I =

{I ..... k}

VT = and

consider

;

Xj = [ Xsj S61

(XM+I' .... XN)

T ;

~

=

(~M+I,M+I'''''~M+I~N)

,

~&{a:V

T

~>I

S

The

feasible

(5.9)

region V

of

T ~-~ s

(5.8) = I

seI

}

>

0

•

--

is g i v e n

8T=(81 .... ~Bk ) is a s l a c k

where

j=M+I ..... N

the p r o b l e m T min V ~

(5.8)

'

by:

s£I;

g

vector.

5 -

The notations X, Vo,~X (V O] of section S, 4 are used agalm~

6 -

See the ~ollowing section,

7 -

V

denote both the point and the [N-M]-up!e, S

443

The meaning of (5.8) is obvious: an o.s. of gives the coefficients of the inequality (5.10)

(5.8)

(~M+I,j'

j=M+I

''" ,N)

~M+I,M+IXM+I+...+~M+I,NXN~I

which, among the ones verified by all the vertices of x but V 0 minimizes the sum of the differences between the two terms of (5.10) evaluated at every element of ~ x ( V 0 ) . As (5.8) has finite o.s. (5.10) always exists. If all (5.7) weakly verify (5.10), it is said a eu~t for (5.1) and our aim is attained. Otherwise, another b.s. distinct from the optimal one is determined. Let it be the following (~M+2,j

/

j=M+I,N)

to which by the definition of the system of constraints inequality (5.11)

corresponds

the

~M+2,M+IXM+I+...+~M+2,NXN~I

analogous to (5.10), which is considered together with it. If every (5.9) weaklyverify at least one equationofthe system (5.10), (5.11~ then this is said c~tt/ng s~tem or briefly c~t for the problem (5.1). Otherwise;another b.s. of the problem (5.8) is determined and then another inequality like (5.11) is generated. As to every b.s. of the (5.8) corresponds a N-M-I face of the polyhedron X, iterating the procedure a finite number of times, a system of inequalities (5.12)

~M+I,M+I x M+I + " . "+~M+I,NXN ~I'

i=I ..... M-M

is determined which is said a cu22/~ S~tem, shortly a c~t for the problem (5.1) and which is added to (5.1b). Then, an o.s. of the new problem is determined; this is of course an element of ~ x ( V 0 )~. Iterating the procedure described above, in a finite number of sheps an optimal solution of Q is reached. The finiteness of the method is justified by the following theorem:

THEOREM 5 1

( [ 8 ] ) . The polyhedra ~ ind

Xo have the s ~ e v e r t i c ~

where X ° is the union of the differences between X and the convex hull of its vertices, and the convex hull of the vertices of X, but V 0. 6 - Determination

of an optimal

solution of Q0

By the theorem (2.3) Q0 has no finite optimal solutions. Nevertheless with a transformation it is possible to obtain a problem equivalent to Q0 having finite o.s. Let Q and Q0 denote respectively Q and Q0i both with the additional constraints: (6. I )

x1+x2+ ..... +Xn+Xn+1=Q0o ;

Xn+1 >_0

Then, by the following:

( [ 8 7 ) . Q has f i ~ t e o.s., i f f a real Qoo exists, such that Q has f i nite o.s. satisfying the inequality Xn+1>0 ;

THEOREM 6.1

444

Q0 can be considered in place of Q0~ w h e n the last has no finite o.s., if Q00 is large enough° T h e o r e m s 3.1 and 3.2 hold for Q too. Remark that, after the above t r a n s f o r m a t i o n , there are vertices of the feasible region of Q0 w h i c h are not v e r t i c e s and then b.s. for Q0" It is useful to e l i m i n a t e such vertices, as the c o m p u t a t i o n increases rapidly w h e n the number of iterations, and then the d e g e n e r a c y of the problems, Q0, QI .-. Qn, becomes larger. This is realized, as indicated in [14], by a transformation, here b r i e f l y described. The f o l l o w i n g p a r a m e t r i c capacity c o n s t r a i n t is added, in place of (4.1) (6.2)

x1+x2 + ..... + X n + X n + l = a t w i t h

Xn+1 ->0

where ~ is a parameter. Let Q0(~) be the p r o b l e m Q0 with ~ in place of Q00, V0(a) an optimal vertex of Q0(a) and X(a) the f e a s i b l e region of Q0(~). All the vertices of X(~) a d j a c e n t to V0(a), w h i c h are not v e r t i c e s of X;are determined. Remark that every vertex of X, is such that (6.3)

lim

ai0 (~) : +~

for at least one i 6 {i=1,...m}. T h e n the set of v e r t i c e s adjacent to every vertex s a t i s f y i n g (6.3) and such that they are vertices both in X and in X(e) , is determined. Such a set is considered in place of ~ x ( V 0 ) ; it is said 0Q~ .... (V^ (e)). The m e t h o d d e s c i b e d in section 5 is [hen applied to J-SX~,, (V0(a)) and a system analogous to (5.12) is obtained. This p r o c e d u r e is j u s t i f i e d r e m a r k i n g that: lim and that the vertices The f o l l o w i n g

X(~)

= X

of X(~)

w h i c h are not v e r t i c e s of X, verify

ProposZ2ion. The polyhedron defined by ~he c o ~ a i n ~ ~ the convex hull of the v ~ 2 i c ~ of x;

(6.3).

of Q0 and by the s ~ t e m (5.12)

holds. In fact every vertex of X belongs to the p o l y h e d r o n and every inequality of the system (5.12) d e f i n e @ a facet of the p o l y h e d r o n itself. 7 - C o n n e c t i o n s b e t w e e n n o n c o n v e x and concave q u a d r a t i c p r o g r a m m i n g problems. The idea p r e v i o u s l y d e s c r i b e d and the subsequent a l g o r i t h m are not so e f f i c i e n t to be applied in the form outlined above. It is then n e c e s sary to i m p l e m e n t the algorithm; to this aim, remark that the central p r o b l e m of this theory, is the concave p r o g r a m m i n g problem, to w h i c h it is always p o s s i b l e to restrict. In factlit is known that for a given n o n c o n v e x q u a d r a t i c p r o g r a m m i n g problem, it is p o s s i b l e a d e c o m p o s i t i o n in convex, and concave s u b - p r o b l e m s E 3 ]. Thus, g i v e n a n o n c o n v e x quadratic problem, it is enough to be able to solve convex and concave subproblems. The s o l u t i o n of the former one can be easily obtained by the known methods. The m e t h o d of the p r e v i o u s sections can be used for the latter one.To this aim consider the p r o b l e m P of section 2; the so called facial d e c o m p o s i t i o n can be used. It consists in a tree like p r o c e

445

dure, b e g i n n i n g with a single node c o r r e s p o n d i n g to X itself; from this node there are branches leading to the (n-l) dimensional faces of X, let they be Fi,(i=1 .... ,k), w h e r e k can be easily determined. [17] If ~(x) is either convex, or concave on each of F i 1 ¥ i = 1 , . . . , k the p r o c e dure can be stopped. O t h e r w i s e l f r o m each of F i, where ~(x) is neither convex, nor noncave, there are branches going to the (n-2) d i m e n s i o n a l faces of X. To these also the procedure described above is applied, until a set of sub-problems of P, eithers convex, or concave is found. Each convex sub-problem, can be solved using one of the efficient existing algorithms, while for the concave ones, the a l g o r i t h m p r e v i o u s l y d e s c r i b e d can be implemented as is shown in the following section. 8 - A sufficient condition for the optimality in a concave quadratic progra/0ming problem. Consider the p r o b l e m P, assuming t h a t ' ( x ) is strct!y concave, and the c o m p l e m e n t a r i t y p r o b l e m (2.3) e q u i v a l e n t to P. Let (x, y, u, v) be an o.s. of the linear p r o g r a m m i n g p r o b l e m associated to (2.3), i.e. a vertex of the p o l y h e d r o n (2.1a,b,c). This always happens, w h e n any problem Q0, QI .... ,Qn, p r e v i o u s l y considered, has been solved. If the o.s. thus obtained, satisfies the c o m p l e m e n t a r i t y condition, a solution of Q, and then of P, is at hand. Otherwise, using the a l g o r i t h m d e s c r i b e d abo ve, such a v e r t e x is cut off, even if (X,~) is o.s. of P. More p r e c i s e l D let Qi l i=0,1,-'',r! be the problems which are to be solved, before obtaining a vertex satisfying the c o m p l e m e n t a r i t y condition, and let (xi, yi, u i, v i) be the c o r r e s p o n d i n g o.s.; (xr,v r) is o.s. of P, but it may be that (x i, v i) is o.s. of P with i
(0)

(0)

(0)

(0)

Let (x , v , y , u ) be an o.s. of the current i-th problem QI and let (x (0) , v (0)) be a n o n d e g e n e r a t e vertex of X. In this hypothe sis, there are n vertices (x (i) , v (i)) i=I, .... n adjacent to it. D e f i n e tO the n-vector of the nonbasic variables, associated to (x (0) , v (0)) and t i i=I, .... n, the vectors associated to (x (i) , v(i)) in the space having t 0 as origin. In the n vertices adjacent to t o let it be:

(8.I)

b°(ti) ~ ( t °) pv

i=I .....

n.

On the straight lines c o n t a i n i n g the edges o r i g i n a t i n g in t 0, consider n points t ~ i l i = 1 , . . . , n t s u c h that: (8.2)

~(t~i)

= ~(t0) I ¥ i=I ..... n !

446

which necessarily Remark that

exist,

(8°3)

as~(x)

t ~i = k~Z t O + ( 1 - k i )t i r ¥i=I ¢,,. ,n

where

(tO)_ 1

and

is c o n c a v e .

then

l

(t i)

t0Dt0+tiDt i

< 0 i -

Wi=1,...,n

i.e~ the p o i n t s t ~i are n o n c o n v e x l i n e a r c o m b i n a t i o n of t O and of t i . C o n s i d e r n o w the h y p e r p l a n e p a s s i n g ~ r o u g h the n p o i n t s t~i; its equat i o n is: i=n (8.4) g(t) = ~ t. / (1-X i) t i -I i=1 l i Remark

that

g(t 0) = -Io

(8.5a)

Consider

now

the

problem:

m a x g(t)

subject

to

(8.5b)

Ax-v

= b

w h e r e (8.5b)have to be s u i t a b l y e x p r e s s e d If the l i n e a r p r o g r a m m i n g p r o b l e m (8.5) C8.6)

gCt')

t O is an o p t i m a l

(x(0) ,

following

v

(0) ,

y

basis

(0) f

if (8.7)

0

as origi~

< 0

for

(0)) U

in the s p a c e h a v i n g t has o.s. t ' p s u c h t h a t

Pli.e.

does

not

(x (0) , v (0)) satisfy

is o.s.

of P,

the c o m p l e m e n t a r i t y

even

if

condition~

g ( t ~) > 0,

a vertex

where

the

following

inequality

may

hold

1)(t) < ~Ct °) e x i s t s , so that the p r o c e d u r e of the a l g o r i t h m of s e c t i o n 5 m u s t rated. A n a l o g o u s c o n s i d e r a t i o n s m a y be done, for the case of d e g e n e r a c y (x (0) ~, v (0)1 ,

see

9 - An al~orithm

solve

be ite of

D4U/.

for n o n c o n v e x

quadratic

problems.

A f t e r the r e m a r k s of the p r e c e d i n g s e c t i o n s h e r e an a l g o r i t h m to n o n c o n v e x q u a d r a t i c p r o g r a m m i n g p r o b l e m s is b r i e f l y o u t l i n e d :

S t e p I L e t the p r o b l e m P be given. Its f e a s i b l e r e g i o n can be d e c o m p o sed, in such a w a y that P is r e p l a c e d by a f i n i t e n u m b e r of convex, and s t r i c t l y c o n c a v e p r o b l e m s . T h e f i r s t ones, can be s o l v e d by any a l g o r i thm for c o n v e x q u a d r a t i c p r o g r a m m i n g p r o b l e m s . To s o l v e the s e c o n d ones, u s e step 2.

447

Step 2 The concave p r o g r a m m i n g p r o b l e m is transformed into the complem e n t a r i t y problem. D e t e r m i n e Q0" Put r=0 and go to step 3. Step 3 The linear p a r a m e t r i c problem of section 6 is solved and the set 0 ~ X ( ~ ) (V0(~)) is determined. Go to step 4. Step 4 A cut is determined; the constraints w h i c h define the present cut, are in addition to Q . Go to step 5. r Ste p 5 Qr is solved (if r>0, an o.s. of Qr is quickly available). If an o.s. of Qr verifies the c o m p l e m e n t a r i t y condition, an o.s. of P is obtained as a subvector of it; terminate the algorithm. Otherwise, go to step 6. Step 6 If the ~subvector of the o.s. of Q r l c o r r e s p o n d i n g to a solution of P, is a vertex of P, go to step 7; otherwise, put r=r+1 and go to step 5. Step 7 The p r o b l e m (8.5) is determined and solved. If the o.s. of (8.~, verifies (8.6) t e r m i n a t e the algorithm; if it verifies (8.7)! put r=r+1 and go to step 5.

448

REFERENCES [I] [2]

-

-

ABADIE J. ~ On the Khun-Tuck~ Theorem. In "Nonlinear programmlng", J . A b a d i e (ed.) ~ N o r t h - H o l l a n d Publ. Co. ~ 1967, pp.19-36. BEALE E.M.L., NumeA/cag Methods. In "Nonlinear Programming", die (ed.), N o r t h - H o l l a n d Publ. Co., 1967, pp.133-205.

J.Aba

[3]

-

BURDET C.A., Genial Quadratic Programming. C a r n e g i e - M e l l o n Univ. Paper W~P. -41-71- 2, Nov. 1971.

[4]

-

COTTLE R .

W.,

The principal pivoting method of quadratic programming.

In "Mathematics of the d e c i s i o n sciences, Part I, eds. G.B.Dantzig and A . F . V e i n o t t Jr. A m e r i c a n M a t h e m a t i c a l Society, Providence, 1968, pp.144-!62. -

and W.C. MYLANDER, Ritt~A's cutting plane method for nonIn "Integer and n o n l i n e a r programming", (ed.), N o r t h - H o l l a n d Publ. Co., 1970, pp.257-283.

COTTLE R.W.

conuex quadratic programming. J;Abadie [q

-

D A N T Z I G C.B. ~ Press ~ 1963.

Linear Programming and Exten~io~{. P r i n c e t o n Univ.

DANTZ!G G.B. , A.F. VEINOTT, Mathemat/cs of the Decision Sciences. A m e r i c a n M a t h e m a t i c a l Society, Providence, 1968. [7]

-

EAVES B.C. , On the basic theorem of complemev~%ity. Progranuming", VoLt, 1971, n.1, pp. 68-75.

[8]

-

GIANNESSI F. ~ Nonconvex quadr~tLc programming, linear complementarity problems, and i ~ l e g ~ linear programs. D e p t . o f O p e r a t i o n s Research and S t a t i s t i c a l Sciences, January 1973.

Univ.

of PISA,

S., The complementarity problem. ming"¢ Vol.2, 1972, n.1, pp. I07-123.

"Mathematical

ITALY. Paper A/I,

[9]

- KARAMARDIAN

"Mathematical Program-

[Io]

-

[11]

- LE~d<E CoE. ~ Bimat~x Equilibrium Point~ and M~thematical Programming.

KUHN H.Wo and A.W. TUCKER, Nonlinear programming. In: "Second B e r k e l e y Symp. M a t h e m a t i c a l Statistics and Probability'; ed. J.Neyman, Univ. of C a l i f o r n i a Press, Berkeley, 1951, pp.481-492. "Management Science", Voi.11,

1965, pp.681-689.

[IC

-

RAGHAVACHARI M. , On connections between zero-one i ~ t e g ~ programming and concave programming und~A linear eonstr~Lnts.

D 3]

-

R I T T E R K. ~ A m~thod for ~olving maximum problems with a nonconcave quadrat i c objective fun~on. Z . W h a r s c h e i n l i c h k e i t s t h e o r i e , Vern. Geb. 4,

1966, pp.

[1 4]

340-351.

Ggobal o p t ~ i z o ~ o n in nonconvex quadratic progrm~ing and

- TOMASIN E . ,

r~ated finds. ces, Univ. [I S]

- TUI HOANG,

Dept. of Pisa,

of O p e r a t i o n s R e s e a r c h and Statistical Scien_ S e p t e m b e r 1973.

Concave progr~Tming under linear constrainbs. Soviet Math.,

1964, pp. 1437-1440.

449

E16]

-

ZWART P . B . Nonlinear programming: Country_examples to global optimization algorithms proposed by Ritter and T~. W a s h i n g t o n U n i v . , D e p t . o f A p p l i e d M a t h e m a t i c s and Computer Sciences School of Ingeeniring and A p p l i e d Science. Report No. Co -1493-321972.

[173 - BURDET The facial decomposition method. Graduate School of Industrial A d m i n i s t r a t i o n C a r n e g i e Mellon Univ. Pittsbrgh, Penn. May 1972.

A WIDELY CONVERGENT MINIMIZATION ALGORITHM WITH QUADRATIC TERMINATION PROPERTY b~GIULIO

TRECCANI, UNIVERSITA

DI GENOVA

I, INTRODUCTION AND NOTATIONS We shall consider methods for minimizing a real valued function of n real n variables~: R _.--> R of the following type : i,i

%

= Xk+l. Xk

= o#

k Pk

were pk ~ the search direction~ is a vector in R

n

and O < k ~ the scalar stepsize~

is a suitable nonnegative real number, Two properties of methods of this kind will be considered~

convergence and quadra-

tic termination. Assume that ~

(x) has an unique absolute minimum point x*; then a method I.I is

said to be globally convergent for the f u n e t i o n ~ ~ if every solution x

i

o£ i.i,

starting at any point x 6 Rn~ is convergent to x*. o Assume now that ~ (x) is a convex quadratic function; then a method I,i is said to have the quadratic termination property if for every initial point x zes ~ ( x )

o

it minimi-

in at most n interations,

In the following it will be assumed that 1.2 ~

(x) in continuously differentiable in R n ;

1,3 ~

(X) is bounded from below ;

1,4

every level set of ~ (x) is a bounded set

1.5

there is one and only one point x ~ R

n

such that grad ~ ( x * ) = O.

Then it can be proved that x* is the absolute minimum point of ~ ( x )

and that

the level sets are conneetedo We remark however that ~ (x) has no convexity property, We shall construct a modification of the well known Fletcher-Reeves conjugate gradient method~ which will be proved to be convergent and to have the quadratic termination property. Even though accurate line minimization is required in our method~ it will be proved to converge without any convexity assumption and search direction restoration~while the quadratic termination property is conserved; this seems to be a somewhat new result respect to the classical conjugate gradient methods of Hestenes-Stiefel, Polak-Ribiere and Fletcher-Reeves, This algoritm has been deduced as an application of the theory of Discrete Semi Dynamical Systems (DSDS)(see

~

) ~ a short surm~ary

451 of definitions and results of this theory which will be used for our purposes is in section 3. 1.6 Basic Notations. G(x)

is the gradient of

n R .

at the point x

gi = g(xi) ' d i = xi+ 1 - x.l " I x I is the~clidean norm of x R n. + R is the set of nonnegative real numbers, + I is the set of nonnegative integers ~is

a metric space,

F(X) is the set of compact non empty subsets of X

2. THE ALGORIT~I 2, I

Xk+ I

2,2

dk=

k

2,3

~

£R

2.4

~

= ----

2.5 p k

Pk :

grad ~0 (xk + ~ Pk

o if ~ - - ~ , T gk gk + 1

=

if

)T Pk = O I

~=Min~ %+1 ~ o

otherwise

,

k

tgk Ilg~ +i l dkT ~ 2.6

k

if

Idkll~I

2.7

Po -- "go

2.8

Pk+l =

" gk+l

+

gk,O

Igk÷II lg~l~

,

Theorem.

=i

k

~k }

ll -

,D e

3. COMMENTS AND IMPLEMENTATION OF THE ALGORITHM 3.1

=O

otherwise,

otherwise

Pk

"

452 n R , is an infinite sequence

The solution of the algorithm 2~ starting at any point x o

L j )xi~

~ such that if for s6me k ~ I+

In addition, if x * f

{xil ~ then ~ i

we have mX'+l = ~ m = ~(xi )

' then x.a = ~"

for i>/ k

is a strictly decreasing sequence of

real numbers. Proof.

Assume that

for i = 0 ~ . . ~

(i)

Pi # 0

(ii)

d.@O

(iii)

#.>o I

Xo,..,~ xk

can be computed by algorit~me 2 and that x. ~ x* l

k. Then the following properties hold for 02i ~ k T Pi oi+l = 0

~

1

(iV)

(Xi+l) < ~(xi> o

These properties hold for i = 0 ; indeed Po hence

~

o ~

0

and

" go

~ o * O ; on the other hand

by 2.7 and go # 0 by assumption, ~ o = i and ~ ( x I) ~ ( x

O) .

Assume that these properties are true for i ~ k , and prove them for i+l. Since T

~l" ) 0 and

gi ~ 0 , P i + l

can be computed by 2.8; t h e n we have Pk+l

2 =

"

~ gi+l

'

Pi+i * 0

f

)

, xi+ 2

gi+l

=

T 0

~ since

gi+i

can be c o ~ u t e d m d

Pi = 0

and

gi+l ~ 0 .

It follows that

di+ 1 * 0 , ~ < x i + 2 ) < ~ ( X i + l ) ,

~ i+l>

0

and the statement follows by induction. Assume now that x = x* for some

k ~ I, while x

J

~ x*

for j < k .

Then by 2.8 we have Pk = 0 ~ which implies dk = O ~ Xk+ I = x k and Pi = di = 0 for every i ~ k . It is clear that algorithm 2 is an ideal algorithm not only because the line search 2.4-2.4 is assumed to be exact, but also because no stopping rule is given and in the computation of ~ even if ~

k

very small quantities can be involved in the denominator,

is not very small.

For these reasons~maintag the assumption of exact line searches~ we propose an equivalent form of the algorithm 2. 3.2. ALGORITHM i. Set i ----O ~ Po = - go

453 + l

:

R

2. Compute ~ . = Min A. = Min

g r a d ~ (xi + ~

l

3. Compute

di = ~ i Pi

4. Compute

x

i+l

= x

i

P i )TPi ----0

+ d I

5. If ,.Igi+lJK g stop, otherwise go to 6. T

6. f~

= i

IgiJ I gi+ll

7. ~

-i

8.

Igi÷l giI

J gil Jpij

Pi+l = "gi+l

J gi+lJ IgiJ 2

+

2I

QIi i- ~ i

Pi

9. Set i = i+l and go to 2. 3.3

THEOREM + # x* , then either there is k E I such that the algorithm stops at a point

If x O

Xk such that J ~ J <

~,

and I Xo,.°°, Xk} belongs to the solution of algorithms 2

starting at x° ~ or the solution of 3.2 is an infinite sequence

x.

such that

l

+ I g i I)l~ and ~ (Xi+l)<~(xi) for every i ~ l

, and{ xil

is the solution of algorith~

2 searclng at x • O

Proof.

k

We have only to prove that

l Pkl

I~tl~l

From the proof of theorem 3.1 it follows that if g i ~ ~ for 0,< i~
T

fgk dkl I~

T

gk = " I ~ T

2

' which implies that :

a~l t~ Pkl

I ~I 2

IgJ

Now we shall prove that algorithms 2 and 3.2 have quadratic termination property• 3.4. THEOREM If ~ (X) is a convex quadratic function of n real variables, then ~ (x) is

454

minimized by 2 and 3,2 in at most n interations. Proof. Indeed if ~ is quadratic~

~

i = 0 and 2 is the Fletcher-Reeves conjugate

gradient method~ which generates conjugate directions and minimizes

~

(x) in at

most n iterations. C.E.D.

4. DISCRETE SEMI DYNAMICAL SYSTEMS. Assume that X is a local compact complete metric space. The norm generated by the metric is denoted by map ~

: X

I " F(X) is the set of non empty compact subsets of X. A

---~ F(X) is said to be upper semicontinuous at x ( X if

lira y--) x 4.1

{

Max zE f(Y)

A DSDS is the triple

Min i z-v[ = 0 v ~ f(x)

.

(X~ ! + f) where f : X ~ i+ ~

F(X) is such that :

(i) (ii)

f [f(x,k),h]

(iii)

+ f is upper semicontinuous in X ~I .

=

f (x,k+h)

~ -~ x E X, h, k [I +

4®2.

A solution of the DSDS through x ~ X

(i)

1+4 ~ C

(ii) (iii) (iv)

is a map

O ~ : 3 -'* X

such that :

I

6- (o) = x (j)~ f C ~ (k), j-k ] ,

¢ j,k C ~

, j>sk

there is no proper extension of ~" which has the properties (i)~ (ii) and (iii).

4.3

A set M -~X

is said to be weakly positively invariant if for every x ( M + there is a solution ~ of the DSDS such that ~ (k) ~ M for every k ~ l .

4.4

The positive limit set of a solution

=

4.5

--

~-~ of the DSDS in the set :

~ k

n

--~

+ ~

, such that 6-(k )-~

A Liapunov function for a solution ~-- is a real valued function

such that

i ~[~

(i)] ~ is a non increasing sequence of real numbers.

The following two properties hold :

n

455

4.6. Property. L+(6 ) 4.7

is a weakly positively invariant set.

Property

A continuous Liapunov

~

for a solution 6" is constant on L + ( ~ ) .

Properties 4.6 and 4.7 will be crucial in our convergence proof.

5. CONSTRUCTION OF A DSDS WHICH IS RELATED TO ~ E

ALGORITHM 2.

Let X be the following set : ~={(xl~x2) ~ where ? :

R n X R n : ~(x2)-<~(Xl) I

Rn

~

R

satisfies 1.2 ~ 1.3~ 1.4 and 1.5. n n Then X is a closed subset of R >~ R and is a locally compact complete metric space with the euclidean metric. Assume that

:

5.1

p : X --~ F(R n) is upper semicontinuous in a nonempty subset X' ~ X.

5.2

a :X ~ R ~

5.3

h :X ~

F(

F(R n) defined as h (Xl~X2) = I

~(e a(xl,x2,P), every

) is upper semicontinuous in X X R n.

PE P(Xl,X2) 1

Y ~ Rn ~ y = ~ p

is such that ~

(y) ~< ~

(x2) for

y 6 h(xl~x 2) and is upper semicontinuous in X ~,X'.

Then it is easy to see that h is upper semicontinuous in X and the map : 5.4

f

:X ~

F(X)

f(xl,x2) = (x2,x2 + h(Xl,X2) )

is supper semicontinuous in X, hence the triple (X~l+,f k) is a DSDS. Assume now that the set : 5.5.

A(Xl,X2,P) =

[ ~ c R + : grad ~ (x2 + ~< p)Tp = O, ~ (x2+ ~(p) ~<~ (x2)

is empty. Then a(xl,x2,P) -----IOIo On the otherhand, if the set we set :

A(Xl,X2,P) ~ ,

456

5.6

a(xl,x2~p) =

The map

~{~ A(Xl~X2,P) : ~ (x2+ ~ p ) < ~ (x2+ p Min A)/

a ; X ~ R n ---9 F(R+) is clearly upper semicontinuous.

We define a map p : X - ~ F(Rn) as follows. 5.7

if x2 = X*

P(Xl~X2) = O

5.8

if x I = x2~ ~

5.9

if x I * x 2 * x* , gT(xl)g(x 2) = 0 P(Xl,X2) = [ y ( R n : lyl~ 1

P(Xl,X 2) = -g(x2)

y = = o(ig(x2) + o(2(x2_xl) , o< 1 ~ 5.10

if x I # x 2 ~ x*

~ gT (xl)g(x 2) @ 0

I g(x2) 1 2

- -

P(Xl~X 2) = -g(x2) +

, (xI_x2)T g(xl) = O

i gT(x I) g(x2) l

(log

ig(h)1 2 5.11

O, ~ 2 ~ O i "

(Xl-X2)

) I g(Xl)l

Ig(h)IIg(h) l

T if x ~= x 2 ~ ~ ~ g (xl)g(x 2) ~ 0 , gT(xl)(Xl-X 2) # O l

I(Xl-X2)~ g(h)l Eh-x2 IIg(xl)l

t gCh)i 2 P(Xl,X2) = -g(x2) +

gT(xl)g(x 2)

i -

I g(xl)l 2 2 g(xl) I (x2-xl), i T ~Xl-X2) g(x I) I 5.12 Property The map P(Xl~X2) is upper semicontinuous for gT(xl)g(x 2) # 0 ~ while the map h(Xl~X 2) is upper semicontinuous for every (xl~x2) ~ X° From the definitions 5.5 - 5°6 of the map a(xl~x2~P) ~ it follows that if we set ;

o

457

(5.13)

.~ (xl,x2)=

then ~ : X ~

R

@(xl)+?(x 2) is a Liapunov function for every solution ~ of the DSDS def~

ned above, 5.14 Theorem + If 6 is any solution of the DSDS defined by ~[~

(k)7 = ~ ( k + l ~

= ~(k+2)J

5.5-5.12 and k E1

is such that

, then ~ (i)= (x*,x*) for every

i~k.

Proof. Let ~ (k) = (Xl,X2),~ (k+l) = (x2,x3) a n d C (k÷2) = (x3,x4) then by hypothesis and by 5.13 ~ (x I) = ~(x2) = ? ~[~

(k)] = ~ [ ~ ( k + l ) ]

implies ~

(x3) = ~

(xI) + ~

(x4); indeed

( x 2 ) = ~ ( x 2) + ~

(x3) ,

Hence it is sufficient to prove that Xl= x* for some i ~ (I~2,3~4). T Clearly, if x3 # x*, it must be Xl~ x2~ g (x2)(x2-xl) ~ O , and also x 2 # x3, since x 2 = x3 would imply ~ ( x 4 ) < ~

(x3).

Then we have Xl~ x 2 ~ x3~ gT (x2)(x2-xl)~ O, gT(×3)(x3.x2 ) = O. Now if gT(x3)g(x 2) ~ O, then by 5.10-5.11 p [ ~ (k+l)] = P(X2,X3) ~ O and T P (x2,x3)g(x3) = - ig(x3) I 2 O, which would imply ~ ( x 4 ) ~ ( x 3 ) . T T On the other hand if g (x3)g(x2) = 0 it would be P2(X2,x3)g(x3) = = ~ i g(x3)

2 +~i2(x3"x2)Tg(x3 ) = - ~ I

(x4) <~(x3)" In any case we have proved that x3

Ig(x3)l ~

O, which would imply

x* implies a contradiction~ hence

xl= x 2 = x3 = x4 = x* • Q.E.D. As a consequence of theorem 5.14 the following property holds. 5.15 Theorem. For every solution 6- of the DSDS defined by 5.5-5.11, L+(6") is a nonempty set and L+(~ ) = (x*,x*). Proof. Since ~ are

is a Liapunov function for any solution, the level sets of ~ (x)

strongly positively invariant, and since by 1.4 they are bounded, L+(~) must

be nonempt y.

458

As ~ (x) is continuous in Rn~ Since by 4.6 L+(~)

is continuous in X and by 4.7 is constant on L'(~')

is weakly positively invariant , if (Xl,X 2) ~ L+(~-) then

(Xl,X 2) = (x*,x*) by theorem 5.14. C.E.D.

6. CONVERGENCE OF ALGORITHM 2. Let

x I be any solution of algorithm 2. Then if we set :

6.1 ~

(0) = (Xo~Xo) , ~

extension of C

(i) = (Xo Xl) , ~ (k) = (Xk_l,X k) then any maximal

is a solution of the DSDS.

Indeed only the cases 5.7~ 5.8~ 5,8 and 5.11 can be verified. It follows that L+(•) x. ~

x*

for i --~

= (x*~x*) by theorem 5®15~ which implies that + ~

.

l

7. CONCLUSIONS.

We have constructed a DSDS whose properties imply the convergence of a conjugate gradient method~ which is a modifification of the Fletcher and Reeves method and has the quadratic termination property. The convergence is global for functions continuously differentiable, bounded from below~ having bounded level sets and one and only one critical point. However these assumptions are not restrictive~

since if the function is simply

continuously differentiable~ the algorithm converges to a local minimum point if the initial point is ~hosen in any bounded level set containing one and only critical point.

459

BIBLIOGRAPHY 1

G.P.Szego and G.Treccani :

Semigruppi di Trasformazioni Multivoche in ~ +. Symposia Mathematica, VoI.VI~ Academic Press, London,New York, 1971, pp.287-307.

2

G.P.Szego and G.Treccani :

Axiomatization of minimization algorithms and a new conjugate gradient method in G.Szego '~4inimization Algorithms", New Yorko

1972 , Academic Press,Inc.

A HEURISTIC APPROACH TO COMBINATORIAL PROBLEMS

OPTIMIZATION

(o)

by E. Biondi,

P.C. Palermo

Istituto di Elettrotecnica ed Elettronica Po!itecnico di Milano

i. Introduction It is well known that the exact solution of large scale combinato rial problems location,

(for instance

travelling

or prohibitive

scheduling,

salesman problem

difficulties

sequencing,

delivery,

plant-

............. ) implies hard

concerning the computation

time and the

memory storage. These problems frequently

are generally approached

developed

tly near-optimal

~ ad hoe", which allow to determine

a general heuristic

approach,

framework of Dynamic Programming,

tive in a large class of combinatorial Some efficient shop scheduling

techniques, efficien-

solutions.

This paper outlines ceptual

by heuristic

algorithms,

successfully

and delivery problems,

based on the co~

which seems to be effec-

problems. tested

in classical

flow-

are discussed.

2. The approac~ Denote f(x,d)

a separable

objective

function,

X

the discrete

set of state variables,

d

the discrete

set of decision variables,

g(x)

= min f(x,d) d

(o) This work was supported by C.N.R. and by foundation

Fiorentini-Mauro.

(National

Research Council)

461 According to the principle of Dynamic Programming the overall opt! mization problem is decomposed into a sequence of linked sub-problems concerning sub-seZs of decision variables. Let dk

be the decision variable (s) at the k-th stage, (k:l,...,N),

D k ={dl,...~d k ~k } x

k

the state variables at the k-th stage,

f~ the 1 k+l

x. l

the feasible definition set of dk~

pay-off

of the

d~ 1

at

the

k-th

stage,

the state variables at the stage (k+l) after the decision d~ 1

at

the

k-th

~k = {d_d 1 _

5k

decision

stage,

_ dk-1}

the definition set of ~k

At the k-th stage the basic dynamic program involves the following computation determine

: g(xk) : min {f~ + min dk l ~k .

f(xk+l , [k+l)} 1 k+l,

= min {f~ + g~x. dk 1 l

:

;} .

(1)

Let k • h(x.k+l.)

i

be a parametric estimate of g(x~ +I) , l where h(x~ +I) is a known (suitably defined) l k+l evaluation function of g(x. ) 1 and

l

I ~ 0 is an adjustment factor of h(x~ +I) , 1

be the (unknown) exact value of the adjustment factor of h(x~+l)l i.e.

g(x k) : min {fk + lk dk 1 1

h(xk+l)} 1

(2)

462

In the followinglproblem Heuristic

Search algorithms

assumptions

2.1

are developed

about the adjustment

(i) by meaningful

Assumption

factors

complexity of

criteria.

1

~1 = I

(k=!~.~N)

The assumption

in accordance with some

k~ , l allow to reduce the computational

Such assumptions problem

(2) is dealt with.

)

(~k)

i=l) ' " " )nk )

is very strong and rather rough~

However it allows to generate a large set S of sub-optimal by an efficient

iterative al~orithm

At each iteration and a solution blems

a different

is determined

solutions

.

value is assigned to the parameter

k

by solving the sequence of sub-pro-

:

min dk

{f~ + ~ h(x~+l)} l 1

(k:l,.~0,N)

,

k given

The minimum cost solution within the set S is selected. The computational tion computed

effort

is limited because the cost of the solu-

by the algorithm

is piecewise

k ) i~e. the same solution may be determined (see fig. 1 and section

constant with respect to in a suitable

3 below).

b

.............

k-range

463

2.2 Assumption

2

k~l = kk

(i=l~''''nk)

This assumption

' (k=l,...,N)

(more plausible than assumption i) leads to a con u

spicuous cut of the definition

set D k

In fact ] ~ (a=l~'''~nk) ~

~

is deleted from D k

if, for every

xk~o, there exists a decision d~ ¢ D k such that 1 k+l, fk + ~k. h(xk+l) < fk + xk h(x. ; (exclusion test I). l i - ] ] Let

R k1 =_ D k

be the definition set of d k after the application of

the exclusion test I. Efficient

Branch-Search

and Branch - and Bound al~orithms may be

developed by applying a suitable branchin~ criterion on the set k R1 . An effective procedure is now outlined. Let U k be an upper bound of g(x k) determined by a classical heuristic method or by the iterative algorithm based on assumption i. Assume that

suk;(o~e~l),

of the parameter

is a close estimate of g(x k) (the value

~ may turn out from empirical analyses of the

quality of the heuristic method used for the generation of the upper bound). Denote

k kLi =

aU k _ f~ l h(xk+l) 1

(d~ E R~)

The following Branchin~ Criterion at the k-th stage the decision

d ko

is now introduced: R1k

such that

464

kk Lo

=

max

dk ~ i aR

ik Li

is selected for branching° The geometric interpretation of the exclusion test I and of the branching criterion is shown in fig, 2 .

J! !

/

/ / /

~ ~ K~

K

,

i l

l t J

2.3. A_~ssumption

ll~ - I~ I < BH

(i,j)

(i,j=l,...,n k)

(k=l,...,N)

This assumption is always true for a suitable value of the parameter

BH, which clearly depends oh the assumed evaluation function

h(x~ +I ) . 1

465

The closer h(xU+l)k 1

containing

is to g(x~+l)k k

the values

Xi,(i=l,...,n

, the smaller is the l-range

k)

,

and a s m a l l v a l u e

8H i s

likely.

According to assumption

3 some d e c i s i o n s

may be d e l e t e d

f r o m Dk.

In faet~ consider the example shown in fig. 3 .

'9

>(~(, ..............

I

I

i

I ~i%

It turns out that d? is not preferred to d? I ]

sin {llk.u 3 - Ik'ull'1lkj _ iLil} ~ I~

if

-

Consequently the following test may be performed d~ is deleted from D k according to assumption 1 a decision d~ e Dk such that ] min {ll k . - ik k u] uiI'iILJ

_ ik Li I} ~ BH

:

3, if there exists

(exclusion test II)

466

Experimental

values of the parameter

delete those decisions

8H may be tested in order to

from D k, which have a small probability

belonging to an optimal The resulting Rk 1

solution. k R 2 ~_ D k is generally

of

subset

found in accordance

with assumption

larger than the sub-set

2 (which corresponds

to

BH : 0). Clearly the risk of deleting a good choice Branch - Search

is less.

and Branch - and - Bound al$orithm~

loped by applying the Branchin~ Criterion, k on the sub-set R 2 ~ (k=l~...,N)

previously

may be devedescribed,

3. Applications The approach can be applied to all combinatorial be solved

via Dynamic

Programming,

stically guided search algorithms. finition of a suitable evaluation According

for medium problems) the computational Experiments

The main problem becomes the d~ function h(x) assumption

or assumption

2 (more efficient,

3 (more founded,

may be taken into account

useful

in order to reduce

effort.

have been performed with reference

ry and flow-shop

that can

Branch - and - Bound or Heuri-

to the size of the problem,

useful for large problems)

problems

scheduling

problems,

to classical

delive-

whose statement and formula-

tion are now briefly summarized. 3.1

Consider the following delivery problem a set of customers,

quirement

for commodity,

by vehicles

each with a known location and a known r~ is to be supplied from a single warehouse

of known capacity.

The objective cost,

:

is to design the routes which minimize

subject to a capacity constraint

the customers'

requirements.

the delivery

for the vehicles

and meeting

467

Let be a set of elements representing

X : {xili:0,1,...,n}

the warehouse mers u:{u..}, 13

(i,j:O,l,...,n)

(i:O) and the custo-

(i=l,...,n),

the set of links available

,

for the

transport, x

k

c

{x-x

--

the set of customers

}

to be supplied

O

after (K-l) stages, dk

the feasible route to be selected at the k-th stage.

Assume h(x~ +I) : l

~X.EX.

]

k+l i

u . o]

The iterative algorithm

[4] and

a branch-search

algorithm [ 1 3 ~

ae

cording to the outlined branching criterion have been experimented in large problems. The computational

results

show that the minimum cost solution a~

rained by the iterative algorithm gives a satisfactory

near-opti-

mal solution of the problem (a typical diagram of the results

is

drawn in fig. i). The quality of the solution supplied by the branch-search rithm is practically

equivalent

putation time is considerably The tests of both algorithms

(or just better),

algo-

while the com

shorter. show remarkable

advantages

with re-

spect to the exact and heuristic methods proposed for the same problem in the literature. 3.2

The classical three-machine

flow-shop

scheduling

problem is

now considered: n jobs must be processed through three machines

(A,B~C)

in the sa-

me technological

order. All jobs are processed once and only once

on each machine.

A job cannot be processed on a machine before it

has been completed on the preceding machine

in the technological

468

order~ The objective

is to schedule the flow-shop so that the comple-

tion time of the last job on the last machine

("makespan")

is

the processing times of job i ~ (i:l,...,n),

on

minimized. Denote ai~ big e i

the machines A,B,C

;

d

the job-set;

dk

~he job to be processed at the k-th stage (k=l,..,n);

~k={d_d I_ ,o_d k-l} Dk

the job-set to be processed at the k-th stage;

B(x k) = max {B(x k-l)

~

C(X k) = max {c(xk-l)~

~ ao} + bo {d_~k+l} ] 1

;

B(xk)} + c. 1

B(X I) = a+b

where a~b~c are the processing times of the

C(x I) = B(x~)+c

first job of the scheduling

;

f~ = C(x k) - C(x k-l) 1

As s ume F

% ~k+l h(xk+l) : max I ~

] b° +

~ a. + dk+ i 3

min

c. -~C(x' k)_B(xk),

min (c.+b.) -k+l ] 3 d

(C(x k) - ~

...... a.) -k+l} ] {d d

A Branch - and - Bound technique has been applied according to the Exclusion Test II and the Branchin$ Criterion.

The results

show that the procedure allow to determine a near-optimal tion in large problems

solu-

in a very effective way. It has been che-

cked that such solution generally corresponds to the optimal one in small size problems,

469 REFERENCES

i

N. Agin

"Optimum seeking with Branch and Bound", Mgmt Sei. 13, 176-186,

2

(1966).

S. Ashour " An experimental investigation and comparative evaluation of flow-shop scheduling techniques"

,

Opns. Res. 18, 541-549, (1970). 3

E. Balas "A note on the Branch - and - Bound PriDciple", Opns. Res. 16, 442-445~

4

(1968).

E. Biondi, P.C. Palermo, C. Pluchinotta

"A heuristic method

for a delivery problem", 7-th Mathematical Programming Symposium 1970 5

The Hague - The Netherlands.

H. Campbell, R. Dudek, M. Smith "A heuristic algorithm for the n job - m machine sequencing problem", Mgmt. Sci. 16, 630-637, (1970).

6

N. Christofides,

S. Eilon "An algorithm for the vehicle dispa~

ching problem", Opl. Res. Q. 20, 309-318 (1969~. 7

R.J. Giglio~ H.M. Wagner "Approximate Solutions to the threeMachine Scheduling Problem", Opns. Res., 305-319 (1964).

8

P. Hart, N.J. Nilsson, B. Raphael "A formal basis of the heu ristic determination of minimum cost paths" IEEE Trans. SSC 4, 100-107 (1968).

9

M. Held, R.M. Karp

"A Dynamic Programming Approach to se-

quencing problems", J. Soc. ind. appl. Math. i0,

196-208 (1962). i0

E. Ignall, L. Schrage

"Application of the Branch - and -

Bound Technique to some flow-shop scheduling problems", Opns. Res. 13, 400-412 (1985).

470

ii

G.B. McMahon,

P.P.

Burton "Flow-shop

scheduling with

Branch - and - Bound method", ~73-481 12

L.G. MiTten

Opns.

Res~ 15~

(1967).

"Branch - and - Bound Methods: tion and properties"~

Opns.

general

formula-

Res. 16, 442-445,

(1968). 13

P.C.

Paiermo~

M. Tamaccio

"A Branch-Search

delivery problem" (1971).

algorithm

for a

Working Paper LCA 7i-4

A HEW SOLUTION FOR THE GENERAL SET COVERING PROBLEM L~szl6 B61a EOV~CS Computer and Automation institute Hungarian Academy of Sciences

1. Introduction~ pa~itioning

The theory and applications

of the set

and general set covering problem is discussed in the

present paper. After the problem formulation, three applications are shown: bus route planning, airline-crew scheduling and a switching circuit design method.

After a short s~rvey of existing methods a

new algorithm is introduced in section 5. The exact algorithm is based on the branch-and-bound principle, but no linear programming is used in determining the bounds. A heuristic procedure is utilized instead, which often gives the optimal or near optimal solution of the problem. The optimality in certain cases guaranted by lemmata 1-3. The last lemma also provides a lower bound for the branch-andbound procedure,

which is close to the real minimum of the sub-

problems after a few steps - as practice shows. A computer program is written for a CDC 3300 compater. The results are promising. The program is being further tested and developed.

2.Problem and terminology.

Let us consider the following problem:

rnin ~T~

A 0 where

A

is a given

~ X v~

4 matrix of

O

and

~

is an _C vector of positive integer elements and e each component of which is 1.

elements is an ~

-vector,

Problem (l~ is known as the set co-

vering problem for the following reason. It is given ~

subsets of

472

the set

~:~

,../ ~

,

c.~',>O

A cost

is associated to each element je ~o

At least one element of each set ~l each set is to be covered a set

i.e,

at a minimal total cost. In other words

~ is to be determined:

Problem tl)

is obtained, if matrix

CL,~~ ~ (,0 The set 1

2 9~ is to be chosen

~

is defined as

otherwise.

is the subscripts of variables

~#"

having the value

in the optimal solution of problem (~ .

The general set covering problem

~'v~fn cTx_

may be interpreted a similar way. The only difference is, that each set J, should be covered is a given

~

~>O

times instead of just ones. Th~s ~

-vector of positive integer components.

The set partitioning problem ~

(3) is ~he same as problem

cww

A_ =e (~,

except that the inequalities are

substit.ted by equations. The same i~terpretation may be used also for problem (3) , the only difference is that each set should be

•

473

fines a partitioning of the set

(3)

~_ of probZem

covered exact ij ones. Each solution •

however de-

Let

otherwise. Then obviously

if

--]o and

Thus t h e

sets

problem {31

~...~

Mm

give a partitioning

may be s t a t e d

at a minimal cost,

of set

~o

• Therefore

as t o d e t e r m i n e t h e p a ~ i t i o n i ~

I t s h o u l d be n o t e d ,

that

apart

~o

of set

from the trivial

case of identical columns in matrix

A

any postitioning uniquely de-

termines the corresponding solution

~

,

if there is any.

3. Ao~lications. 3.1

Planni~

of bus routes.

with an attached cost c~

.

It is given o

There are ~

possible bus routes

bus stops and ~

is the

incidence matrix, i.e. if route J

goes through bus stop

~

otherwise.

bus network of minimal cost is to be determined in such a way, that at least one bus route should go through each bus stop. If variable ~

has the value

1

or

0

realised or not, then problem

depending on whether bus route (1)

j

is

is the mathematical model for the

b~s route planning.

~.2

Airline-crew scheduling.

assigned to each of the g i v e n m

Exactly one crew should be

flights in such a way that each crew

obtaines an acceptable full assignement /limited number of flights, not too long working time etc./

The objective is to minimize the

474

number of crews actually ~sed. To solve this problem let us determine a great nu]nber of acceptable crew assignements and calculate matrix

A." ~'

if flight

is included in assignementj

otherwise

and the meaning of the variables

X,,-~

if assignement

~

is accepted

otherwise.

set

Then a

pastitioning problem

(3)

is obtained with

cj'---d

is called truth f~nction, if both the variables

u~;.-.) UN

the function

and i.

0

~

may take only the values

is also referred to as the

off

on

or truth value true.

and

The value

position of the switch or the

truth value fals.__~e.Similarly the value position

0

Cj=~...,~).

1

is also interpreted as

The costs of

M~D gate and

OR

gate are given. The problem is to realize the truth function (u~)

;~n) at a minimal cost. Let us suppose, that the function

is given either in a tableau form or in a disjunctive normal form. The latter

is the disjunction of different terms, where each term is

a conjunction of a subset of variables form

A-u~ Let

;..,~

~k

F

and their negated

d-%. denote either variable

Definition. The conjunction of function

U~) " 2 U ~

if

U~,~k~'" ~ k j ~

u~ or its negated form 4-u~ •

Q = ~k~ ~ L

~kr

implies, that

is a prime implicant ~ (~,.-.,~N)=4

475

and no part of

Q

has the same property.

Then our problem may be transformed into a set covering problem by the following steps:

(i)

Determine all prime implica~ts

and their attached costs

C ~ CL}

C~

%)~Zr'v~

of function F

/The number of conjunction

in the prime implicant times the price of the AND gate plus the price of the OR gate counted once, becaase these prime implicants will be connected by the sign of disjunction/

and define the matrix

A

:

t Then the truth function

~

0 otherwise. may be written as

F= and the problem becomes 1he set covering problem If the function

~

takes the value 1

for more then half

of the values of the argument, then the function ~ - F

may be de-

termined instead. There are also other tricks to decrease the size of the problem.

4. Survey of methods.

Practically a version of any integer

programming method may be tried for the set covering problem. On the other hand a great number of papers are already devoted to the subject, thus only a partial list of publications is mentioned here in which

476

each direction is represented by one or two papers.

First of all a version of the Gomo~y [SJ by Martin ~l~

is reported to be effective to set covering problems.

A paper of House, Nelson and Rude [7] of a

cutting plane method,

is devoted to the development

special algorithm only for the set covering problem. The sub-

stance of the method is the construction of additional rows to matrix A

for the exclusion of solutions not better than the best one ob-

tained so far during the algorithm. The paper of Bellmore and Ratliff ~3] also falls to the category of cutting plane methods with a substancially differen~ type of cutting method.

A typical example for the use of branch-and-bound method is the paper of Lemke, Salkin and Spielberg [ ~

.

The main difference

between their and our approaches/discussed in section 4/ is, that they are solving linear programming

subproblems for obtaining bounds

and in the method of the present paper no linear programming is used at all. Most probably the structure of branching is also different.

Heuristic methods play an important role in large problems for several reasons. The airline-crew scheduling is solved by a heuristic method by Arabeyre, et all [l]. other

Garfinzel and Nemhouser [4J

apply

heuristics for the set partitioning problem. A rounding process

and consecutive fixing is used to decrease the size of the problem. The smaller problems so obtained are solved by existing integer programming methods. The group theoretic approach of integer programming is u~ed by Thiriez Ill]

having the special advantage, that in most

cases the determinant of the optimal basis because of the

5.

O-1 elements in matrix

B

is usually

small,

A.

A new solution for the seutcoverin~ prob!em_~. The description

of the method consists of two part. The first part gives an explanation

477

of a h~uristic method. The second one describes an exact method using the branch-and-bound

principle and also the heuristic method for

calculating bounds.

5.1, A he uristi p method. rows,

Ik

Let us define the set of uncovered

and the set of subscripts of unused variables,

executing iteration ~

.

~& after

At the beginning all rows are uncovered

and no columns are used:

Io

,

Let us introduce the column coumt at iteration ~ k

and the e v a l u a t i o n

o f maused v a r i a b l e

, ,-f

as

at i t e r a t i o n

V=o

is nothing but the row covering cost for each individual row, thus it gives an evaluation of variable

At iteration

~÷ ~

the variable

~"

at iteration ~

Xjk.~

.

with the best

evaluation is chosen:

The new sets are easily calculated

where

Mk+4

able

x~'k~ 4 :

~$ the set of subscripts of the rows covered by vari-

The p r o c e d u r e i s c o n t i n u e d ~ n t i l

either

~

or

~t

becomes

478

empty. In the first case we have obtained a feasible solution of the problem. In the second one no feasible solution exists. Let us suppose that a solution is obtained, i.e.

Xi-- t 0

I~ = ~)

otherwise.

Then the following lemmata show how good this solution may be:

Lemma

then

4

~

l°

If

is an op¢imal solution of problem (lJ.

Proof. Consider any solution the following notation

We can suppose~ that

E

of problem

(~

and introduce

:

~-

~

.

Furthermore let

The n

The two expression may be handled seperately:

by the definition of

~j~

and

~=

On the other hand

479

The first inequality holds because of the definition of the number r

and the second one is also valid, because

of problem (1). (6) -(8) results, cv ;

is a solution

that

s c~'~ '

feasible solution of problem

for any

~

(1)

,

thus the lemma is

proved.

The following lemma gives a stronger result:

Lemma 2.

(9)

Let us define the numbers

~,.~'a~ -~ , ~i ~.~M~,

~

~"

for each row:

~'¢ ~A)...~s--~-

If

(~o) 2,,,,~ ~,.j ~ ~ ~

~o~ ~

O<j.~

~,

~~...~ j~.

4

then ~

is an optimal solution of problem (9 •

Proof. Consider any feasible solution ~ Because of the definition of numbers

of problem

1 .

~

%'--4

On the other hand, using the definition (5), and the fact

that ~

is a

the supposition

feasible solution of problem

,

(lO) i.e.

480

We obtain

~,~.u,. 5 Then ~ll1

and

co .

=c_x.

C12) together gives the desired result, that

c<~

s

c~'Z

which proves the lemma~

The following lemma provides a lower bound for the branch-andbound procedure if lemmata

1

of the feasible solation ~ The optimality of

j

and

2

did not prove the optimality

obtained by the above heuristic algorithm.

may also be proven sometimes by this result,

see the notes after the lemma.

Lemma 3.

1

4

whe re

Denote the optimum of problem

Z

__ >

~

=

{

(~

by

Z~ .

Then

)

'---A

{~"

is the

"minimal covering fraction" of row i:

{°i Proof.

Define the following new problem

I

(¢= 4 ; . I~ 9

481

~q

~n

~'---4

k--4 v'~'

~t"=4

k=/

whe re

Ij if j:k J

otherwise.

To any feasible solution

x

there exists a solution of problem

l l5)) in such a way that the objective functions are the same, e.g.:

On the other hand the minimum of problem (15) is obviously ~

.

This proves the statetement (131 of the lemma.

Notes

:

1°

If

cw~=_ _ ~

4

, then ~

is on optimal solution of

problem (1) .

2o .

If

_crY_> F

timum of problem (l)

,

then ~

gives a lower bound of the op-

which may be used in a branch-and-bound pro-

cedure.

5.2

A branch-and-bound procedure fo r solving problem

i .

Many papers are devoted to the discussion of branch-and-bound procedures. See e.g. the survey of Lawler and

Wood [~

. On the other

hand the computer program is now developed further, thus only some basic caracteristics of the present

1°

LIF0

/last in first out/

comptur

program is given here:

rule is used for memory saving

482

20

Fast termination of s~bproblems

is guaranteed if no so-

lution of the subproblem exists. /The subproblem s form then problem (1), at

0

or at

3o

are of the same

only some of the variables are fixed either

1.

Fast recanstruction of the next subproblem to be considered

is made in a special way.

~o

Good solutions are usually given in case of abnormal

termination.

6 m _ T h ~ computer program in FORTRAN.

is written for a CDC 3300 computer

The program is being tested and further developed. It

solves problems up to

lO0 variables in a few seconds. A version of

the program is suitable of solving the general set covering problem(2) without transforming it to ~he form (~ o used to solve also problem [~

,

but

A similar

approach may be

most probably in this case

a somewhat different evaluation of variables

/or group of variables/

is more useful. References

Arabeyre~ et all The Airline-Crew Scheduling Problem: A Survay. Transportation Science, 3 /1969/ No 2.

[2]

M.L.Balinski, On maximum Matching, ~inimum Covering and their Connections. Proceedings of the Princeton Symposium on Mathematical programming . /1970/

3]

[~]

503-312

M.Belmore and D.Ratliff, Set Covering and Involuntary Bases, Management Science 18 /1971/ 19~-206 R.S.Garfinkel and G.L.Nemhouser, The Set-Partitioning Problem:

483

Set Covering with Equality constraints. Operations Research 17/1969/ 848-856. R.E. Gomory, An Algorithm for Integer Solutions of Linear Programs, 269-302 in [6~ . R.L.Graves and Ph.Wolfe /eds./ Recent Advances in Mathematical Programming, McGraw-Hill /1963/ R.W.Ho~se, L.0.~elson, and T.Rado, Computer Studies of a Certain Class of Linear Integer Problems. Recent Advances of Optimization Techniqmes. Lavi and Vogl /eds./ Wiley /1966/ pp. 251-280.

[a]

E.L.Lawler and D.~. Wood, "Branch-and-Bound Methods: A Survey", Operations Research, l~ /1966/ 699-719. C.E. Lemke, H.Salkin and K.Spielberg, Set Covering By Single Branch Enumeration with Linear Programming S~bproblems Operations Research /to appear/ G.T.Martin, An Accelerated Enclidean Algorithm for Integer Linear Programs, 311-318 in [6J K.Thiriez, The Set Covering Problem: A Group Theoretic Approach Revue Trancaise de Recherche 0perationelle

v-3 /1971/ 85-10~.

A THEORETICAL PREDICTION OF THE INPUT-0UTPUT TABLE

Emil KLAFSZKY Computer and Automation Institute, H~ngarian Academy of Sciences

l~l The pr0ble~m. Denote

!l,I2,.®.Ii,.o.I m

J1,J2,...Jj~.~J n

sinks.

going from

Jj ~

Ii

of

Ii

A

and call

to

used ~p by

output of

it

zi )

Let

~2>0

sources and

be the amount of the quantity

in other words the quantity of the production

Jj~ We shall denote the

mxn

matrix

in~_~_zout~_ut__tab_~le.~ The sums .~o<%/. E_~,~

and the s = s

< the torsi

(ofcj)by

(the t o t a l

input of

J j ) a~

eb4

usual are called the in~_~t-__~oE~t~_ut_maErgi_na_~l_v_al_~es _. The fundamental problem treated in this paper is : what we can say about a new input-output matrix cribed marginal output values values

C

: (~,a,

rent matrix

. . ,

~.)>o

X = ( ~ij')

which has the pres-

/~ : ( / 3 , , / % . . . ~ ) > o

a~d i~put

respectively, when we know the our-

A : C~j ) ,

More exactly what additional hypothesis can be set by the aid of the matrix

A

to ensure

a

"sufficiently good"

system of equalities

(i) O o ,,~,...~),

which evidently has many solutions in general.

solution of the

485

2. The method RAS.

A well-known method

[3-7]

which ensumes an

sufficiently good solution works with the following hypothesis: The variables

(2)

~j

are of the form

~,.> o ,

(,_-,,~,... ~),

5.>o, O:q,~,...,~). We s h a l l

~;

denote t h e d i a g o n a l m a t r i x

(c:,,~, . . . . )

ana ~-0--',~, . . . . )

So the assumption

consisting

by

R

o f t h e numbers

and 5

respecti~ky.

(2) can be written in the form:

X=~ Hence the name of the method. This method is sometimes called as S~koys~i_~-

or

Fratar method

and also as the Gravit~ method

/after the first users of it/ /after the principle of gravitation

which was used to explain the hypothesis/. It is easy t o see t h a t

if

there is a s o l u t i o n

o f the system (1)-(2)

then the system (i) has a solution with the following properties

}ci = O, (3)

if

~,./= o

and

Conversly the above condition is also sufficient for the solvability of (1)-(2)

as the following well-known theorem asserts.

T H E O R ~ 1. I There is a solution - unique in ~,~ -- of the systems

I (i)-(2)

if~ the~ is a solution o~ (i) satis~ing (5) •

486

I~ the

followi~_~r__a~£9~h__we__£h__a!!_5££_~a£2~£E_h~o_t_~££!£_2~£a__o__n

some information theoretic considerations which also leads to the method

RAS.

This treatment ~ields the above theorem as an eas~_

corollary. If we denote the solution

X

ensured by theorem 1

#~(A~,c)

by

then beca~se of the uaiqueness we get easily the following property of the method

COROLLARY

RAS.

Suppose ~hat for

~he system

holds~

i.e.

(A,~,c)

(1)-

(2)

the

(A, ~, c~)

is solvable. Then the r e l a t i o n

"step by step" prediction has

the same result as the

3.

and for

"one step"

one.

Treatment of the 9roblem using ~he geometric pro~rs~ming.

In the following for the sake of simplicity of the treatment

we

may suppose withouth the loss of generality that the equalities

hold. We shall use the hypothesis as follows:

The p r e ~ c t i o n

X ° C ~ j ) i s considered to be

if it satisfies the system (1)

and the

"good"

I-divergence

/information divergence or information surplus by Shannon-Wiener/ is minimal;

of the table

i.e.

~'--4 j=~

X

related to table A

if the function

°(9'

has the minimal value on the set of solutions of system(l).

487

The continuous extension to the whole non-negativ orthant of the function

~

~

is zero if

~).=o.

Let us introdmce the following notations:

and

Since the function C@ ) has finite infinum on the sol~tion set of (l)

only if

.~,~. -o

~'=0

i~

implies

4 =0

, thus we have to choise

( ~,j ) ¢ Q.

Using this restrictionwe get a mathematical programmimgproblem: the dual problem of a geometric programming as follows.

~inimize the function

(sj

.Z__..],~j~

<:d

on the solution set of the system

/

~,;i->-o,

6;j) E cR,

,~,4. ~ ~, ~-,,~,...~). c,;i)e~ We may write the primal of this dnal geometric programm [2] : Find the s,~prem~m of the objective f~uction

on the solution set of the system

488

t In what follows we have to distinguish two cases. First we shall investigate the problem and secondly when

A.,

(5) - (6) when it is canonical

it is not canonical but consistent

The canonical property of the problem

the system (6) ~as a soiutio~ of prope~y

<5)-

¢A.,)

CB.,).

(6)means

that

};j>o ¢ O V J ~ Q )

.

However this assertion is equivalent to the (3) •

Using some results of the geometric programming we get the following assertions:

Ci)

For the dual geometric programm is canonical (and so consistent ) and the feasibility set (6) is bounded, so the objecUive function (5) attains its minimum. This minimum is unique because of the strict convexity of the objective function.

(ii)

The canonical property of the problem and the bo~ndeness of the dual objective function
( ii)

The pair of solutions

pp. 169 Theorem I.).

are optimal iff the

e q~alit ie s

fo= all

(Z,j)E

489 I hold(J2] pp. 167 Lemma 1.) . 1 I

I As ~ q = ~

therefore the above assertion is equivalent

I[ to dd~g~the following system of equalities i

t(8) i

~)'=~"

~)d)E~.

for all

l

Using the above observations we get the following

THEOREM

2.

Suppose that the prediction problem (with the parameters ~i ~, c ) dition (3) •

fulfills the canonical con-

Then the hypothesis of RAS method

and our hypothesis based on the minimization of I-divergence yield the same result.

~_oof_~..

~

~¢,e,,~

~

a sol~tio~ of

&

solution of the problem by the method

~ d the n=bers }'i'~"5 they also satisfy (7)

So ~,~,~,b~"

f~fill (8).

The numbersVfulfill (6)

It is easy to see that

because

~g~t~g1~ be now a solution of the optimization

problem. So they satisfy (8)

The Theorem 1.

RAS .

are a pair of optimal solutions.

On the contrary let

ted in the form

(i) - (a) i.e. the

RAS

and this means that they are represen-

•

is now the eas~_c~onse~ence of the Theorem 2. and

the assertion ~ i ~ . B.,

Let us suppose that the feasibility set (6) is consistent.

Denote

E = (el)=(er~))-- (~j)

an rzaXz~

matrix, where

490

Denote further It is clear that

I~E if=~ IIe~. II=~_lleci)II,

Let ~s consider for with parameters

the modified prediction problem

E =>o

whe re

~, ~ i c~

+ IIE II

+ Jl#"tl

(9)

It is obvioas that if

where

g-~o

~ :°;-

then #.@)--~/3~J)

and

~5°~=

The modified problem is a canonical one for all optimal solution

~

.

We shall show that the

can be written in the form

g>o RAS

and so its /Theorem 2./

"small modificatlon " " of the problem yields

"small modified" optimal solution, namely the theorem is tr~e a follows:

THEOREM

3. ~ The optimal solation is a continuous function

_Pr_£oof: Indi~ectly~ let us suppose that there is a sequence ~j~)... E~°o. -~ o

for which

XE~-~

and

X ~ X;

°

Then, as the optimal solution of the original problem (g= o) is unique we get the relation

491

O.o;

¢ (xD < ~(~) we have the limit

Because of the continuity of the functiomal relations

~(,C+~,EJ---e(*:), and

~,(x,[)

This and

(lO)

-

¢(ej,

<,---o.

imply the existence of an index

~o

for which

the relation

ecg+ oE)<

)

holds. This however contradict the optimality of Xo +

There

XEIo because

the

is a feasible solution of the problem modified by ~ o

is

a well-lr~o~m

method

~0 ~m~_~St_e.~ha~_ Ill mization

of

the

square

-

different

from

that

of

RAS -

due

which works with the hypothesis of the minicontingency:

I The prediction X = (~ij) is considered to be "good" I I if it satisfies the system (1) and the "distance" of I

I the table

X

from the table A

is minimal;

i.e.

if the

I function

,

>(x;=.ZZ

~,,

I has ~he m i n i m a l v a l u e on the s e t o f s o l u t i o n s

There is a close connection between the functions

~

o f system (Z),

and ~

.

Namely if we take the first term of Taylor series of the function ~ - ~ ~?

arouad of point

~

instead of the function ~

492

into the function of

~

~hen we get exactly the function ~

which proves our observation. References Deming, W.E and Stephan, F.F. "On a least squares adjustment of a sampled frequency table when the expected marginal $otals are know" Ann.math.Star isls, ll/19¢O/ pp. ¢27-JJ~. Duffin, R. J. and Peterson, E.L. and Zener,C. Geometri 9 ~ , John Wiley, New York, 1966.

[3]

D'Esopo,D~A~ and Lefkowitz, B. ~'An algorithm for comp~ting . intersonal transfers using the gravity model." 0pns. Res__~., 1963, ll.No.6, pp. 901-907. Fratar,Thomas J. ~'Vehic~lar Trip Distribution by Successive Approximations" T r a f f i c ~

[6]

pp. 53-65 /January 195¢/

Stone,R. and Bates, J and Bacharach, M: A ~ro~ramme for Growth Input,output relationsships 1954-66. University of Cambridge, 1963. Stone~ Re and Brown, A.: "A long term growth modell for the Britisch Economy" /In the book Europes F~ture in Ed. Geary, R.C./

AN IMPROVED ALGORITHI~( FOR PSEUDO-BOOLEAN PROGRAN~LING Stanislaw Walukiewicz, Leon $lomi~ski, Marian Faner Polish Academy of Sciences Institute of Applied Cybernetics Warsaw, Poland I.

II~RODUCTION

Problems of nonlinear integer programming have been developed recently as applications to facility allocation [13,15], capital budgeting [12], transportation system management rl~] and networks design [10]. The methods for solving such problems have been considered more rarely than the methods for solving linear tasks, so today we have three prominent methods for handling the above mentioned problems. They are as follows: (il the implicit (in particular case explicit~ enumeration method proposed by Lawler and Bell in 1967 [11]; (ii~ the the method of transformation into equivalent linear problem developed by Fortet in 1959 [5 ] and Watters in 1967 [19 ] and (iii) the pseudo-Boolean programming described by Hammer and Rudeanu in 1968 [8,9,18]. The efficiency of the first two methods has been proved in practice [11, 17]. Many authors [7, 17] have pointed out the following disadvantages of pseudo-Boolean programming: (i) procedure is hardly suitable for automatic computations because of the low degree of the formalization; (ii) test used to determine the families of feasible solutions are quite weak; (iii)the method falls into two independent parts (constructing the set of feasible solutions, determining the optimal solution or solutions) and the amount of computations seems to increase directly with both the number of constraints and the number of variables. The H-R algorithm for pseudo-Boolean programming, given in the paper, has not the first of these disadvantages. The remaining two are reduced. This algorithm may be considered as an application of the branch and bound principle to discrete programming.In sections 3 and 5 we give a short description of the H-R algorithm. Bradley [2, 3] has shown that every bounded integer programming problem is equivalent to infinitely many other integer programming problems. Then we may ask question which of the equivalent pr6blems is the most suitable for branch and bound algorithm. We will discuss this question in section 4, and now we note, that the answer contains

494

also the estimation of the efficiency of the method (ii). A summary of the computational experience with the H-R algorithm is presented in section 6. 2. THE PROBLF~ Let Qn be an n-dimensional unit cube, i.e.

a set of

= (Xd, x2, o..~ Xnl, xj ~ (~,II, j ~ (I-~), and let R real numbers. By a pseudo-Boolean functions we shall

points

X

=

be a set of mean any real

valued function f(xl, x2, ..., Xnl with bivalent (0,1) variables,i.e. f : Qn --~ R. It is easy to see that such a function can be written as an polynomial linear in each variable [8J. We don't distinguish, by means of a special graphical symbol,

the

logical variables, which take the value true (1) and false (0) from, the numerical variables, which take only two values 1 and 0. It will be clear from the context that either the variables x, y are logical or numerical. In addition there exist well known isomorphism between logical and arithmetical operations: x V y = x + y - xy,

(2.1)

x A y = xy,

(2.21

x = I - x. (2.3) As the pseudo-Boolean programming is a numerical program we suppose that in each given pseudo-Boolean function at least a substitution (2.1) was done. Now, it is possible to formulate a pseudo-Boolean p r o g r a m ~ n g

prob-

lem. It is as follows: Find such X ~ = (x~, x~, ..., x~)K Qn (one or all) that

f(x ~) = where

Min

f(x),

(2.4)

X ~ S

S = { X ~ gi(X) ~ O,

i~

(1,--~}.

(2.5)

In this formulation f and gi' i ~ (I-~) , are pseudo-Boolean functions, S - a set of feasible solutions. If S = ~ then the system of constraints (2.5) is inconsistent. We call the pair (m,nl the size of (2.@), (2.5) and S ~ - t h e set of optimal solutions.

3. THE H-R ALGORITHM The problem (2.~)~ (2.5) may be solved

by means of

the

explicit

enumeration as follows: Let fak be an incumbent (the best feasible solution yet fiund) .For each X g Q n we check, if X fulfils the system (2.5) and afterwords, f(X] ~ fak" The branch and bound principle allows: (i} to execute this enumeration on the level of subsets of

Qn;

if

495

(ii) to omit in this enumeration some of subsets of

Qn

without

loss of any X ~. It is obvious that in each application of this pronciple we may distinguish two main operations: branching and computation of bounds. Generally speaking, the computational efficiency mainly depends on the number of branching and in turn this number depends on the accuracy of the bounds computing. On the other hand, the time devoted to the bounds computing shouldn't be too long,otherwise the applications of the branch and bound principle is not efficient. The H-R algorithm presented below, is a result of both theoretical considerations and practical experiment s. 3.1. The General Description of the H-R A15orithm O

Let S i be a feasible solution set of the i-th constraint and Si_ 1 be a feasible solution set of the first (i-I) constraints, i E ( T ~ ) . For the sake of definitness, it is assumed that S O = Qn" Then #

S i = Si_ I ~ Si,

i ( (T~).

(3.1)

Joining the system (2.5) in the (m+1)-th constraint fak - f(X) >I 0

(5.2)

where fak is a parameter, we have converted the pseudo-Boolean programming problem into the problem of solving the system of (m+1) pseudo-Boolean inequalities. It should be noted that it is not necessary to determine whole the set S (S') . It is sufficient, if the following relation hold S~ C S i,

i ~ (I-~).

So furthermore by S i (S~) we shall mean the set which, may doesn't contain all the feasible solutions of (2.4), (2.5) but contains all optimal solutions.

(5.3) be, which

In other words, the H-R algorithm determines the feasible solutions set of g l ( X ) ~ 0 at the first step, and s/ter by checking which of the X ~ S 1 are satisfying g2(X) ~0, i.e. it determines S 2 and so on up to determining S m. Solving (3.2) for X E S m we obtain all optimal solutions of (2.4) , (2.5) • 5.2. The Branching Process We can bring each of the constraint of (2.5) to the following form (we omit index i here) a O + alC I + a2C 2 + ... + akC k ~ 0 , where ao, aj 6 R, Cj - a Boolean conjunction (product) of some of

(3.4) the

496

variables x 1~ x2~ o~o~ ~n~

j E(T~

~ and

!at! >~ la2i ~ ... >i iakl-

(3.5)

The H-R algorithm requires all sets to be described by means of characteristic functions and eash characteristic function must be the Boolean disjunction of the disjoint reduced conjunctions defined in the following way: A logical expression h(X) = K DID 2 ... ~

(3.61

is called a reduced conjunction in Qn' if: (i) K is a product of some of letters ~j, i.e. ~J ~ LjIFxj'xj)'J~ ~(~). We not exclude the case K = 1. (ii) Each D k is a negation of a product (disjunction) of some of letters ~j, j ~ ( ~ ) , such as that the letters appear in K don't appear in any Dk, k E ( T ~ ) . The variables appear in K will be called fixed variables (in K). It results from (3.6) that, if K ~ O, then h(X) describe in a univocal way, some set G C Qn' therefore we will sometimes write h(G) instead h(X). A set G we will call a family (of solutions}. For the sake of definitness, it is assumed that if K = O, then the corresponding set is empty. Let H(Si) be a characteristic function of the feasible solution set of the first i constraints of (2.@), (2.5), i ~ ( 1 , - ~ ) . Then H(Si~ = hl(X ) v h2(X ) V

... V h q ( k 3 ,

(3.7)

where hi(X)A hi(X) = O, if i ~ j and q - a number of disjoint families of solutions. By branching a given family G by means of a conjunction C we mean computing the reduced conjunction for the following products: h(G1) = h(G) C,

(3.8)

h(G2) = h(G)~.

(3.9)

So we obtain two new disjoint subfamilies by branching onee

the

given

3-3. The Computation of Bounds Let (3.4~ be a (i+l)-th constraint,where for simplicity we omit index (i+1). To obtain Si+ 1 we branch the family corresponding to hl(X) in (3.7) by means of C I according to (3.8) and (3.9). Next we branch each of these subfamilies by means of C 2 and so on up to branching by means of Ck® We repeat that action for each family in (3.7) • The essence of the H-R algorithm consists in excluding some fami-

497

lies from above mentioned process. For each family we compute bounds which play a role of an argument in various fathoming criteria. In linear tasks we may use linear progrsmmlng to compute the bounds as exactly as possible, but in nonlinear problem the computation of such bounds creates certain difficulties.Below, we present some fathoming criteria, which efficiency was investigated. 3.3.1. The Logical Criterion LC If, for example, in accordance (3.81

Lc: h1(GI) = h(X)^C = O,

(3.10)

then we say that G I has been excluded by logical criterion.In such case

a

h2(G 2) = h(X) = h(G). (3.11) Therefore we may make an assumption,that each family is not empty. 3.9.2. The Global Numerical Criteria GNC These criteria are computed recursively for given constraint gi(X}~O, i g(~,m), without using information of Si_1.Let I(G) (u(G)} be a lower (upper) bound of gi(X) over the family G.We consider (3.4) and at the begining we have l(Qnl = a0 +

~ , ai< 0

a i,

(3.121

u(Q n) : a O +

~ • ai> 0

a i.

(3.13)

After branching Qn by means of conjunction C 1 we obtain, ing to (3.8) and (3,9}, two families G I and G 2 for which if if

a I~ O aI ( 0

(3.14)

if if

aI > O al
(3.15}

~l(~l , if I(G2) = [l(Q n) - % , if

aI > o %<0

(3.16}

aI > 0 a1 < 0

(3.17}

l(G1) =

l l(Qn) + a I, l(Qnl ,

accord-

~

U(Qn ) , u(G1) = [U(Qn) + al,

u(G2) =

U(Qn ) - a I, Lu(Q n} ,

if if

For each conjunction Cj, j ~ (2-~), and a family G we obtain similar formulae, i.e. in (3.14}-(3.171 we replace Qn by G and a I by aj. In results from (3.14)-(3.17} that in the branching process a lower bound of gi(X} >I 0 doesn't decrease and an upper bound of it doesn't increase. The GNC consists in checking of two conditions for each family G:

498

GNCI :

l(G) ~ O,

(3.18)

GNC2: u(G)< O. (3.19) In GNCI is satisfying, then all points of G are solutions of gi(X) >/O, therefore we may exclude G from branching process. In GNC2 is satisfying, then G doesn't contain any solution of gi(X) >i 0 and we may exclude it from branching process. The families which are neither satisfying GNC1 nor GNC2 are branched in above described way. 3.3.3. The Local Numerical Criteria LNC These criteria answer the question if branching of given family G of S i by means of the conjunction of gi+l(X)>/0 is necessary or not. Let G be a family of S i and let (3.6) be its reduced conjunction. We introduce two sets of index % = { j : cjK = o } , (3.2o)

J1

{J CjK

= ' = K). Let L(G} (U(G)) be a local lower (upper) bound of gi+l(X) a family G then we have

L(G)

u(G)

= z(G)

+

g

,

aj

-

/

, aj,

J e J1

J ~ JO

aj > 0

aj< 0

= u(G) ÷ _ _ ~ aj aj~ J eJ1 J ~Jo aj< O aj~ 0

(3.21)

0 over

(5.22)

(5.23)

where l(G)~ u(G} we compute according to (3.12) and (3.13).Now we construct criteria very similar to (3.18) and (3.19) by replacing l(G) (u(Gll by L(GI (U(G) I. We will mark them by LNGI and LNC2 respectively • If LNCI is satisfying, then G is obviously the set of solutions of gi+l(X) ~ O. If LNC2 is satisfying, then G cannot contain any solution of gi+l(X) >i O. 3.3.4. The Incumbent Criterion IC In practice we often know a relatively good estimation for fak.Let F(G) be a lower bound of f(X) over G. If

it:

F(G) > fak'

(5.2~)

then G may be excluded from further considerations. ~. THE EFFICIF~CY OF THE H-R ALGORITH~I Let (3.7} be a characteristic function of a feasible solutions set of the i-th constraint, i.e. we're replacing now B i by Si, i C ( ~ .

499

Since all coefficients then

of

considering constraint are real

there exist infinitely

number,

many formulation of this constraintthat

H(S °) is sill a correspondin~ characteristic

function of it.

fore there exist infinitely many formulations of -Boolean programming problem.

a

There-

given

pseudo-

It is known [5, 19] that every pseudo-Boolean programming may be converted into an equivalent linear problem. This consists in replacing each nonlinear component appearing function or/and in constraints by new variable and

joining

problem

convertion in goal two

ad-

ditional linear constraints. So the efficiency of such linearization depends on the number of nonlinear components. According to [7] we may solve every integer linear problem with about 70 variables. For instance, in example 8 from [9 ] solved without computer,we have r = = 18 nonlinear components m = 3, n = 7. So an equivalent problem has n + r = 25 variables and m + 2r = 39 constraints. For comparison, the biggest problem solved in C17] by means of IBM 7040 (m = 7, n = 30, r = 10) has 40 variables and 27 constraints. We may reformulate the question that we set in Introduction in the following way: what factor does the efficiency of the H-R algorithm depend on. Basing on the up-to-date results we may establish two such factors: an order of solving constraints and a decomposition of system of constraints. 4.1. The Order of Solving Constraints We introduce the

concept of

the optimal order

of solving

con-

straints by means of so-called "strenght" of given constraint gi(X) Let (3.#) be such a constraint. We define the strength of it as Wi

=

-a 0 -

a

laj

, i6(1,-,-~.

(4.1)

aj4 0 If Wi( O, then gi(X) > O, is redundant, i.e. gi(X) ~_. 0 for each X E Qn" If W i > I, then gi(X) ~ O is inconsistent and S = ~.So we may consider only such constraints for which

o < wi < I (4.21 because for cases W i = 0 and W i = I the unique solutions are known. We can see that (%.I) is an estimation of the number of families in S~. Since in the H-R alforithm the enumeration is executed on the level of families,therefore we try to choose as the first,such a constraint for which the number of families is as small as possible,i.e. we choose gi(X) >~ O for which W i is the nearest either I or O.So we can give the following priority rule in pseudo-Boolean programming:we

500 put constraints in order according to nonincreasing priority and ~b(tVi - 0.5) for o.5~< w i ~ I P(gi ~ [(0.5 - W i) for O < Wi~< 0.5

P(gi ) ,

(4.3)

We introduced coefficient b > l in (4.3) because if W i ~ l then each family has more fixed variables. In practical computation we assume b = 2. In section 6 we present the computational results which illustrate the usefullness of putting the constraints in optimal order. 4.2. The Decomposition of Systems of Constraints The idea of decomposition of a system of constraints consists in separating it into subsystems in such a way that constraints in each subsystem contain as much as possible common variables. Such separation make possible fathoming the great number of families by LC. Let qij be a number of components of gi(X)>i 0 in which the variable xj appears~ i E (1-~) , j ~ (1-~) , gij >10. The correlation coefficient of the i-th and k-th constraints we compute as n Tik = ~ ~

qijqkj,

i,k E (1--~}

(4.4}

j=1 We separate

(2.5)

according to

the

following proceduz, e.

Let

gs(X)>~ O be the strongest constraint. We compute Tsi for all i 6 (1-~), i ~ s, and create a subsystem from these for which Tsi takes the largest value. Next we find strongest constraint among remaining ones and follow the above described way. Let Z be a subsystem containing m z constraints.The priority of the subsystem we define as P(Z) = W(mz/m) ,

(4.5)

where W is a mean value of strength of constraints belonging to Z. We begin computations with subsystem for which P(Z) is the largest. 5. THE I~[PLEME~{TATION OF THE H-R ALGORITH~ The H-R algorithm was written i ALGOL 120~ language and implemented for the second generation, Polish computer ODRA 1204 (storage capacity 16 k, access time 6 3~sec, addition time 16 )~sec) . For comparison IBM 7090 - 32 k, 2.2 9sec, 4.4 psec. The H-R program is an adaptive one with two parts ~ZSTER and EXECL~OR. The first part obtains the information about the problem just solved and upon this information it controls some parameters of the second one. For example, ~h~STER checks, if the solved problem is linear or not. For linear problem it checks, if it is the covering prob-

501

lem or not. MASTER also computes Wi, i ~ (7---,m).andputs the constraints in optimal order,and separates (2.5) into subsystems according to section 4.2. On the ground of this information we may automatically change: the order of applying the fathoming criteria and the procedure of multiplication of characteristic functions. If for example 0.3 ~ W i ~ 0.6, i ~ (~,m), then we consider (3.2) as the first constraint. In the H-R program the multiplication of characteristic functions is done in the following way: we determine all families for the first (with the greatest priority) constraint. Then we take the last family with the greatest number of fixed variables and solve g2(X) ~ 0 over it and so on up to solving gm(X) ~ 0 and computing fak or improving its value. Next the H-R program checks a list of created families and if this list has run out, then it takes the next family from S I. Such organization of the multiplication requires small storage capacity to execute it. 6. COMPARISON OF TPiE CO~PUTATIONAL RESULTS The efficiency of an algorithm should be measured by means of number (or its estimation) of properly defined operations needed solve suitable chosen typical tasks.But it is very difficult to

to de-

fine such operations in discrete programming, therefore the computer time needed to solve the subjectively chosen examples is considered as the measure of the efficiency of an algorithm. This time obviously depends on these examples, the computer and the language.This creates such a situation as described in [1]. The partial solution would be a statistic approach to comparison of results for example in such a way as in [13, 15]. In order to reduce subjectivism of our results we took all numerical examples from the references [6, 11, 16, 17]. Table I presents the comparison of results for linear tasks and Table 2 for nonlinear ones. We can see that the H-R algorithm is the best especially for nonlinear problems. We should observe that the examples [17] are rather specially constructed for Taha's algorithm and in spite of all the H-R algorithm, is faster that the remaining one.The example from the second part of Table 2 cannot be solved either by Taha's or by Watter's algorithm. Table 3 shows how optimal order of constraints influences the efficiency of computation. We can observe that the majority of these examples has wrong order of constraints and putting the constraints in

502 TABLE 1. Computational results for linear problems Problem Number

Computing time in seconds for method

Size Source m

n I

Lawler and Bell

Mimstep

1

13_21 (b)

1

71-227

4

16

H-R algorithm

t ....

1

10

10

2

15

15

3

20

20

10-60

481-3.577

22

4

3O

3o

650_1325 (c)

6658

136

5

40

4O I

6

15

15

7 (d)

35

15

8(e)

50

32

SZomi~ski

9 (e)

87

48

tl

t

2_6(a)

354 Haldi

921

3o

Computer:

27

711-1195

~7

796

29

24960

IBM 7090

2383

ODRA 1204

(a) Results for different examples. (b) Results for different versions of the algorithm. (c) Computation non completed. (d) This problem was solved by Freeman [4] by means of the IBM 7044 computer in 150 seconds. (e) This problem has a quadratic goal function.

optimal order is especially efficient for large problems times)~

~tu p

to

10

503

TABLE 2. Computational results for nonlinear problems Computing time in seconds for method

Problem Numbet

Size m

1 2 3

3 3 3 7 7 7, 7 7 6 6 6 6 6 7 7 10 10 10

8 1 11 12 13 14 15 16 17 18

n

5 10 20 5 lO 20 30 5 10 10 10 10 10 23 23 27 27 27

Source

Taha

H-R alLawler and Bell gorithm

Watters

r

5 5 5 lO lO 10 10 10 ~ 15 15 15 15 15 75 " r-i 105 ~ O 106 ,~p~ 156 ~m . 316 ~

0.3 0.2 0.2 1.9 0.6 3.3 1.0 5-1 2.1 8.3 2.4 3.4 11.2

41 41 1 <1 1 2 3 <1 2 1 1 1 1 68 89 187 298 407

0.6 8,3 919,4 0.7 3,8 > 3000,0

0.5 0.6 3.6 4.7 2.8 21,1 19.0 3.0 2.9 6.3 6,5 8.7 14.1

5.6

13 24 438 634 716

Computer:

0DRA 1204

IBM 7040 For problems 14-18 IBM 7090 TABLE 3. Efficiency of solving the equivalent problem

Problem Numbet 1 2 3 4

O

5

©

O

8 9 10

Computing time in seconds Size

the best order

references order

the worst order

m

n

15 20 3O 15 2o 30 50

15 20 3O 15 20 3o 32

4 22 136 4

6

8

10 57 29

23 411 88

39 632 235

7 10 7

23' 27 20

89 298 2

273 3000 3

347

5 38 3O0O

22 235 50O0

w

3

504

fi. CONCLUSIONS I. The H-R algorithm can be generalized in such a way as it was done in ES~ with pseudo-Boolean programming. 2. On the ground o£ recent results we may guarantee to solve any problem with about 50 variables and for the problem of special structure with considerably more variables. We may observe that the efficiency of the H-R algorithm considerably increases in case of the implementation it for highly parallel fourth generation computers such as i!~IAC IV. 3. Many authors [2, 3, 7] have pointed out the fact that the equivalent problem corresponding to a given one may considerably increase the efficiency of the known algorithms. But this question hasn't been investigated so far. On the ground of our results we may say that the equivalent problem need not be a linear one. REFERENCES I. 2. 3. ~. 5. 6. 7. 8. 9. 10. 11. 12. 13. I@. 15. 16. 17. 18. 19.

Anonymous~ ~athem. Pro rammin 2, 260-262 (q972). Bradley H.G., ~anag. Sci. 17, 35#-366 (1971~. Bradley H.G., Discrete Math. 1, 29-45 (1971). Freeman R.J., ~perat. Re s. 1#, 935T941 (1966) Fortet R., Cahiers d~--~ntr9 'd e RAcherche Operati_onnelle,l,5-36 (1959). Haldi J., Working pap. No. #3.Graduate School of Business, Stanford Univ. 1964. Geoffrion A.M., Marsten R.E., ~anag. Sci. 18, 465-491 (1972). Hammer P.L., Rudeanu S., Boolean methods in operation research and related areas. Nerlin 1968. Hammer P.L., Rudeanu S., 0oerat. Res. 17, 233-261 (fl969). Hu T.C., Integer programming and networks flows. New York 1969. Lawler E., Bell M., Operat. Res. 15, 1098-1112 (1967). ~ao J.C., Walingford A . B ~ ~lanag. Sci. 16, 51-60 (1968). Nugent Ch.E°~ Vollmann T.E., Ruml J., ODerat. Res. 16, 150-173 (fl968). Randolph P.H., Swinson G.E., Walker M.E. In: Applications of mathematical programming techniques. London 1970. Ritzman L.P., ~anag. Sci. 18, 240-248 (1972}. Slomi~ski L°, ~-roc. of ~he Conf. on Control in Large Scale Systems of the Resources Distribution and Development,Jablonna 1972 (in Polish)° Taha H., ~ . 18, B 328-343 (1972) Walukiewicz S., Arch. Autom. i Telemech. ~5, ~55-483 (1970). Watters L.J., Op~ra~. Res. 15, 117flJ117& (1967).

NUMERICAL ALGORITHMS FOR GLOBAL EXTREMUM SEARCH

J. Evtushenko

In many problems of eperation research in which systems containing uncertain parameters are designed, optimum solutions often are based on minimax strategies. Utilization of such an approach for solving practical problems is restricted by lack of ~umerical methods. To this time, as far as I know, there do not exist any general numerical methods for obtaining minimax strategies in multistage games. Some first results in this direction obtained (the numerical solution of a class of one-step processes). In present paper a numerical method for determinatiom of minimax estimation is presented. This method is based on numerical method of finding global extremum of a function [I, 2 I. I@ Suppose that a function whlth constant

C

~ C ~ ) s a t i s f i e s Lipshitz conditions

:

We shall consider the problem

)2where

Xis

a compact set,

We shall call the vector

a solution of problem (I) if

(1)

506 ,~]C where ~

-

(2)

- is the accuracy of a solution.

If for any sequence ~ ,

is found then for all

~

"''~ ~C the value

C~ belonged to the spheres

the condition (2) holds. If the spheres (3) entierly cover the domain ~ t h e n

the problem (I) is solved and m a g n i t u d e ~

proximate maximum o f / ~ o n

S.

In~1]

is an ap-

the simplest algorithm of

such covering is presented. In the case when ~ )

is differemti-

able function the local methods of maximum searching are used as auxiliary methods which essentially accelerate computation. Similar approach was used for finding global extremum of a function which gradient satisfies Lipshitz conditions. The programm was made which used A L G 0 L - 60 for computing. 2. We shall now consider determination of minimax estimation for ~ / ~

J=

steps process:

~

*,,~-~ ~

~-,~

where OC. - has dimension ~(~')and

a o<~ W o O

di.n~ional

... , ~ m ~=

~

]Cc~)

(4)

( '~d.' Og.~ , ...g &~Z) - is

vector. ~'he e~tremum with respect to ~n~ r"

vector X ~ is searched on a compact domain Z ~

C~i,

where <

is

i

an

£ -dimensional Evclldian

space. The function ~ g ~ J s a t i s f i e s a

Lipshitz condition on the domain

, ~ - ~'f X ~'~ X ' ' ' ~ X Z ,

The method of seeking a global extremum is used step by step for solving problem (~). This numerical method permits us to solve problem (@) to an arbitrary fixed accuracy. A number of modifications are developed which use local methods for acceleration of convergence.

507

In the Computer Center of the Academy of Sciences special programs have been developed for solving (~). As examples of problems solved by this method we can mention the two following problems

Numerical computations

show

that these methods work effeciently.

3. Consider the game problem

where W

'#

- probability measures defined on sets X

tively. F u n c t i o n ~ i s

assumed to be continuous on X x y .

easy to show that for any probability measure ~

Let

~(r)

denotes

Function ~ ]

,Y

, #

respecIt is

:

the solution of p r o m b e s :

is convex. Instead 0£ problem (5) we can solve

proolem (6). #or any m e ~ u r e ~

we shall find ~ )

finding global maximum. For minimization ~ )

using method for

we can use the local

numerical method, which we shall discribe now° 4. Using approximation, we can our problem put into the following mathematical form: Find a vector

~ = (~d;Z~#..) ~ )

which

minimizes convex function

z subject to If

a~ ~ Z =

{ ~." ~

Y{Z) is differentiable

the system:

(7) 0

; ,'C z , . ~ k ~ .

then for'solving

problem we consider

508

'~e can prove : eeI 3]), that the lim=t solution

g~

g(e,, O

the system (8) is the solution of problem (7) for any If

~g~is

nondifferentiable,

ff

but differentiable

of

.

in any di-

rection (as in our case), we can use following disoreteversion:

where the vector

~I

belongs to the set

~

I of support functi-

onals :

k sequence O6g such that

~=a:

~...~ + ~

If 6~ is sufficiently small then the limit of sequence a solution point for any

~e ~

~

~ ~is

.

References i

EBTymeHEO

D,P., ~ypHa~ BNUZC~TenBHO~ MaT~N&TIEM Z ~&TOM&~-

qecEol @~3zK~ II, 1390-1408 2

EBTymeHmO D.F:, ~yp~az BNqMC~MTe~BHO~ MaTeM&TMEM M ~a~e~aT~~ecEoR ~ S ~ E z

3

(1971), MOCEBa.

12, 89-104 (1972), MOOEBa.

EB~ymes~o D.F., i~a~aH B.F. ~ypHa~ B ~ q ~ C ~ e ~ H O l

~a~e~a~

Ma~eMaT~qec~ol @isz~z 13, 588-598 (1973), MOC~BS.

GRADIENT TECHNIQUES FOR COMPUTATION OF STATIONARY POINTS E . K. B l u m Department

Let u ¢H

J

u, t h e n

U. of S o u t h e r n C a l i f o r n i a California 90007

b e a r e a l f u n c t i o n a l d e f i n e d o n a s u b s e t of a H i l b e r t s p a c e

is a stationary

particular,

of M a t h e m a t i c s , Los Angeles,

if u

p o i n t of

is a minimum

the derivative

is zero.

J

if s o m e d e r i v a t i v e point of Thus,

J

of

J

is z e r o at

and the derivative

extremum

but of course the converse need not be true.

of

J

u.

H. In

exists at

points are stationary

points,

W e shall present s o m e gradient

m e t h o d s for determining stationary points of a rather general type.

W e consider

sets of non-isolated stationary points and non-convex functionals and give conditions for convergence of the gradient methods.

W e then give applications

to the optimal control p r o b l e m of M a y e r and to the generalized eigenvalue problem

A x = k Bx, w h e r e

A

and B

are arbitrary bounded linear operators

f r o m one Hilbert space to another. The gradient methods presented here are based on the intuitive idea that convergence can be expected w h e n e v e r there is a neighborhood of the stationary

point u in which the cosine of the angle ~ (x) between the gradient the vector x-u

is bounded a w a y f r o m zero.

?J(x) and

F o r a p r o b l e m with equality

constraints, the angles ~i(x) between the gradients of the constraints and x-u

also enter into consideration.

equality constraints. Let

R

W e shall first consider the p r o b l e m with

denotes the inner product.

be the real line and let D

gi:

D -~ I~, 1 < i < p, be real functionals.

and

C =

be a subset of H.

Let

The sets C(gi) = ~x e D : gi(x) = 0}

P ~ C(gi) are called ,'equality constraints". i= 1

Let

$: D - b R

another functional, called the "objective (or cost) functional"° Freshet (or strong) gradients of these functionals at x by

be

W e denote the

?$(x) and

vgi(x ).

(See [2], [5], [6] or [8] for pertinent definitions. ) T h e n the differential of J at x

510

with increment

h is

d J ( x ; h) = J ' ( x ) h = < VJ(x), h> .

spanned by the vectors

{Vgi(x) }

orthogonal complement, H=

G x @ T x and

T x,

The subspace,

G

x

is called the "gradient subspace" at x.

is called the "tangent

subspace".

VJ(x) = VJG(x ) + VJT(X ) . W e call

Its

Thus,

V J T ( X ) the "tangential

component " of VJ(x). Definition i.

u ¢ D

is called a stationary_point of (J, gi)

if VJT(U) = 0 and

u~C. If u is a local m i n i m u m under appropriate conditions

point of J on the equality constraint

u is a stationary point of { J, gi} . This

consequence of the Lagrange Multiplier Rule. rule.

h

Let

neighborhood of u.

N(h) of 0 ¢ R p+I in t at t = 0. such that

VJ(y)

h ¢ Tu

Let

and

and

Proof:

where

N

is s o m e

tp) is in s o m e neighborhood

Vgi(y ) be continuous in t ~N(h) and

P Z i

+

u

VJ(y) continuous

k i vgi(u) = 0.

Furthermore,

if (Vgi(u)}

not all zero is a linearly

% 1..... k P are unique and not all zero.

Thus,

u is a stationary point of (J, gi} .

See

Notation

r i LZI.

~ = x/

Definition Z.

!I x!! is the n o r m a l i z e d

and

(iii) .

VJT(X) / 0 the gradient

if

the following four

vgi(x ) are uniformly continuous in x

vgi(x ) / 0; (ii) the G r a m

nonsingular (normality);

is called quasiregular

N = N(u) such that for x ¢ N

conditions are satisfied: (i) VJ(x) V J G ( X ) / 0,

vector.

A stationary point u of ( J, gi}

there exists a neighborhood

When

,q C,

If VJ(u) /0, thenthere exist real }~0, kl ..... k p

~0 VJ(u)

VJT(U ) = 0.

u

~Tgi(y) exist at all points y = u +to h +

and t = (t0,tI , ....

independent set, then k 0 = I and

zero.

There are m a n y versions of this

J(u) -<- J(x) for all x ¢ N

Let

P I~ ti Vgi(u) ' w h e r e

when

is a direct

W e state one in the following theorem. Theorem

and

C, then

For the angle

matrix

(< Vgi(x), ?gj(x) >) is

@(x) = a r c s i n

V @ (x) exists and

(tl 7JT(X) II/ I t V J(x)!t),

[I V @ (x)[l is bounded a w a y f r o m

VJT(X ) = O, the one-sided differential

d @ (x; h+)

exists for all

511

h cH

d e ( x ; h +) uniformly as Ilhll - . o ; (i~) F o r x ~ N

and d e ( x + h ; h +)

not a stationary point, let stationary point to x

and c~1• = a r c c o s

U x

on line s e g m e n t

< Vgi(x),

(0 <

c~i, ~ < w. )

>y

if V J T ( X ) /

= { stationary points [x,u] } . F o r

P E 1

0 and

u is the closest

u ¢ [Jx ' let A X = x-u

< v e (x),

gX > , ~ : a r c c o s

T h e n there is a constant

such that

u

ax>

7 > 0 such that

P E i

cosZC~ i + cosZ~

Z cos

~i

>Y

if VJT(X) : 0 .

A gradient p r o c e d u r e for determining quasi-regular stationary points is given by the following f o r m u l a s .

Xn+ 1

= x n + sn h(Xn)"

(1)

h(x)

: hG(x) + hT(X).

(Z)

hG(x)

= -

P

Z

I

gi (x)

- tan e (x)

l!vo(x)!1

hT(X)

vo(x)

if vJT{x) / o

:

(4) 0

d/Z

(3)

Vgi(x)

IIvgi(x)ll

< sn < d , w h e r e

W e call this the

VJT(X) : 0.

0
p.

(5)

"angular gradient procedure" or a " m i x e d strategy procedure,,.

A s an example,

consider

bounde d self-adjoint operator. g(x) = IIxIIZ -i;

if

i.e.

unit eigenveetor of A

C

J(x) =

where

A : H ~ H

~Ve i m p o s e one equality constraint, C, defined by

is the unit sphere.

if and only if

Iris easily proved thafi u

V J T ( U ) = 0.

=

!lxlt ! < A x ' x ' > !

ItVJT (x)

llax]l 2

is a

(See [Z].) In [2] it is also

s h o w n that

re(x)

is a linear

IIx!l z

512

if ~TJT(X ) / 0 and cos

9 (x) / 0.

The following t h e o r e m can be proved. (See

[z], [3], [4] ) Theorem

Z.

Let A

is an eigenvalue of A spectrum of A,

be a bounded self-adjoint operator on

of multiplicity

1 and

H.

If # /0

h is an isolated point of the

then any unit eigenvector belonging to X is a quasiregular

stationary point of {J~ g} , ( It appears that the angular gradient mefJqod can be used to c o m p u t e such intermediate eigenvalues and eigenvectors even w h e n

~. is not isolated°

In the

latter case, the stationary points are not isolated. ) F o r example, if Ax :

1 ~ K(t, s) x(s)ds is an integral operator with a s y m m e t r i c kernel ~0

continuous on the unit square, then A

I<

is a c o m p a c t self-adjoint operator on L Z.

Its spectrum is at m o s t d e n u m e r a b l e and a non-zero eigenvalue m u s t be isolated. B y a suitable choice of x0(s), the angular gradient procedure will converge to intermediate eigenfunctions. Th___eorem 3.

Let

T h e general convergence result is as follows.

u be a quasiregular stationary point of {J, gi} and N

a quasiregular neighborhood of u.

There exists

r > 0 and positive k < 1 such

that for any initial vector in the ball B(u, r) the angular gradient procedure

converges to a stationary point u* and IiXn u~:~II_
See [2].

A s a second example, consider the M a y e r optbnal control problem. are given the differential equations of control dx/dt u ¢ R q, the boundary conditions function J(a, x(a), b, x(b))°

:f(x,u), with x 6 iRm

~i (a,x(a), b,x(b)) = 0,

We and

l < i < p, and the cost

It is required to determine a control function u(t)

which produces a trajectory x(t) satisfying the boundary conditions and minimizes J.

We

shall restrict u(t) to be piecewise continuous

in s o m e open set of R q.

on

[a, b] and have values

To simplify the example, suppose ~#j and

x(a) is prescribed.

Then

x(b) depends only on u : u(t) and

J b e c o m e functionals of u.

Take

H to be the cartesian product of q copies of Lz[a,b] with inner product

513

1 q

< u,v>

:

Z ui(t)vi(t) dr. A s 0 i=l

VJ(u) = Jx(t) (Sf/Su) 0

g~i(u ) = }'ix(t) (Sf/SU)o,

and

matrix of partial derivatives the differential equations,

is well-known, the gradients are given by

(Sfi / 8 % )

(Sf/SU)o

is the

evaluated along a solution (x(t),u(t)) of

Jx(t) is the solution of the adjoint equations

dz/dt = - (Sf/ox)Tz with values

Jx(b) = (SJ/Sx(b))0,

solutions of the adjoint equations with gradient

where

and the

,~ix(t)are

~ix(b) = (8~i/Sx(b)) 0 . However,

~78 (x) would be too difficult to compute.

Therefore

the

the angular gradient

procedure is modified by taking

hT(X) :

- (i/lq vJo(x)IL [ < re(x),

va~(x) > r )

and approximating the differential < V @ (x), V JT(X ) >

[e(x+s

VJT(X),

by a finite difference

v Jr(X))- e(x)]/s. To obtain convergence of the modified angular gradient procedure w e

replace the condition (iv) in Definition 2 by the following: P 2 E cos ~i + cos d 0 cos ~ / I cos ~ 01 > 7, 1 and c~0 = arccos < VYT(X),~x > .

where

(ivT)

~ 0 = arccos

This m e t h o d has been applied successfully to

o p t i m u m rocket trajectory ( m i n i m u m

fuel) problems.

It has also been tested

successfully on the classical brachistochrone problem. Now, w e consider a related method for the unconstrained problem [10],[iZ]. To motivate it, w e consider the generalized eigenvalue p r o b l e m

A x = k Bx, w h e r e

A

to another

Let

and

B

are bounded linear operators on one Hilbert space

J(x) = O / Z )

N A x - ( / < B x ,

Bx > ) Bx !Iz, w h e n

H

H'.

Bx / O. r h e ~ it i s

not difficult to show [I0] that J(x) : 0 if and only if VJ(x) : 0.

Thus, the

eigenvectors are precisely the stationary points of J. Since there can be a subspace

E l of eigenvectors, the set of stationary points need not be isolated.

514

To compute

such stationary points w e can u s e the gradient m e t h o d ,

x n + ! = x n 4 h(Xn), w h e r e

h(x)

= -

Z ~(x)

VJ(x)

(6) .

II v:<xttt 2 A straightforward calculation yields that R(x) = / < B x , B x >

VJ(x) = (A-R(x)B) ~" (A-R(x)B)x, w h e r e

and ~denotes the adjoint operator.

b e e n tried successfully on the finite-dimensional case, w h e r e s q u a r e matrices.

T h e m e t h o d has A

and

It should be especially effective in case w h e r e

A

B

are

and

B

are b a n d or s p a r s e m a t r i c e s and w h e n only certain intermediate eigenvalues are being sought. T h e step in (6) is a special case of a m o r e

general m e t h o d w h i c h

.can be

applied to find certain kinds of non-isolated stationary points. Definition 5. if for

~> 0

A set E of stationary points

there exists a n e i g h b o r h o o d

N of E

is a C-stationary set for and a constant

J

e > 0 such that

for x ~ N, ? g(x) is continuous in x and there exists a unique nearest point x~': -" ~ E

and for

x ¢ N-E

the following conditions hold: (i) V J(x) ~0;

(ii)

Z cos

co(x) > c,

(iii)

f cos

where

c¢ = a r c c o s

<

~7$(x), A x >

~(X)- 2(J(X)- $(X~;'))/HVTJ(x)II!!Axll

I <

and e.

A x = x - x ~< ; 1"4 i s c a l l e d

a

C - stationa_aF_~-~ n e i hborhood. T h e following c o n v e r g e n c e t h e o r e m Theorem

3.

Let

E

G-stationaryneighborhood

is p r o v e d in [10].

be a C-stationary set for of E.

For

x ~N

-2 (J(x) - J(x~:~))2

J

and let N

be a

define

vJ(x)

if vJ(x) / 0,

iIvJIxl!! h(x> =

i I

[

and let X n + I = x n + h(Xn). positive constant fix -Eli < ii n I_

(7) 0

otherwise. T h e r e exists a n e i g h b o r h o o d

k < i s u c h that for a n y initial vector

k n ! I x o - E !i holds for

point in the closure of E.

n > 0

Furthermore,

IV[ of E

and a

x 0 ~ Iv[ the inequality

and the s e q u e n c e (Xn) c o n v e r g e s to a for arbitrary

8 > 0 the n e i g h b o r h o o d

515

M

can be chosen so that

I k z - (i-

inf x ~M-E

cos z =(x)) I < 6.

Of course, the step h(x) in (7) is only computable if w e k n o w (In the generalized eigenvalue problem, approximation to the value suffice.

However,

in which

J(x*) is known.

J(x*).

J(x;~) = 0. ) It is possible that a close

J(x;', ~) would be available in practice and this might

pending further study, application is limited to those problems

The step-sze in (7) is an approximation to the distance in the gradient direction f r o m

x to the point nearest to x;~-'. Thus, the method of theorem 3

could be called a '~gradient m e t h o d of closest approach. " in this respect, it differs from the steepest descent method and other gradient methods [Z], [5], [6], [7], [9].

Its application to eigenvalue problems in infinitedimensional

spaces (e. g. integral equations) would generally

involve discretization errors

of s o m e kind [i], [ii]. This requires further investigation.

References I.

K. Eo

Atkinson, Numerical

Integral Operators, Z.

TAMS

Solution of Eigenvalue P r o b l e m for C o m p a c t 1967.

E.I<. Blum, Numerical Analysis and Computation (Ch. 5,1Z), AddisonWesley, 1972.

3.

, A Convergent Gradient Procedure in pre-Hilbert Spaces, Pacific J. Math., 18 , 1 (1966).

4.

, Stationary points of functionals in pre-Hilbert spaces, J. C o m p .

5.

Syst. Sci Apr. 67.

J. W. Daniel, The Approximate Minimization of Functionals, PrenticeHall, 1971.

6.

A. A. Goldstein, Constructive Real Analysis, Harper 67 .

7.

E.S. Levitin and B. T. Polyak,

Constrained Minimization Methods,

Zh. vychisl. Mat° mat. Fig., 1966 (Comp. Math and Math. Phys).

516

8.

M . Z . Nashed, Differentiability and Related Properties of Nonlinear Operators - in Nonlinear Functional Analysis and Applications, ed. L. B. Rall ACo Press 1971o

9.

S. F. ivicCormick, A General Approach to One-step Iterative Methods with Application to Eigenvalue Problems,

J. C o m p .

and Syst. Sci.

Aug° 72° 10.

E. K. B l u m and G. Rodrigue, Solution of Eigenvalue Problems in Hilbert Spaces by a Gradient Method, U S C

11.

H. Wieiandt, Error bounds for Eigenvalues of Symmetric Integral Equations Proc. A M S

12.

G. Rodrigue, A x = X Bx

13.

Iviatho Dept. Prepring Apr. 72.

Syrup. Applied Math., C, 1956.

A Gradient Method for the

Matrix Eigenvalue P r o b l e m

Kent State U. Math. Dept. Dec. 72.

S. M c C o r m i c k

and G. Rodrigue,

A Class of Gradient Methods for Least

Squares Problems for Operators with Closed and Nonclosed Range, Claremont U. and U.S.C.

Report

PARAMETERtZA~ON AND GRAPHIC AID IN GRADIENT METHODS Jean-Pierre PELTIER Office National d'Etudes et de Rccherehes A6ro~patiales (ONERA) 92320 - Chatillon (France)

Abstract The first part reports an experiment in which a graphic interactive console was used to operate a gradient-type optimization program. Some indications are provided on the program sturcture and the requirements for the graphic software. Conclusions are drawn both upon advantages and difficulties related to such project. The second part deals with parameterization of optimal control problems (i.e. solution through non-linear programming~. A local measure of the loss of freedom pertaining to such technique is established. Minimization of this loss leads to the concept of optimal parameterizatian. A first result is given and concerns the metric in parameters space. PART I : Console and gradient 1. Introduction In the past, interactive graphic display consoles have been used, in the field of optimization, to select the desired model (i.e. state equations, constraints...) and initiate computations (i.e. provide initial values as in STEPNIE,W]SKI 1969). The experiment carried on at ONERA is original in that the interaction deals with the optimization procedure itself. A conclusion of previous computational experience had shown that, in general, a rapid solution of large, highly non linear optimal control problems requires sequential use of several numerical techniques although each of these is, on the paper, sufficient. This is why ONERA developed a fairly large optimization program, TOPIC (Trajectory OPtimizatio~ for Interception with Constraints), offering a range of options. Options can fall into seven categories : 1} controls (how will they be modelized) 2) constraints (penalization, Lagrange multipliers ...) 3) search direction-local analysis (metric choice, semi direct technique ...) 4) search direction-global analysis (takes past step into account e.g. variable metric} 5) step size (fixed, linear search techniques ...) 6) convergence index (Kelley 1962, Fave 1968'...) 7) technical options (e.g. integration procedures}. Although some options seem to be non-independant, it is best to program them as if they where, the present tread being to re-introduce in various algorithms possibilities which, at first, seemed not to be compatible. In order to facilitate comparisons of methods and speed up computations, a graphic display console has been interfaced with the program. The console has a treble action. (i)

monitor computations so that an operator can juge of their worthiness ;

(if)

aid an operator diagnostics and enable direct action ;

(iii)

facilitate edition of results.

2. Ggneral structure of the program The presence of a graphic package together with an already voluminous computing program results in a quite

518

large memory requirement (about 400 k octet.s) so the program structure h a s to be carefully studied and be compatible with overlay t e c h n i q u e s . TOPIC structure is showed in figure I. It is entirely written in FORTRAN IV language. E a c h of the block can be divided in a set of overlayed segments except for MAIN, MAIN CONSOLE and MEMORY b l o c k s . The MEMORY block s t a n d s for labelled commons which contains all variable values such as : current control, s t a t e and gradient h i s t o r i e s , algorithm memories and so forth. T h u s t h e s e values are preserved during overlay operati,ms. The MAIN program is reduced to a switch and c a l l s for initialization and input routines, to the console and optimization driver (s) in block 1.

I

MAIN MAIN ]abeiled

Fig.

] -

commons

Program structure

computin subroutines

:f.__J

I

I MAINconsoleII l

"lS 77" DRIVER

l

plotter 1

I console

~-

I

®

__I

L

f-

GENERALPURPOSE ONERA SOFTWARE _ _

plotter 1

--I

console

I

-

output

F

i!ol

_ _

/ 3. Computing programs A very general gradient program has to perform the following twelve t a s k s : 1 - build a design vector from a c t u a l parameters plus control functions (if required). 2 - L o a d initial state. 3 - Forward integration of :~tate (load s t a t e tables). 4 - Compute performance index from final state. 5 - Algorithm d e c i d e s wither to go on 6 or 11 CALL TO CONSOLE. 6

- Compute final adjolnt.

7 - B a c k w a r d integration to compute gradient (or direct finite differences). 8 - Projection of functional gradient into design vector space. 9 - Algorithm d e c i d e s to go on ]0 or 11 (or to exit) C A L L TO CONSOLE. 10 - Search direction modification (non local methods).

i/-

519

i t - Step computation. 12 - Control or design vector modification. Of these tasks 1, 2, 3, 7 and 8 can be deleted in the case of a function extremalization. It has been sound efficient to schedule calls to console after points 5 and 9. The list of tasks gives a good idea of program(s) in block 1, i.e. driver program(s). Such routines are ~ust big switches ~nd call specialized routines to perform each of the 12 tasks. When the console package returns to the driver, a flag enables to restart computations at any of the 12 points. Block 2 in figure I contains algorithm routines, integration routines, model routines plus least squares, linear system solution and so forth. Block 3 routines are divided into two classes : - specific : routines handling control functions and design vector; - general purpose : matrices, vectors and functions handling, such as performing + X

II- [I

operations on such

elements. Such structure leads to an unusual programming of algorithm routines which cannot call directly subroutines to perform tasks but instead return control to the driver with a proper set of flags to indicate what is needed.

4. The graphic package It is, in fact, composed of two distinct packages : (i)

The ONERA general purpose graphic package : SYCEC ;

(ii)

a special package, IS77, specific to the program. SCREEN

TITLE

NUMERIC

LOGICAL KEYS 0 0 o

0 o

0

0

CURVES

/ o:o,ooooo:o

0 o

o

I ~

ALPHANUMERIC KEYBOARD

Fig. 2 - Console system

LIGHT PEN

520

SYCEC divides the console screen into 4 areas, as shown on figure 2. We are going to review the use IS77 makes of the 3 more important areas, with emphasis on the monitor set up which is more delicate : A) The curve area : 3 curves are plotted against the number of function evaluation : • the current value of the performance index (thin curve), • the current value of the unaugmented (i.e. true), performance index (dotted curve), . for visual aid, the best value reached for the performance index (heavy curve). Comparing the first two histories, the operator sees how penalty terms are behaving. Comparing the first and the last curves tells him immediately if an improvement is currently achieved : the two curves merge, After each gradient computation sensitivity {unctions with respect to controls are plotted on the sereeu. Of course in intervention phase (i.e, when the operator has a manual control) all interesting curves such as controls, state, pseudo-state (constraints) ~.. should be available for examination and even for manual changes (through alphanumeric keyboard or directly with the light pen). For parametric optimization problems, histograms are plotted instead of functions. t3) The numeric area : it allows 20 numbers, real or integers, to be displayed w&hout mnemonics. It has been found necessary to reserve 8 lines for flag displays showing main options on the algorithms, printout volume, iteration count. Thus the operator knows what is actually performed by the computing program. Remaining 12 lines display the performance index, the unaugmented performance index, values of final state and pseudo state, final constraints and the current step size value for the linear search. To compute a distinct pseudo-state for each of the currentstate constraint has been found more informative than to lump them up all in a single variable (which is possible), in spite of the extra dimension requirement in the integration procedure. Here again, in intervention phase IS77 brings upon request in the numerical area any significant number of the problem for examination and modification. C) The message area performs an important task of giving to numerical data on the screen their full meaning by keeping the operator aware of where these data come from. So diverse standard messages let the operator know : where the console package was called from in Block 1 ;

-

- which IS77 subroutine is in action ; -

if an action is expected from him (e.g. depress a key from a given set) ;

- if no action is expected. This is used for lengthy input/output operations. The message is supposed to keep the operator cool. To conclude this description of [$77, subroutine ~Maln console" has just a logical function mad calls, either on automatic or manual mode, specialized subroutines ; Block 3 routines should be available to the graphic package for simple computations as required by the operator.

5. Difficulties encountered (i) TOPIC is a program specific in its formulation to one optimization problem, and 1S77 is specific both to the problem and the program. Develop a non specific graphic package (i.e. depending neither upon problem formulation nor upon the constitution of the memory block) would require a big programming effort. Values should be passed through arguments instead of commons. IS77 would have its own input file (it has one, but short) to define, through a set of flags priorities for the monitor option. And as a handbook could not indicate (as it is presently done) the signification of numbers, this would have to be done on the screen. A file of mnemonics would have to be loaded and each numerical value shoed appear with its name. For such a work, the console software would have to be rebuilt from scratch.

521

(ii) At it is, on the IBM 2250 console which is used, the possibility for manual intervention goes through a program stop demanding an operator action. Thus there is no possibility to have a (normal) monitoring phase interrupted unexpectedly by the operator, although this would be the best use for the system. Actually a simple 2 positions switch would be sufficient for this purpose, but such switch does not exist on the 2250. (iii) The economy of the system is delicate : the time tg necessary to update the screen, in monitor option, is constant (about 3-4 seconds) while the time necessary for computations varies widely with the problem. A good reference is the central unit time per function evaluation, t e. If t e is too small compared to tg, the use of a complex console system is unjustified to monitor (on pro° duetion) such unexpensive calculations. If t c is too long ( > 1 minut) the screen is static and the operator (a specialist) time is wasted although computations are expensive enough to be closely followed. For the problem deIt with through IS77, t e varies from 10 secs to 1 minut which has been found reasonabl.e.

6. Conclusion After a year and a half of use, the graphic system has proved it's efficiency for : -

acquiring rapidly an insight into a new problem,

- building a reasonable first guess for the algorithms. - getting fast results (a night

on

the console is equivalent to a week of normal procedure ! ).

Moreover it helped us to reach some interesting conclusions upon constraints handling, global algorithms, integration methods and parameterization of optimal control problems (such conclusions might have been reached without the graphic system but may be less rapidly). However developmeot of a really general graphic software is an expensive task which now awaits the conclusion of a present phase implementing a general purpose, versatile, multiple option optimization program.

PART II : Parameterization of optimal control problems

1. Introduction Optimization problems in functional space and, among these, optimal control problems aim to determine, in general, a vector valued function U*of, let us say, time over a given (finite) interval ct~ , which optimizes some performance index. Any problem of practical importance in the field has to be solved on a digital computer and this eventually transforms the function into a set of many parameters. Already this approximation raised some theoretical examination (KELLEY DENHAM 1968). Then, variable metric algorithms, which have represented a major improvement in the field of parametric optimization, with their convergence properties, tempted several groups into using these techniques to solve optimal control problems (BRUSCH 1970 - JOHNSON'KAMM 1971, SPEYER & al 1971, the author 1971). Of course this technique impose a modeling of controls (figure 3) involving a limited number of parameters (an order of magnitude less than the average number of discrete points in functional programs). Of course such a technique delivers only a suboptimal solution V*n which depends upon the way controls are parameterized and the number n of parameters. A first question which arizes is to know wether or not one can find n(~-) large enough sothat ~ I U " - V ~ )~,(~.

i.e.

if

V~---~U* when

n ~ .

522

U(t)! !

"~'~ -~n - I

l

!

! 1

I

i

i

J

I

I

~o

t2

LIn

J4

t3 t 4

tn_ 2

tn_ 1

tf

t

Fig. 3 - Parameterization example This question of theoretical importance has been solved by CULLUM 3_972. The contribution o~ this paper is to try to select, for a given number of parameters, the best parameterizatlon for the control.

2. Steepest slope in a hifbert Let U ~

and J(.) an application of bll into ~

. Theslope/3

of

J(.)

in direction dU and at point ~ is defined

as

/ .~--,.o) I1~ au tl "[ whe~e,~S.~ and li.~ is ~he selected ~or~ in ~ . WheneveraFrechet derivative ~(~)['7

exists at point~,

II d0 ~1 Let U~ be Hilbert and ~ ( . ) be an application of ~d onto symmetric and coercive. Let ~ ,

~*, it's a dual space. ¢~(.) is linear, continuous,

o> be the duality product (i,e. the application of L~x (,~* into ~

defined by"

U*(U)) and define a dot product in ~ as

), .>

I.,.

The norm in ~ will be the associated norm II. ~ . A classical step in the gradient technique conmsts into defining a steepest slope direction dU* which maximizes/,~ . This direction is found to be (4) and ~ e corresponding slope is d-

(5)

<

Io), o ,,

523 Let us define (6)

dlj) ~

? ( ~.

/ (dO)

3. The V(.) application Similarly, let ~ be hilbert with a metric associated to application c~(.) and K (.) an application in ~ , ~.,.~ the duality product and V(.) an application of ~1 into I~ . At point ~ ~ , let .~ V (-~)[.] be the Frechet derivada tire of V. The steepest slope direction of K(.) in /~ at point ~ , da*, is transformed through b~Va (a--3[.1 ,into direction

(7)

aU

Lv c~)[

=

-' ~

at point'U = V(a). As example, suppose UI to be ~ 2 ( ~ ) , ~ to be-£~: V(a, t) is some control model depending upon a n-vectar ~a~ Dot products are respectively defined asJ~ ~ .[M(t)].,> dt and ~ . [ N ] . , > where N and M(t) a r e symmetric, continuous, positive definite matrices. Let ~ (a) [.]= ~'gk(g)I ,'-~ and

~(~)['-] where~. I " > , not w be taken for ~ ,

-

£~",@,')1 . > e ~

.>, is the uanal euclidian product in ~n.

Notation HU is used by similarity with optimal control problems but, so far, is just a notation. In this example,

~v c~,~q[N-'] ~ where appear terms

/~,

'

~V'T-

4. Pararrreterization efficiency In the case where K(.) is defined as the composed application

the chain derivation formula gives

~here ~V~" ~ .")[.] i, the adjoint of

~~V

(.~[.].

>

~

524

In the special ease where V(.) is binnique (i.e. ~ and ~

can be identified), it ~s possible to select in ~O~ the

~ P ( . ) metric so that

be identical to dUb(see (4)). Then .~ : L In the general case (and in the ease of parameterizatian) ~ live V(.) application b V* is not so. Therefore r~7 V

and ~ cannot be identified and even for an injec-

-1 ( -b. ~V* (.))) cannot be identified to o ~ ' 1 ( . ) and dU/~ dU*

However, it is sufficient that ~}V(.) be injeetive in order to confer to application ba

all the required properties, starting with coercivity, to be associated with a metric in / ~ . Moreover W ( . ) precisely defines the image-metric in the case of identifiable fl~-~l spaces. It is non-local (i.e. constant ~" ~'~. ¢~ ) if V(.) is linear. In any case we shall define the parameterization efficiency at point'ff as the value of ~ (dU),(where dU is defined in (13)fi which is less thane one. ~(dU) can be considered as a local measure of the loss of freedom induced by parameterization of the functional problem. In the above defined example of suboptimization in ~ n of a problem defined in ~ 2 ( ~ ) ,

and the value of ~ is given by

5, Optimal parameLerization Having built a index of quality for a parameterlzation, it is normal to try to maximize it. Maximization of ~ can be carried on on several steps :

i)

being given, select rift.) in

so that

is maximum.

2) For a given dP(.), v(.) can usually be imbedded into a family of transformations depending upon a given set of parameters ~b~ which are not subject to optimization i.e. do not belong to the t a'~ set. It is possible to solve an accessory optimization problem, maximizing 7 ( V ( ~'~'>, to

I b'~, .), dU ( | ' E ~ ,

] b ~ , .)) with respect

lb~.

3) Evantaally it is possible to compare optimum values of ~ for various kinds of V(.) transforms and select a preferred parameterization technique. The problem is ~hat as ~ is locally defined such accessory optimization problems will lead to local solutions and local conclusions. However, if it can be done rapidly, the parameterization can be modified whenever the algorithm does not use past step informations. A first conclusion can be reached on optimal parameterization and it is a non-local conclusion : in order to maximize ~ with respect to 0~7.), one should select o~.) Z 6 ~ ' (.) as defined in (14) which, incidentally, simplifies (15) as, in our example [Yl ,has to be equal to [ N'] .of (9).

525

6. Conclusion A quality measure of the parameterization of a functional optimization problem has been introduced with the efficiency coefficient ~

. This notion will enable comparisons of parameterization techniques, hopefully. A first

conclusion has been reached on optimal parameterization which leads to the selection of the proper metric (when it exists) in the reduced space of controls. Proofs have not been given and will await applications to practical examples for a more complete development.

Acknowledgment May Mr C. Aumasson, [rom ONERA, find here the author's gratitude for his many discussions and criticisms over parameterization. The author is also indepted to Miss G. Mortier, from ONERA Computer Center, [or developing special features in SYCEC and [or her help in the gr~hic project.

References PART I FAVE,

J. -

Crit~re de convergence par approximation de l'optimum pour la m6thode du gradient, in computing methods

in optimization problems, springer verlag 1969, p. 101-113 (proceedings of the 2nd International Conference on Computational methods and optimization problems, S~:n Remo, sept. 1968). KELLEY, H.J. - Methods of gradients,in Optimization Techniques, G. Leitmann ed., Ac. Press, 1962, p. 248-251. STEPNIEWSKI, W.Z. , KALMBACH, C.F. Jr. - Multivariable search and its application to aircraft design optimization. The Boeing Company, Vertol division, 1969. PART II BRUSCH,

R.G.

, SCHAPPELLE,

R.H.

-

Solution of highly constrained optimal control problems using non-linear

programming. AIAA paper 70-964 and AIAA Journal, vol. 11 n° 2, p. 135-136. CULLUM Jane

-

Finite dimensional approximations of state constrained continuous optimal control problems.

SIAM J. Control, vol. 10 n° 4, Nov. 1972, p. 649-670. JOHNSON, LL.

, KAMM,

J.L.

-

Near optimal shuttle trajectories using accelerated gradient methods AAS/AIAA

paper 328. Astrodynamics specialists conference, August 17o19 1971, Fort Landerdale Florida. KELLEY,

H.J.

, DENHAM,

W.F.

-

Modeling and adjoint for continuous systems 2nd International Conference on

Computing Methods in Optimization Problems, San Remo, Italy, 1968 and JOTA vol. 3, n o 3, p. 174-183. PIGOTT, B.A.M. - The solution of optimal control troblems by function minimization methods. RAE Technical Report 71149, July 1971. SPEYER, tl.L. , KELLEY, H.J. , LEVINE, N. , BENHAM, W.F. - Accelerated gradient projection technique with application to rocket trajectory optimization. Automatica, vol. 7, p. 37-43, 1971.

LES ALGORITHMES DE COORDINATION DANS LA METHODE MIXTE D'OPTIMISATION A DEIFX NIVEAUX

G. GRATELOUP Professeur

A. TITLI Attach~ de Recherches au C.N.R.S.

T. LEFEVRE Ing~nieur de Recherche

! - INTRODUCTION Pour contourner les difficult~s th~oriques et de calcul qui se pr~sentent lors de la r~solution des probl~mes d'optimisation de grande dimension, un moyen efficace est certainement l'introduction de m~thodes d'optimisation ~ deux niveaux, utilis~es notamment en commande hi~rarchis~e dans les structures de commande g deux niveaux. Pour cette t~che, que l'on supposera ~tre l'optimisation statique d'un ensemble de sous-processus interconnect~s, on peut alors utiliser la notion de "division horizontale du travail" faisant apparaltre des sous-problgmes r~solus de fagon locale, les actions locales ~tant coordonn~es par le niveau sup~rieur de commande, de fa~on ~ obtenir l'optimum global. Or, chacun des probl~mes inf~rieurs ~tant d~fini par 2 fonctions (modUle du sous-processus et erit~re associ~), il y a trois modes possibles de d~compositioncoordination : - par l'interm~diaire de la fonction crit~re - par l'interm~diaire du module - par action sur les 2 fonctions. Ce troisi~me mode qui est gtudi~ ici, utilise comme grandeur de coordination les variables d'interconnexion entre sous-syst~mes et les param~tres de Lagrange associ~s~ Dans cette communication, les sous-problgmes loeaux d'optimisation sont d~finis, et diff~rentes possibilit~s de coordination sont propos~es pour le niveau sup~rieur de cormnande. Sont examines notamment les coordonnateurs type gradient, Newton, ~ iteration directe et gradient-iteration directe. La r~solution d'nn probl~me de r~partition optimale des ~nergies, dans un systgme de production hydro~lectrique, permet de mieux comparer certains de ces algorithmes coordonnateurs.

Laboratoire d'Automatique et d'Analyse des Systgmes du C.N.R.S. B.P. 4036 31055 TOULOUSE CEDEX - FRANCE

527

II - DECOMPOSITION DANS LA METHODE MIXTE II. I P osStion du probl~me (Probl~me lls~parable") Supposons que le processus eomplexe ~ optimiser soit divis~ en N sous-syst~mes comme eelui repr~sent~ sur la figure I. entr~es globales

Ui

entr~es de couplage

Xi

Y.

sorties d~finitives

Sous-syst~me 0

>Z.l

n° i

sorties de couplage

~M. co~m~andes i Figure I Ui' Xi' Mi' Zi' Yi sont des vecteurs g mu , mx , mM , my. composantes respectivement. 1 i i i Pour un vecteur d'entr~e globale U donn~, le sous-syst~me est eompl~tement dgerit en r~gime statique par les ~quations vectorielles : Z i = Ti(M i, X i)

(I)

Yi = Si(Mi' Xi)

(2)

L'interconnexion entre les sous-syst~mes est repr~sent~e par : X i = Hi(ZI...Zi...Z N)

additive".

(3)

La fonction objectif du syst~me est suppos~e donnge sous forme "s~parable, N F = ~--- fi(Xi, Mi) (4) i=l Le probl~me global est de meximiser (4) en prgsence des contraintes

figali~ (I) et (3). ANCe probl~me d'o~timisation, on peut associer le Lagrangien : f i (xi, i) ÷

÷

"zi"

La solution optimale doit n~cessairement satisfaire les conditions de stationnarit~ de ce Lagrangien, ~ savoir :

~fi ~Ti )T LX. = 0 = + ( ~ i + £i l ~Xi 9Xi

= o = ~f_i + (~7 i )T >i LMi

~Mi

(6)

(7)

~ Mi N

LZ.~ = 0 = - ~i -~(.= ~zi~HJ)r ~j

L~i

=

O

=

T.

i

-

Z.

i

L~i = 0 = X i - Hi(Z I...Zi...ZN) II.2

(8)

(9)

(]0)

D~eompgsition, formulation des sousTproblgmes (G.Grateloup,

A.Titli) Cependant, pour simplifier la r~solution de probl~mes de grande

528

dimension, on r~partit le traitement de ces ~quations entre deux niveaux de commande, non arbitrairement, mais de fagon $ obtenir une forme "s6parable" des 6quations au niveau infgrieur° Dans la m~thode propos~e, une telle r6partition est obtenue en choisissant et Z eomme variables de coordination, c'est-~-dire eomme variables transmises pour utilisation au ler niveau de eommandeet modifi~es au niveau sup~rieur jusqu'g l'obtention de la solution globale reeherch~e. Pour e et Z donn~s, le Lagrangien prend alors la forme "s~parable" suivante N

N

L = Z ei = i=I

fi(Xi,Mi)+ ~iTxi - eirHi(Zl ..

.Zi...ZN)+~iT(Ti-Zi )

(ll)

L'examen de L i permet de formuler chaque sous-probl~me en termes de eommande optimale ; ainsi, !e sous-probl~me n ° i s'~crit : max

. fi(Xi,Mi) + e i X i - eiTHi(Zl , Z i, ZN)

sous Ti(Xi, Mi) - Z i = 0

(12)

pour ~ et Z donn~s.

Ii apparalt bien dans (12) qu'~ la fois crit~re et module sont utilisgs pour la coordination. Sur le plan analytique, la r~solution de chaque sous-probl~me correspond au traitement des ~quations (6), (7), (9), les ~quations restant ~ r~soudre au 2e niveau ~tant :

L ~ (x, z) = 0 Lz(Z'

F

,e)

= 0

(13)

L'~quation (9), qui doit ~tre compatible pour ~ et Z donn~s, impose reX. + m M . ~ i I

mz. i

(14)

Le transfert d'informations n~cessaire entre niveaux de commande est repr~sent~ fig.2

x ~ ( e ~

I coordonnateur ~ ~, ~x~(e~,z)

~ 2 ~ ~ f .

.

.

.

C2u -probl mel I_ n°i I

I

~~~~I ~) sons-probl me L t

2e niveau de eommande

1deeo an e ler niveau

I

F_~ure 2 : Transfert des informations dans la m~thode mixte 11.3 Dgcomposition des probl~mes non s~parables (A. Titli, T. Lef~vre, M. Richetin) Dans l'hypoth~se de "non s~parabilit~" du probl~me retenue ici, e'est-~-dire lorsque le couplage entre les sous-syst~mes intervient non seulement par les gquations d'interconnexions classiques entre les entr~es et les sorties, mais aussi, par l'interm~diaire des fonctions erit~res, on aboutit ~ la formulation suivan-

529

te du probl~me d'optimisation : N '~N max F = max ~ fj(X'j, M'j, W)

j=1 sous Z i = Ti(X" i, M" i, W)

(15)

X" i = HIi(Z)

i = I ~ N

WX, = H2i(Z) l en mettant en @vidence le vecteur W form@ avec les composantes de X i, M i qui assurent ce couplage suppl@mentaire. Apr~s regroupement,

si N'~< N, de certaines @quations de module

eouplage, il est possible d'@crire max

~

(15) sous la forme ci-dessous

et de

:

fj(X'j, M'j, W)

j=1 sous Z W j = T'j(X'j, M'j, W)

(16)

X'j = H'j(Z')

j = I ~ N'

Wx. = H".(Z') J J

La d@composition de ce probl~me global d'optimisation peut alors Stre obtenue : - soit en ins@rant W dans les variables de coordination Z et - soit en ajoutant au probl~me initial des contraintes de la forme •

= O, j

~

Ji

ensemble des

Mj

blames

problgme

Dans ce cas, un terme de la forme ~

.E

~f[~!

indices

en i n t e r a c t i o n

des sous-proavee

le

sous-

i.

-[~i] ]

est ajout@ au Lagrangien global et les variables de coordination peuvent ~tre Z, ~ ,

III - COORDINATION DANS LA METHODE MIXTE III. ; Coordonnateur

type gradient

:

Par analogie avec la m@thode de Arrow-Hurwicz pour la recherche d'un point col, on peut utiliser l'algorithme coordonnateur

d~ =_ Le de

~,

~[~=

suivant :

~[~) _ K L e

Diff@rentes @tudes de convergence de ce coordonnateur ont @t@ faites (A. Titli). III.2

Coordonnateur

type Newton

II est possible @galement, pour assurer la coordination,

d'appliquer un

530

algorithme de Newton-Raphson

~ la r~solution de l'ensemble des 2 ~quations vectoriel-

les :

LZ = O

en ~crivant

dW ~FdLW] w~ - I : ~-~ = - I--7~I LW LUWJ

et

m

= 0

.~'i

LQw

On montre

L~

J

(A. Titli) que si cet algorithme est applicable,

il est asymptotiquement

stable, III°3 Coordonnateur

~ iteration directe

Dans le cas d'un eouplage lin~aire certaines conditions

sur les composantes

N

(X i = 7

sont satisfaites,

CijZj) , et si j=l

(~-_ mx. = ~. mz ), il est alors possible de calculer Z ~ partir de L ~ =

O et

i .l de ~Z =i0, mettant ainsi en oeuvre une m~thode ~ iteration directe. partlr Une ~tude g~n~rale de la convergence de ce mode de coordination a ~t~ effectu~e

(T. Lef~vre). III.4 Coordination mixte

: iteration directe-gradient

Darts ce genre de coordination,

:

qui ne ngcessite pas ~ X L = ~ z - ~ - -

certaines gquations du niveau sup~rieur permettent une d~termination directe de certaines variables de coordination par un algorithme de type gradient

(BI) , les autres variables

(B 2) ~tant d~termin~es

:

BI i+l = Fl(Ai) B2i+l = B2 l' + ~ i :indice d'itgration,

K >

(19) . K . F2(Ai)

(20)

O, ~ = ! I suivant la nature des variables B 2 (variables

physiques ou param~tres de Lagrange). Remarque

: Cette m~thode mixte peut ~tre g~n~ralis~e au cas des probl~mes d'optimisa-

tion statique avec eontrainte in~galit~ et d'optimisation dynamique

IV - EXEMPLE D~APPLICATION

(A. Titli).

: REPARTITION OPTIMALE DES ENERGIES DANS UN RESEAU DE

PRODUCTION HYDROELECTRIQUE

(T. Lef~vre)

IV.! Formulation duo.problgme Nous traitons ici un probl~me similaire g celui abord~ par MANTERA, Soit le r~seau s~rie-parall~le de deux rivi~res~

d~crit par la figure 3, et compos~

comportant chaeune deux centrales

sent leur ~nergie g u n at 3 non modulables

; les quatre centrales fournis-

systgme de 5 charges dont 2 modulables

("conforming loads")

("non conforming loads").

Ces centrales sont couplges au syst~me de charges par un r~seau ~lectrique devant fournir une puissance PD" Chaque centrale est d~finie par un module

(Fig. 4) :

531

qn

L

= gnL(Pn L)

i L=

{], 2~

n

~l, 2}

Les pertes de production PTGL sont donn~es par : PTGL = PTGIL + PTG2L

2 avec : PTGL i = ~ PTGL i n=! Les pertes en ligne sont de la forme : 4+3 4+3 PL = 7 - - ~ (pi)T BiJ pj i=! j=! Le probl~me global est :

rain

[PL + PTGLI + PTGL2 ~

pL ~ L = ~I, 2~ n n tl, 2~ SOUS : PL + PD - PT = 0 I

!

!

q22 - q!2 - Q!1 - Q121 ~ 0 t Contraintes sur les dgbits.

q2

- q|

Qll 2

QI22 ~ 0

1

0 ~< P l x( P11 max

o .< P2 .< p21 O ~

Contraintes sur les puissances.

P12 ~ P! 2 max

0 ~< P2 2 ~ P2 2 max e t a pour lagrangien :

z

z

Pour arriver ~ une d~composition de ce probl~me en deux sous-probl~mes relatifs au~ deux rivi~res, il est n~cessaire de d~composer PL' ce qui peut se faire par introduction de deux pseudo-variables P' 12 et P'22 et de deux contraintes ~galit~ suppl~mentaires : P! 2 - P'l 2 = 0 P22 - P'22 = O ( ~, A ~ seront les paramgtres de Lagrange associgs ~ ces contraintes). PL prend alors la forme s~parable : PL = PLI + PL2" Si pour une m~thode mixte, nous choisissons : P'I 2, P'22, ~ , ~ et ~ t cormne variables de coordination, le lagrangien (21) prend alors une forme sgparable L = L I + L2, avec :

532

~..

2.

eorrespondant g deux problgmes d'optimisation au niveau inf~rieur. IV.2

ModUle lin~aire

La r~solution num~rique de ee problgme de r~partition optimale de l ~ n e r g i e dans un r~seau hydro~lectrique a ~t~ faite sur un ordinateur II30-IBM travai!lant en simple precision. ~iE~_i~K~

:

au niveau infgrieur, deux m~thodes ont ~tg mises en oeuvre pour

r~soudre les probl~mes d~optimisation relatifs aux deux sous-problgmes : une m~thode lagrangienne utilisant un algorithme quadratique (Newton-

-

Ralphson)

; - une m~thode de pgna!isation utilisant l'algorithme de Davidon.

N~xeau_~2~!d~nna~e_u~ : deux algorithmes ont gt~ utilisgs pour traiter les variables de coordination : - un algorithme de type gradient - un algorithme mixte (gradient-iteration directe). R~s~ta~

: I~ A l ~ r i t h m e

du gradient

L'~tude de la convergence de cet algorithme montre que la valeur optimale de la constante d'itgration K est : K ~ = Au bout d~une centaine d'it~rations,

0.00065

l'erreur totale

est de l'ordre de : ~ _.~ 0.85 et d~croit tr~s lentement (figure 5) ~ cause des , 2 ~ 2 pseudo variables P I , P 2 et des param~tres de Lagrange associ~s ~ , ~ , qui varient extrSmement peu A chaque iteration. Pour rem~dier ~ cette faible rapiditg de convergence, on a utilis~ un algorithme mixte : gradient sur te sur les variables dont l ~ v o l u t i o n est tr~s lente : A ~

~

et iteration direc-

A

Ce~te m~thode s'av~re tr~s efficace, comme on peut le constater sur les r~sultats pr~sent~s ci-dessous° 2 = Al$orithme mixte Cet algorithme converge en un nombre minimum d'it~rations pour K ~=

0.0004.

L'erreur totale est toujours d~finie par :

533

La figure 6 montre l'@volution au cours de la convergence de l'erreur (pour PD = I0 MW). La lin@arit@ suppos~e du module des centrales simplifie beaucoup la r@solution des optimisations locales. Mais un module plus proche de la rEalit@ (modUle exponentiel ou de classe CI) peut aussi ~tre utilis@. IV.3

ModUle exponentiel

Au niveau inf@rieur, les deux m~thodes d~crites pr~e@demment sont utilis~es. Cependant, la m@thode de p~nalisation est plus performante. Au niveau sup@rieur, seul l'algorithme mixte est retenu et les r@sultats obtenus sont, ici encore,

tr~s performants (cf. figure 7 qui donne l'~volution de

l'erreur). L'utilisation de ce modgle non lin~aire,

de type exponentiel, est plus

r~aliste et n'introduit aucune difficult~ suppl@mentaire de mise en oeuvre. Elle conduit m~me g u n

temps de convergence plus faible. IV.4

ModUle de classe C1

Au niveau inf@rieur, seule la m~thode de p@nalisation utilisant l'algorithme de Davidon a @t@ retenue, car les m@thodes proc@dant ~ un calcul direct du Hessien, sont oscillantes sur cet exemple. L'algorithme mixte donne, ici encore, de bons r@sultats (cf. figure 8).

V - CONCLUSION Dans cette communication, nous avons pr@sent~ une m~thode mixte de d@composition-coordination des probl~mes d'optimisation de grande dimension et d@fini les t~ches de chaque niveau de commande. Nous avons montr@ que le coordonnateur type gradient, toujours applicable, prEsente des conditions de stabilit@ et que le coordonnateur type Newton est toujours convergent, s'il est applicable. Les conditions d'uti~isation d'une coordination ~ iteration directe ont @t@ d~gag~es. Cette coordination apparalt int@ressante pour le traitement des probl~mes non separables. La resolution d'un probl~me de r@partition optimale des Energies dans un systgme de production hydro~lectrique (probl~me hautement non s@parable et d@lieat rEsoudre), nous a permis de mieux comparer, sur le plan des applications, eertains de ces diff~rents coordonnateurs. En particulier, l'efficacit~ de l'algorithme de coordination ~ iteration directe ou mixte (itEration directe + gradient) en Evidence.

a @rE mise

534

!

Charge

CentrQle ? Centrale 6 \ \

~ ~ ~ ~- ,'- ~ i J ~ C~ e n t r a l e , .~\,, tt Charge / 1 ~-~

\~/ / /

"9, "%

,,-q

Centraie 4

/

//

/

\

\\ \

5

Centrale 2 i,] -~ iI \

\

Centrale 3 ,,

7

/

20 miles ~

Cent~le 1

Figure 3

3200.

t ICentrale

2

2dO0. 2400.

2o00 1600. ,<.,..,

Figure 4 1200. 800.

4 0 0 . ~ 00 20 40 60 80

i

100 puissance produite ( M W ) des c e n t r a l e s - Cerecteristiques

535

~T 6

4

K = 6,5]0 -4 PD = IOMW

\.

2

÷

I

i "+--J+ - - + - + - +-+--+--+--+--+--+--+--+--+--+--+--"

0

I 0

10

N

20

30

40

50

60

70

80

90

I00

Figur ~ 5 : ModUle lin~aire, Algorithme du gradient. Evolution de l'erreur,

4i' "I!t

1,40 12

+ K=3.10 -3 * K=4. ]0 -3

x K=7.10 -3

90'flit 70 56

\ ....

42

,,, X

28 14 0 0

X*..

]" .x. x ÷-i ~

~

.

.

.

.

" - " X --=-he

20

- - ~

N V

~

V

3O

v

v

v

v

~

%

v

v

v

40

Fisure 6:ModUle lin~aire. Algorithme mixte. Evolution de l'erreur

~

= 1omO 50

(PD

536

~ &T 24 21

781 15 +: K = 7.10

12

-3

×: K = ]3.10

-3

e: K = 20.10 -3

3

N 0

5

~T ~Figure

I0

7:ModUle e x p o n e n t i e l . A l g o r i t h m e

15 mixte.Evolution

20 de l'erreur

(PD=]O MW)

70L .

,

8 7

x: K = 5.10 -3 -2 +: K = 1(

6

•: K = ]

.10 -2

5 4 3 2

N 0

5

10

Figure 8 : ModUle de c!asse C I. Algorithme

15

2O

mixte. Evolution de l'erreur

(PD = 10 MW)

537

BIBLIOGRAPHIE

ARROW K.J., HURWICZ L., UZAWA H. : Studies in linear and non linear programming. Stanford University Press. 1964.

GRATELOUP G., TITLI A. : Combined decomposition and coordination method in large dimension optimization problems. A paraltre dans International Journal of Systems Science.

LEFEVRE

T.

:

Etude et mise en oeuvre des algorithmes de coordination dans les

structures de commande hi~rarchis~e. Thgse de Docteur-lng~nieur. Universit~ Paul Sabatier, Toulouse, d~cembre 1972.

MANTERA

I.G.M.

:

Optimum hydroelectric-power generation scheduling by analog

computer. Proc. I.E.E. Vol. ]18, n ° I, January 197~.

TITLI

A.

:

Contribution ~ l'~tude des structures de commande hi~rarchis~es en rue

de l'optimisation des processus complexes.

Th~se de Doctorat gs-Sciences Physiques.

Universit~ Paul Sabatier, Toulouse, juin 1972.

TITLI A., LEFEVRE T., RICHETIN M. : Multilevel optimization methods for non-separable problems and application. A paraTtre dans International Journal of Systems Science.

APPLICA~ONSOF DECOMPOSITIONAND MULTI-LEVEL TECHNIQUE,~ TO THE OPTIMIZATION

OF DISTRIBUTED PARAMETER SYSTEMS

Ph. CAMBON

L. LE LETTY

CERT/DERA ~ Complexe A~rospat~al TOULOUSE FRANCE

ABSTRACT The resolution of optimal control problems for systems described by partial differential equations leads, a~ter complete Cor semi) discretizatlon, to largescale optimization problems on models described by difference (or dlfferemtlal) equations that are often not easy to solve directly, The hierarchical multi-level approach seems to be well suited to a large class of synthesis problems for these complex systems, Three applications will be presented here : - minimum energy problem for the heat equation with distributed i s t h e c l a s s i c a l t e s t example) - minimization

o4 a s t e e l

index for

the parabolic

equation with

input

[this

one-sided

heating - reheating ~urnace with a non linear boundary condition

[radiation)

INTRODUCTION Solving an optimal control problem Cor an identification problem) for systems described by partial differential equations leads, after complete dlscretlzatlon, to a large-scale optimization problem which is often difficult to solve directly in a global way. The decomposition and multi-level hierarchical techniques with coordination seem well suited to the solutlon of a large class of these complex synthesis problems, Applicat±ons o~ these techniques wil! be made here to optimal control problems with the aid of two-level hierarchical structures,

539

I - GENERAL PRINCIPLES OF MULTI-LEVEL TECHNIQUES Let us briefly present here the basic concept and the general principles of the hierarchical techniques. Associated first with the names o{ M.D. MESAROVIC as for the control aspect by hierarchical structures and L.S. LASOON and also some other authors as for the matehmatical programming aspect for large-scale optimization problems, these techniques are now actively studied for compgex dynamic systems. In distributed parameter systems, some applications have also been studied by D.A. WISMER and Y,Y. HAIMES. The baslc idea consists for a too complex "system-objectlve function problem" to be solved directly in a global approach to define a number of subproblems by subsystems and subcriterla sufficiently simple to be efficiently treated by classlcal methods end algorlthms and to coordinate the interconnected set by hlgher levels [one or more higher levels), The control structure [or the optimization structure] consists then of unlts at several levels giving a pyramidal hierarchical structure [f~gure I)

A I

\

/

Kth Level

/

'

\

,[ I c

\

/

\

/

\

/ 2 th

\

/

Level

/[

\

/

/

lStLevel////

\

%

C~__

\ \ \ \ \

Z L

it hio bj syb~ySt~umnorion OF&

I I

I GLOBAL SYSTEM AND OBJECTIFE FUNCTION FIGURE

1

540

This structure is called '~several levels-several objective functions" ~ several objective functlons for the control units at a given level have different objectives which can be moreover in conflict due to the fact that the subproblems are separately and independently solved at a lower level while in fect interconnected, The aim of the higher levels called higher level controllers is then to coordinate the set of operations of lower level in order to achieve the optimal solution of the whole problem at the top o~ the structure. The classical structure is a two-level structure with a single unit at the secono level. It is o£ course the simplest one but it is sufficient for most o{ the problems encountered.

II

- TWO-LEVEL HIERARCHICAL

STRUCTURE,COORDINATION

Let us consider a system

S

METHODS

which is decomposed into subsystems S i, i

= 1 to

L. The system

[1]

--IZ" = ~ i

S i is represented by its m ~ e l. . .equation ...

[Xi'Mii---

;

~i

~ R pi

,

M,i c Rmi

~

:

--iZ' e R qi

where --iZ'are the outputs, ~i the local inputs and X. the coupling inputs from the --i other subsystems, Z. --i

_

coupling inputs

S. z

~t

2~Outputs

~Iocal

inputs

The whole system ms reconstructed by taking into account between the subsystems.

{2~

-~x =a!. i

cij ~j

'

i,j

the coupling equations

= ! to L

REMARKS 1) We assume hare that the coupling equations between the subsystems are linear, This is not necessary ; non Linear equations ~i = Ci(Z) ere equally possible but in some coordination methods [as the non admissible method ) a separable form is needed :

x. = Z

--m

cj

j~i

{z.

--J

2] The comdition j#i [no internal coppling] ly the case.

Objective

is not necessary but will be usual-

function

We assume that we are given an objective function in a separate form which is decomposable on the subsystems : L

c~i ,

~i,211

which can be written

[by [1]]

J =

2

i=I

Ji

541

L

[3]

J =

~ I=I

3i

[Xi-- ' --IM']

The global problem which is : "Minimize J under the constraints (1] and [2], i = I to L" leads to the determination of the saddle point of the Lagrangian : L = L [~i' ~i' ~i' ~i' ~i]i=l,L

C4]

L i=l ~ [Ji[~i'Mi] + ~i (~i - ii [~i' ~I ] + Pi (X i_ -j i Cij ~j]]

We assume that Ji and T i are continus and have continuous derivatives with respect to the variables.

Then, the e q u a t i o n s are :

ixi

° ~x_/ ~_xi) ~Ji

i

Ei

+

_oi

= o

is)

=

f

c61

=

0

[7]

o

[8]

0

(9]

T

~i : ~i -L~TIL-~I) f i i=1,L pj j~i

!o i

X. -i

-

ji

f Cij~j j#i

The application of the principles of two-level hierarchical techniques will consist here to split the treatment 0£ these statlonarity conditions into two levels in order that the first part correspond at the first level to a set of separated optimization problems end that the second level realize the coordination which is here the resolution of the remaining equations. The second level controller will be of iterative type using the first level information after solving the sub-problems which will be done either also by an ±teretive scheme or by e direct method dependinz on the problem. Three now well-Known methods will be used. I)

Admissible" method

[or ooordinatlon by the ~i, also called coordination by the model]. The ~i, i = I,L, are given to the first level by the second level controller.

~_~_!~£~_~,

we s o l v e :

542

giving

:

i = I~L '

~

2i(!)

L Ei

pi[z)

This first Min Ji with

pert of the squations

the subproblems

:

(Ei'~i)

:

X.

Z

~ --i

C,.

j#i

Z.

'

~i

given

ij -j

The local model "admissible", nation global

represents

and coupling

equations

are satisfied

justifying

the name

The optimal Z remain to be found (coordination by the model] ~ then the combio4 the soluEions af the sub-problems will achieve the optimal solution o4 the problem with : L = E L i and J = Z Ji' i i

At the second level, the equations L7~ = 0 are then solved ~or-exampie~-a-steepest descent metho~ --

by an iterative

scheme,

+

~i

Z - k L--Zi --i

or a Newtcn-Rslphson

'

method

or some other optimization ~Z i ~2ch

=

p~ - Z

is calculated

The applicable

Dimension condition 2)

level

T h e £i

from the values operating

(M i) ~ Dimension limits

admissible"

(coordination First

algorithm,

with

of the variables

until .__ .ilbill ! E

glven by the
, This coordination

level.

method

[~i!

its applicability, method

by the ~i' also called

coordination

:

are fixed

~i

:

:

which

"Non

:

CT

algorithm is when

i = 1,L

= ~ i =-tzl =

L..y~ = 0

gives

:

by the objective

function],

is

543

I

~i = ~i [~)

~i (£]

ki

Second level : Determination of new values of p_ by the algorithm +

Ei

Ei

÷

~[

)

!oi

i

,

=

l,t

where

~i:x'

-~ --z

cij!j

J#i

In this method, when the optimal solution Is not yet obtained, we have : e~d' i 0 : the coupling equations are not satisfied which justify the name "non ~iss~ble". The global lagrangien can be written :

i

j

i

and i s of a s e p a r a t e form f o r t h e v a r i a b l e s r e s p o n d i n g optimization sub-problems are :

I

Min [Ji

+

T Pi

Xi _ Z T --i J~i

considered at the first

level,

The c o r -

Cij -pj]

with Z. --3.

= _Ti

[Xi,~i? - -

- -

This method does not imply any dimensionelity condition on the variables in the general case where the model equation is non linear in ~ and M i [second order terms are however neededj excluding non linearities of the form IX~jl or IMiji). In the case where linearity occurs in ~i and ~i, the objective function fi needs to include non linear terms in ~i and M_/ up to the second order at least. In practice, it is necessary to examine the compatibility of the equations or to reformulate the coordination method. 3) Mixed method (or coordination by the ~ and the ~]. We have :

At_the_~i£st_level

L~i

~

[~ and ~ fixed )

giving

~

h

=

~i

(Z,p)

544

f !.

:

~i-

=

£i

kz&zl +

,

i

: I,L

kp &pi

In the case where Z Dimension i method can be used to solve : ~o i

= 2

for the i

L_Zi

:

for

[X i) = Z Dimension -i

[Z.} a non iterative direct --i

and

2

the £

from the values of the variables given by the first level.

III

- APPLICATIONS. RELAXATION SCHEMES

We will give applications of the different coordination methods to optimal control problems arising from systems described by partial differential equations. In order to gain in memory requirements on the computer and to gain also in convergence speed, we have been led to ovoid the application of the multi-level approach in its usual conception by using relaxation schemes for the resolution of the first level equations. We use then for these equations the "new" values of the coordination variables ~i and P_i for the next first level sub-problem (Si+ I, Ji+1 ),

iV - FIRST PROBLEM.'OPTIMAL DISTRIBUTED INPUT FOR THE HEAT EQUATION WITH MINIMUM

ENERGY This i s the c l a s s i c a l

t e s t problem whose s o l u t i o n

i s w e l l known and could be

obtained by easier and faster ways (either using the adjoint equation or approximating the problem on a truncated basis of the eigenfunctions of the operator ~2/~x2). It is also the example given by O,A. WISMER in a multi-level approach using the maximum principle after semi-disoretlslzation and decomposition,

I ) The problemq and d i s c r e t i z a t i o n

C ~t = I

0 ~2xY +

u (x,t]

y[x,O) = y ~ ( x ] L Y[o,t)

= Y [l,t)

= 0

in

@

x E [0.1]

in

~

t c [O.T]

in

Z

Final c o n d i t i o n

~ y Ix,T]

=

Cost f u n c t i o n

;

llUI2LZ(Q]

After discretizo~ion

:

J(u) ti[i

=

= 1,N)

Yo[x)

;

xj [j = I,M], we hove :

545

Yij

- Yl-l,J At

= a

(Yl,j+l where

Ylj = Yoj

- 2 Yij

)+u..

+ Yi,J-q

ij

i = 2,N J = 2,M-1

q = D/(Ax) 2

initial condition

Yil = YiM = 0

:

boundary conditions

YNj = Ydj = final desired condition -

i

j

la

2) Decomposition and coordination. . First solution A first decomposition is to consider esch node (i,j) of the discretization grid as a subsystem Sij Xij ~

Sij

I

zij

with :

Then,

I

Zij = Yij Xij

Yi-l,j/At ÷ o CYi+j+ I + Yi,j_l )

Mij

uij

we h e v e :

Zij = [Xij + uij+/m ,

1

m = ~-~

÷ 2q

Xij = Zi_l,j/At + q (Zi,j+ I + Z±,j_d i = 2,N j = 2,M+I The objective functMon ±s : N M J[u) = E Z z Ax At -i:I j=1 u~j equivalent to : J(wl

=

~

r.

i

j

u2

iJ

>

546

The global lagrangion is written N MMI F

:

L = i=2 ~ j=2 ~

[Xij*

M-1

j=2

}

I u~J + ~iJ [Zij ~Nj FZ ~ Nj _ y dj.]

+~

z uij

uij)/~] + PiJ

[XLj-Zi-I/At-~[Zi'j+I+Zi'j-1)]~

on the boundaries

i=I,

j=l,

j=M

The initial and boundary conditions are taken into account in the coupling constraints by their particular values, respectively #or i = I and j = I and M.

The

stationority conditions for

1 < i < N

are :

f

bxij <

-~ij / ~ + 'Oi°j

Luij Lzij

2 u , -

LpLj

Zij

- [Xij

Lplj

×ij

- Zi-l,j

/~

iJ ~ij Pij - Pi+1,j/At

- ~ (Pi,j-1

+ uij]

Oi,j+t )

+

/

/ Z~t - c [Zi,j+ I + Zi,j_ I]

The admissible method can be used as

Dim. Mij = Dim, Zij~

We have then :

At the first level : After immediate direct resolution i

:

Xij = Zi_1, j /& t + ~ [Zi,j+ I + Zi,j_ I] uij

~ zij - xij = [zij - zi_1,j]/At

Pij

2 ~ uij

Plj

Ulj/~ = 2 uij

- o (zi~j+ I - 2 zij + zi,j_ I]

At the second level +

zij = zij - K LZi j where : 2 Lzij = A--T [ulj - u i+l,j ~~ - 2 o [ui,j*l - 2 u,. iJ + Ui, j-1 ) with ulj [Zij - zi_1,j]/&t - d Ill,j+ 1 - 2 zij + zi,j_ I) There remain to treat - for i=N the supplementary condition LXN j = ZNj - Ydj = 0 by ~Nj = XNj ÷ ~[ZNj-Ydj} -

the terms ~ u~j on the boundaries i=I, j=1, j=M. Here~ we obviously hove : uij = 0

We have then IN-I) [M-2] coupled sub systems. Several examples have been done, with different step sizes : N=31,61 ~ M=41,61, For N=31, M=41, we have 1170 sub-systems and 2370 variables (state variables end inputs} where 1170 variables ore coordination variables. In the relaxation scheme for the admissible method, e new Zij is calculated at each node [i,j] ¢ { i=2,N , j=2,M-1 } and the ~ already calculated are used in LZij.

547

Note here that we can solve for the Zij from Lzi j : 0 . The memory requirements

only and

:

Tableau Z±j

Vector

are

( i = 1 , N ~ j=1,M)

XNj ( j = 2 , M - I )

The problem has been solved on an IBM 360-44 computer ~or diqqerent step sizes (cf. tables). For N=31 and M=41, 125 iterations are needed for a 10 -3 relative precision on :

Sup

Lzij

i,J

Zij

and

Sup

XNJ

J

~Nj

which is too long.

3) ,S,,e,,condsolution. New d,,e¢omposition and mixed second-level controller A more interestimg decomposition consists to consider each column of the diseretisation on the time axis as a sub-system with : ~i

= {YiJ

'

j=2,M-I}

-M. -!

= {uij

,

j=2,M-I}

Then, we have now for the vectors Z i, ~i, X_.i the same model and coupling equations as i n the first solution :

X~ = Z i _ I / A t + ~ ( Z i , j + I + Z l , j _ I ) The mixed c o o r d i n a t i o n method g i v e s : At the f i r s t

=

(A --~Z')

level

~i

° Lui : ~ i ° 0 which gives the explicit solution

:~

P_i = ~ P_.i

-~i ~ gi/2 x i ~z_

i - _ui

A__t__t_h_e__s_e_c_o_n_d_l_B_v_e_l, we have : i=2,N-II

Lzij

-- m P i j

" PL+I,j/At

- ~ (Pi,J+1

+ Pi,j-1 ) = 0 j=2,M-I

I Lpi j ~- @

~ Zij

.Zi_l,j/At

q (Zi,j_ 1 e Zi,j+ 1) - uij = 0

LZNj = ~ PNJ/2 + XNj - ~ (PN,J+I + PN,j-I ) = 0

Then, a direct solution consisting of a sweep forwards on S i from S I to S n with ~i = ~o giving the Z_i {rom_~ol = ~ and of a sweep bachwards from S N to S 2 with ~Nj given by LZn j and giving the p.i from ~Z i = ~ gives the solution. The Iteretive scheme is then on XNj : ÷

XNj = XNj + ~ (ZNj - Ydj)

~

I z

.....E..

548

4) R e s u l t s S e v e r a l cases have been s o l v e d some o f w h i c h a r e shown in t h e t a b l e s , The initial condition yo[X) is a symetrlc triangle about x = 0.5 and the ~inal condition Yd(X] is a symetrle polynomial o{ order 4 on x. The advantages of the relaxation scheme over the usual conception can be briefly described by the
V - STEEL INDEX OPTIMIZATION I) ~ r o b l

~y

x ~ [0,L]

: o ~x-~-~

y{x,O] x =0

Z~Y E

x= L = u ( t )

Cost { u n c t i o n 3(u]

t £ [O,T]

= y~ = cat

~Y

where

WITH ONE-SIDED HEATING AND INPUT CONSTRAINT

em

~2y

~

Final conditions

/T

=]0

[the

y{O,T] y{L,T]

(u[t) - y](~ d t

R[t]

= TM = TS

~' ~=2 ~0 i~ u[t] -< y =I.I i$ u [ t ] > y

REt)

This is the heating of a slab with a decarbonation performance index. An oxydation criterion o~ the same nature could be used in the same way. The quantity Pd = X J--f R(t]

(u(t) - yl 2

dt

represents

the decarbonation

depth.

o The numerical

values are

y = 850°C

uM = 1250°C

,

: TM = 1150 ° C

2) D i s c r e t i z a t i o n . The previous

with

.

second decomposition

I ~i

{YiJ

, j:I,M}

~i

{YLj

, j=I,M}

Si =

.

Decomposition

.

.

1

will only be considered.

,

T = 6000 s

We have

:

--1

•

Xij

- T (Xi,j+ I + Xi,j_ I)

= Zi_i, j

j=2,M-I

Xll

- 2 T Xi2

= Zi_l, I

j=l

T = d x At

t =

TS = 1200 ° C

and c o o r d i n a t i o n

~ {Ri(XiM-y]2xAt i=2

+ £

+ £ iT [A ~i - Z . ]

1

+ ~i

~ = 1 + 2~

Penal

+ ~T| [Zi

(ui]xAt ~ f.

Xi]}

[XN1 - TM]2 x At + X M [XNM - TS]

549

where

Penel [u±) = ~ 0

|

i#

ui SuM

(ui-u M) if

u i > uM

We penalized the input constraint and the flnal condition XNI = TM The variables are : XIM , {XLj,Zij ; j=I,M-I}, Pij (j=I,M), Pij (j=2,M-I]. The mixed method has been applied for : i < N, J=I,M-I and i=N, j=2,M-I and the non-admissible method has been applied ~or : i = N, j=1 First level : i
LXiM = - PiM - T P i , M - 1 LXi,M-1

= -Pi,M-I

LPi J = Z i j Lxij gives

+ ~ 0i,M-1

- Xij

= -~iJ

~. z + ~Xi'----M - T Pi,M-2 j = I,M

= 0

+ ~ PLj

- T (Pi,j+1

+ PL,J-I

j = 3,M-2

]

Lxil = -~il + m Pil - T Pi2 ~ ~ ~ X i=N2 LXN I = m Pil - T Pi2 + ~ [XNI - TM) LXNM = -PNM + ~

gives

XNI and XNM

Second

level

+ XM

:

i < N LZLM = ~iM

LzlJ

= ~ij

Lpi j

= m Xij

- Pi+1,j

j

J

- T [Xi,j+

I + Xi,j_

= 1,n-i

I ] - zi~ j

J

j=I,M I

Lpi I = m Xil - 2 T Xi2 - Zi_I, I I i = N LZNj = ~Nj

LXM

=

XNM

;

J = l,M

TS

-

The solution {with g and g i fixed) is then as follows : I) ~M and PN1 fixed, ~ and Z are given by a direct

forward-backward p~ocedure.

It

÷

2] Then :

I pNI + = PN + ~ LpN I iM

= A M + ~ [XNM - TS)

RESULTS An example is given in Table II (temperature at the surface and at x=O, input u[t]). The results are not sensitive to the step sizes. A comparison is made with a theoritlcal solutlon obtained after Laplace transformation and transforming the problem into a minimization of e time integral criterion with an integral constraint.

550

Vl - NON LINEAR CASE. REHEATING FURNACE WITH,A RADIATION BOUNDARYCONDITION I) T h ~ r o b l e m . De.composition I ~Y - n @Zy ~t - ~x-~ ~

Final conditions

y(x,O)

Constraint

Tx y

= yo[x,}

:

y(O,T) : TM y[L,T) : TS

u(t) ~ uM = 1400°C

x=0 = 0 f

: ~ r[uct] + 273]"

I By I x=L

J{u} = /0 T

[YIL.t) + 273) 4]

R[t) x ~y[L,t) - y]2 dt

= I 0I

R{t)

I# iT

y(L't) ~ y Y y (L,t) >

Th~ same decomposition gives analogue model and couplin~ equotlons, The Le~ronzCan is : N L = Z fi + ~i [XiM - Xi,M-I " XA x (u i + 273) 4 - [Xin+273) "]

i=2

+ Z

{

i=2

N

Z j =2

Plj

Xij - T[XI,j+ I ÷ Xi,j_ I) - Zi_1,

M + pil

!~Xil-

{jo2 rij +

2T Xi2

Zi-l'1]

-xij]

L

[XNI - TMJ 2

[zi.,-uil +

IM

[XNM - TS]

2) Coordination by the. M.ixed Method The mixed method is used for i~ N, j = I , M-I and i = N, j = 2, M-I. The maln difference Is here we have, after first level resolution , to treat at the second level the terms : L

uI

= 2

E

Penal

(u.) x At - 4~X~x

LXlM = -~ PI~M-I L~ i

=

Zi M

÷ ~I

I)

-

~f

u I 7/ uM~

(u i + 273) 3

[I + 4XA x (ZiM + 273)9

ZI,M_ ! -

The following procedure

x

XAx [(u.f + 273) ~ - (ZIM + 273) ~]

is used :

~

= 0

and we have u i

ZiM

Is obtained from

L~i =

0

~i

is obta!ned fro

LXi M = 0

= uM

(non linear equation)

~i and 2i ore obtained by the some #orword-bockword example.

2) If u1 < uM ZiM is obtained from LXiM = O [non linear equation), ning : Penol {ul)X~At ~i = ~XAX[ui+273]~

.= 0

u i is obtained from L~ i = 0

~rom

procedure as in the previous

Pi,M-1 being known, o t t e r obtai-

Lui = 0

[non linear equation]

3) If u i thus calculated is ~ uM If u i < uM, go to 2. The stop criterion is on LpNIO

we go o~ain to I).

551

~he results (shown on the figures III) have been obtained on an IBM 360-44 computer. With N=61, M=11, and a satisfactory initialisation of PNI the execution time for the non linear case has been 7mn 24s where it was 3mn-'7s for the linear case. Non linearity increases of course the number of iterations and the computing time but it however difficult to say to which accuracy the non linear equations at the second level should be solved during the evolution of the iterative procedure for the whole problem. The results are also little sensitive to step sizes variations,

VII

-

CONCLUSION

The application of multi-level hierarchical techniques seems to be very interesting to solve optimal control problems for systems described by partial differential equations. The difficulties in optimization problems related to the large dimensionality of the systems obtained after discretization of partial differential equations makes this approach promising (cf. also Sr. WISMER). Moreover this approach allows to attach in the some way problems with constraints, performance indexes different from the most usual case of quadratic functionals, and also non linear problems which will be often the case in pratical situations. Here the applications have been made for optimization on a digital computer. It will be very interesting to approach these problems in view of a hybrid computation or in wiew of parallel computing. This will introduce different techniques of decomposition and coordination.

BIBLIOGRAPHY

"Optimization of large scale systemS" Congr@s IFAC Varso-

El]

R. KULIKOVSKI vie 1969

[2]

D.A. WISMER "And efficient computational procedure for the optimizat~n of a cZass of d ~ b u t e d p a r a m ~ systm~s" J o u r n a l of B a s i c E n g i n e e r i n g June 1969

[3-]

Ph.CAMBON - JPoCHRETIEN - L . L E LETTY - A. LE POURHIET "Commode~t optimisation des syst~m~ ~ param~tres r~poyutis" Convention D.R.M.E n ° 70 34

166, Lot n ° 4 - C.E.R.T. - D.E.R.A. [4]

Ph. CAMBON "Application du calcul ~ a r c h l s ~ ~ la commande optimale de syst~mes r~gis par des ~qu~tions aux d ~ r i v ~ p a ~ " Th@se de Doc-

teur Ing@nieur - Universit@ de TOULOUSE, Juillet 1972 [5]

[6]

A.TITLI "Op~m~aY~on TOULOUSE 1971)

de process~ complexes" (Cours donn@ ~ i ' INSA de

G.GOSSE - M.GAUVRIT - D.VOLPERT - J . F . LE MAITRE "IdentificaConvention D.G.R.S.T. n ° 70 7 2507 Soci@t@ HEURTEY - D.E.R.A.

R. KISSEL

-

tion et opt~misation des f o ~ "

"Ap~gus su~ la commode ~ c ~ e "

[7]

A. FOSSARD - M.CLIQUE s Mme N.IMBERT Revue RAIRO - Automatique n°3 + 1972

[8~]

Y.Y,HAIMES "D~composition and multi level approach in the modeling and management of water reso~ces syst~mes" NATO Advanced study Institute on Decomposition as a tool for solving large scale problems CAMBRIDGE July 1972

552

t - PLANCHES [ : PREMIER EXEMPLE COMMANDE I~PARTIEI~QUATION DE LA CHALEUR

O°C 3OO

APPLICATION DU CALCUL HIERARCHISE METHODE ADMISSIBLE (pr&cislon 10-3 ) Temperature 0 (x,t) t=O, ~-, ~ , T [.1 - Evolution de la temp*~rature - M~thode admissible

200

=

IO0

L~ 0

151

0,5

1x

APPLICATION DU CALCUL HIERARCHISE METHODE ADMISSIBLE C. . . . . de U.(x,t)t:O, T ~,T

[.1 b i s -

Evolution de l a commando

t=O 0,5

1

INFLUENCE DU PAS EN t (N) --N=3t M=41 - - - N=61 M=41

0

1/2

1

~

1

\ \

\

\

-

R6chauffage

\

\ \ ~ , ~

V,,2"

20°C ~

//

I 1000s

I.

~ -

THEORIQUE

~

J

/

i

'

~ TS

"'t-

,

I> Tx

~ T M

ta

ison "rayonnement" et

I 3000s

-

~

"simple conduction" par rayonnement

3 ooo s

Temperature "four"

..................................

- -

PAR APPROCHE

R6solutlon par approche theorlque

------ Rechauffage

1ooo s

II.1

"four ~ t heorique

850~C - - - _T_emperature ~surface ~

1200~C 1150°C

1400°C

85OO£

U~

Temperature

RESOLUTION

H - PLANCHES H : DEUXI~ME EXEMPLE CRIT~RE MI~TALLURGIQUE - SIMPLE CONDUCTION

850

115C

uN

®°C'

120(]

II

~ /

0

T~

t&M

o0cl N=61 M = 11

PAR RAYONNEMENT

= 850~ =1200~C TM =1150°C UM = 1400°C

N=

- -

61

N = 121

------

M = 11 M=11

DU PAS EN TEMPS M R RAYONNEMENT

3 O00s

Ill.1 - C h a u ~ e ~ r rayonnement M~thode mlxte

INFLUENCE CHAUFFAGE

1 O00s

/

CHAUFFAGE

H I - P L A N C H E S III : T R O I S I ~ M E E X E M P L E DE RI~CHAUFFAGE - RAYONNEMENT

FOUR

Tt

~J1

ATTEMPT

TO SOLVE

A METHOD

A COMBINATORIAL

IN T H E C O N T I N U U M

BY

OF EXTENSION-REDUCTION

E m i l i o S p e d[cat£o~ G i o r g i o CISE,

PROBLEM

Segrate,

Milano,

Tagliabue

Italy

ABSTRACT CombinatoriaI programming

optimizaHon probiems

problems

in

the large unconstrained

in

(n-l) 2 variables

optimization algorithms

tic assignement problem

is c o n s i d e r e d

programming

are formulated

and n 2 constraints.

as nonlinear

Methods for solving

problem generated are considered,

s i s on c o n j u g a t e - g r a d i e n t

the nonlinear

n variables

b a s e d on t h e h o m o g e n e o u s m o d e l . as an a p p l i c a t i o n

withempha The quadra-

example and results

from

approach are discussed.

GEN ERALI TIES L e t P be an f

:~ x rl p e r m u t a t i o n

is s u p p o s e d to d e s c r i b e

matrix

t r i x P i s sough~ w h i c h m i n i m i z e s in p r a c t i c e

l u e s of

n.

the following

way: we formally n x n

selected) starting

is n o t f e a s i b l e ,

(generally

model c o n s i s t i n g

of

interchanges

when

define

matrix

f

in

however,

f

small van! p e r m u -

has special structu-

matrix;

matrix X

o

matrix

P

in i t s d e -

a s e t of c o n s t r a i P t s ( r e d u c t i o n )

w e g i v e an a r b i t r a r y

and minimize

space

to o p e r a t e in the c o n t i n u u m in

R n2 ( e x t e n s i o n ) r e p l a c i n g

X; we introduce

ele-

c a n be d e t e r m i n e d by e n u m e -

even for relatively

the m i n i m u m of f

n

and a ma-

t e c h n i q u e s u s i n g o n l y a s u b s e t of t h e

s o l u t i o n must be a p e r m u t a t i o n

problem;

In p r i n c i p l e ,

It is p o s s i b l e ,

w h i c h m a k e X to b e a p e r m u t a t i o n

Every

p,l.

of P . T h e f u n c t i o n a l

of s u c h t e c h n i q u e s is that t h e y o p e r a t e in the d i s c r e t e

matrices.

by a g e n e r a l

sensibly

f

exist fop estimating

of t h e p e r m u t a t i o n

finition

.

such a procedure

res; the main feature

of a d i s c r e t e

The matrix P describes

Optimal and suboptimal

tation matrices

f=f(P) a functional

some property

m e n t s w h i c h m a y be i n t e r c h a n g e d .

ration;

and

f

( o r in s o m e f a s h i o n

s u b j e c t to the c o n s t r a i n t s , .

and a local minimum for the continuous

i f t h e I o t a ! m i n i m u m is a l s o a g l o b a l m i n i m u m t h e n it is an o p t i m a l s o l u t i o n not unique!)

to t h e o r i g i n a l

combinatoriai

problem.

If it is o n l y a l o c a l

non g l o b a l m i n i m u m ,

t h e n it m a y b e o r n o t b e a s u b o p t i m a l s o l u t i o n .

A s e t of c o n s t r a i n t s

defining a permutation

matrix

is t h e f o l l o w i n g :

555

(1)

x..(x. -1) = 0 JJ

i,j = 1,2,. ....

n

U

t%

(2)

~:×. =

1

j=1,2

{x..= H

1

i = I , 2, . . . . . .

I,=1

(3)

(14.)

~x..

n-~

n-1

= n

• Ij b~:L

Conditions

.......

IJ

(1-4) ape necessary

and sufficient

f o p X to be a p e r m u t a t i o n

matrix;

see

the Appendix for mope about them. System

(t-4) consists

of

u s e d to e l i m i n a t e 2 n - I

ng+gn-1

equations,

c o m p o n e n t s of X .

Therefore

p r o b l e m w h i c h is a f u n c t i o n of a p e r m u t a t i o n um b y a n o n l i n e a r In r e a l i s t i c

programming

In o r d e r

fop unconstrained

l i n e a r e q u a t i o n s m a y be

every combinatorial

can be e x p r e s s e d in the c o n t i n u 2 in ( n - l ) 2 v a r i a b l e s a n d n c o n s t r a i n t s .

and the associated

as q u a d r a t i c

nonlinear

assignement

programming

CONJUGATE-GRADIENT

in v e r y m a n y v a r i a b l e s

ALGORITHMS

FOR

n

m a y be

p r o b l e m is a v e r y

to deal w i t h it t h r o u g h t h e p e n a l t y f u n c t i o n a p p r o a c h ,

minimization

optimization

matrix

p r o b l e m s of p l a n t l a y o u t f o r m u l a t e d

m o p e than one h u n d r e d ~ large one.

problem

and the 2n-!

algorith'ns

h a v e to b e d e v e l o p e d .

UNCONSTRAINED

MINIMIZATION

Basic formulas Algorithms

for unconstrained

with continuous first

minimization

derivatives

g=g(z)

search methods) or also gradient values. first

kind,

of a f u n c t i o n F = F ( z )

of them ( s a y N e l d e r

O (m 2) s t o r a g e cient algorithms

requirement.

H e r e we d o

blems because both storage tely a rather

not c o n s i d e r

Pate of c o n v e r g e n c e

and M e a d 1 and P o w e l l f s

Newton's and Quasi-Newtonls

using gradient values,

(5)

Zk+1= Zk-aks k

a n d s o m e of t h e

2 m e t h o d s ) h a v e an methods are very effi-

and time pep iteration

(often (m+l)-superlinear)

requirement can be obtained using conjugate gradient algorithms 3 k i n d . T h e s e m e t h o d s a p e b a s e d on t h e i t e r a t i o n

Reeves

m e t h o d s of t h e

b u t t h e y h a v e t o be d i s c a r d e d

requirement

f a s t Pate of c o n v e r g e n c e

m variables

may use only function values (direct

because they often have only linear

most e f f i c i e n t

of

are

for large pro-

O (m2). F o r t u n a

and limited storage of t h e F l e t c h e r -

556 ',~ere fined

is a s c a l a r such that F{zk+ 1)~F(z k) and the nsearch v e c t o r ~p s k is de-

ak as s

=g o

o

(6) Sk = gk + bkSk-!

(k=!, 2 . . . . .

)

and b k is a s c a i a r . The f o l l o w i n g f i v e choices of b k a r e considered here: T gk gk (7)

bk -

(Fletcher-Reeves 3 )

T

gk-i

gk-1

(gk-gk 1)Tgk Tgk-1 gk-1

(8)

bk=

(9)

(gk-gk_l)Tgk bk . . . . . ~

( P o l a k - R i b i e r e 4- )

{Sorenson

5

)

(gk-gk_ | ) Sk_ 1

(10) b k -

gk gk T gk-1

Fk

(Fried 6 )

gk-I

where the s c a l a r

t kTs the nonzero solution of equation

Fk

tk=

(11)

ak-1Sk-1 g k - I 1

Fk_ I

=

2Fk_ I

a v a r i a t i o n of formula, (10) is o b t a i n e d w h e n

equation

the s c a l a r t is the n o n z e r o

solution of

itkL+°k-:Sk-:gktkl:,

(12)

Methods using f o r m u l a s {7),{8), (9) deLermine the minimum of a p o s i t i v e d e f i n i t e qua_ d r a t i c function in no more than m i t e r a t i o n s when a k s a t i s f i e s the tTe×act l i n e a r s e a r c h u condition T (13)

gk+1

sk = 0

557

In s u c h a c a s e v a l u e s of

bk

t h e y d i f f e r on n o n q u a d r a t i c

d e f i n e d by f o r m u l a s functions

(7),

(8),

(9) a r e t h e s a m e ,

o r w h e n e q u a t i o n (13) d o e s not h o l d . F o r m u -

l a s ( 1 0 - 1 1 ) a n d ( 1 0 - 1 2 ) a p e t h e s a m e if e q u a t i o n (13) is s a t i s f i e d formula

(7) on q u a d r a t i c

geneous function m o p e than

m+l

functions.

F -

1 2r iterations,

T h e y a l l o w to d e t e r m i n e

(x T K x ) r ,

where

K

if e q u a t i o n (13) h o l d s .

Formula

the m i n i m u m of a h o m o in no

(10-12) is a variation

is t h a t if e q u a t i o n (12) is s o l v e d in

s t e a d of e q u a t i o n (11) t h e n the l a s t t w o s e a r c h v e c t o r s

ape

K-conjugate

e v e n if

s e a r c h is n o t p e r f o r m e d .

T h e f i v e m e t h o d s h a v e been i n c o r p o r a t e d

in a p o l y a l g o r i t h m

t a p t o t h a t one a d o p t e d in a ~ ; t u a s i - N e w t o n p o l y a l g o r i t h m particular

and c o i n c i d e w i t h

is a p o s i t i v e d e f i n i t e m a t r i x ,

t h a t we p r o p o s e to F r i e d ~ s m e t h o d ; i t s r a t i o n a l e

exact linear

but

ak

is determined

storage requirement g ven elsewhere

by a p a r a b o l i c

experiments

Extensive

comparison

is s i m i -

described

s e a r c h b a s e d on F i e l d i n g l s

on the I B M 1800 is l e s s than

Numerical

whose strategy

previously

8

7

; in

method;

2500+/4 m w o r d s ; d e t a i l s a r e g i -

of t h e f i v e c o n j u g a t e g r a d i e n t

methods described

in the a b o v e

s e c t i o n w a s made d u p i n g t h e d e v e l o p m e n t of t h e p o l y a l g o r i t h m . Whereas a detailed 10 a n a l y s i s c a n be f o u n d e l s e w h e r e , t h e f o l l o w i n g c o m m e n t s a r e in o r d e r : -

more accurate

precision

linear search

in d e t e r m i n i n g

r e d u c e s t h e n u m b e r of i t e r a t i o n s ,

the minimum.

T h i s is e s p e c i a l l y

homogeneous functions where exact linear search - r h e r e f o r e if strongly

-

m a r k e d on q u a d r a t i c

r e c o m m e n d e d to u s e h i g h p r e c i s i o n that the search

and

is a c o n d i t i o n f o r t e r m i n a t i o n .

m f u n c t i o n e v a l u a t i o n s a r e e q u i v a l e n t to one g r a d i e n t

b e r of f u n c t i o n e v a l u a t i o n s on

keeping the same

evaluation,

it is

in the l i n e a r s e a r c h b e c a u s e the n u m requires

is s u b s t a n t i a l l y

independent

m. the five algorithms

Ribiere

behave similarly

and the Sorenson

tion over the original

with a marginal

superiority

methods over the Fletcher-Reeves

method.

is evident when high precision

The superiority

of the Polak-

a n d of F p i e d V s v a r i a -

of t h e l a s t m e t h o d s o v e r t h e o t h e r s

l i n e a r s e a r c h is u s e d ,

but is l o s t w h e n l o w p r e c i s i o n

is a d o p t e d . -

s o m e c a s e s s h o w that~ w h e r e t h e o r e t i c a l l y

termination,

in p r a c t i c e

stantially

less when

algorithm

on f u n c t i o n s

m

this number is l a r g e . in v e r y

is

larger

m

i t e ~ ' a t i o n s w o u l d be r e q u i r e d

when

m

This isapromisingresult

many variables.

fop

is s m a l l b u t m a y be s u b f o r t h e 9 o o d n e s s of t h e

558

-

strict

t e r m , : n a t i o n [s s t r o n g l y

Table 1 clearly

sensible

evidences this for

to e x a c t n e s s of the l i n e a r s e a r c h .

Fried~s

method.

The function minimized was

a homogeneous function with x

O

= (1,2~ . . . .

~ m} a n d

m

r = o , k . . = d . . (d.. t h e K r o n e c k e r d e l t a ) , s t a r t i n g p o i n t U ~J ~J e q u a l f o u r o r t w e n t y . T h e e x a c t v a l u e f o r a k is g i v e n b y

formula

T

&-~ (14)

a k = __(2PF k)

gk Sk

4. SkTK s k

-

zero.

If

q ~0

we c a n o b t a i n ( m + g ) - t e r m i n a t i o n

scent, s e a r c h e s , r

I m o d e l i s the f u n c t i o n F = " 2 7 ( x T k x ) r +

Fried~s functional

and

-

q. H o w e v e r

to F ~ ~ 0 ,

Friedls

making initially

with

q

identically

two steepest

de-

a n d u s i n g t w o e q u a t i o n s of t h e t y p e (11) to s o l v e s i m u l t a n e o u s l y experiments

of t h e m e t h o d ~s n o t r a d i c a l l y verges

q,

s h o w that e v e n i f

q

is

changed. A theoretical

then Fried~s

~0

a n d l a r g e the e f f i c i e n c y

explanation

is t h a t if F k c o n -

m e t h o d t e n d s to b e h a v e as F l e t c h e r - R e e v e s

method may be interpreted

method.

Termination

is k e p t s c a l i n g

in F l e t c h e r - R e e v e s formula. k in t h e s a m e w a y t h e P o l a k - R i b i e r e p a r a m e t e r and t h i s

modification

gives marginal

improvement

TABLE

I

~

as s c a l i n g

for

to t h e a l g o r i t h m .

E f f e c t of e x a c t n e s s ~n t h e l i n e a r s e a r c h

FUNCTION

METHOD

m=4

Polek Ribiere

m=4

Friedls variation

m = 20

Polak Ribiere

a / a (~) k k 1 1 .01 1. 1

F

ITERATIONS

1 E-6 2E-5 2E-8

.5 4 /4

1 1.01 1. 1

3E-38 ?E-9 4E-6

/4 4 4

1 1.01

6E-6 gE-6 3E-6

13 13 lg

5E-6 5E-6 9E-6

11 tl li

I. I Friedls variation

m = 20

b

I 1.01

I. I APPLICATION The quadratic n

different

TO THE QUADRATIC

assignement problem arises

locations

in o r d e r

ASSIGNEMENT

when

n

PROBLEM

e l e m e n t s m u s t b e a s s i g n e d to

to m i n i m i z e a c o s t f u n c t i o n w h i c h c a n b e w r i t t e n

as

559

(15) where

~-

~____ x x a . i j fk i l b k j

x,, = 1

the matrix easily

1

f=

if e l e m e n t s i

X .~ f x i

A compact writing

f

The matrix

ape interrelated, matrix

x.. = 0 otherwise.

whereas

A-~ a l l

TP~

f

1

The quadratic values,

of

%

D

fdij

where

dij

ddfx : ~ c a n be w r i t t e n Ij

f

generally

is n o t p o s i t i v e

e v e n if it a s s u m e s n o n n e g a t i v e

on t h e s p a c e of p e r m u t a t i o n In t h e u s u a l c o m b i n a t o r i a l

say.

(larger)

framework

it can assume negative

(f = 0 o n l y f o r a d e g e n e r a t e d

is n o t u n i q u e ! ) ,

assignement.

the quadratic

techniques.

Such are implicit

linear

values

definite;

problem)

matrices.

w i t h by o p t i m a l a n d s u b o p t i m a l

twenty,

in

(AXB+ATxBT)

function

(which generally

can

xTAxB]

of d e r i v a t i v e s

D =

B="=- b k

is f o r i n s t a n c e

the f o r m (17)

and

Clearly

as I~exchangeH a n d I~distanceH m a t r i c e s .

of t h e c o s t f u n c t i o n

21

-

j

is a p e r m u t a t i o n

be i n t e r p r e t e d

(16)

and

assignement

The first

Suboptimal

when

n

methods and Lawlerls

techniques

is d e a l t

ones give a global minimum

but t h e y a p e u n f e a s i b l e enumeration

problem

ape generally

is as l a r g e as 1! r e d u c t i o n

to a

b a s e d on h e u r i -

stics; w e l l k n o w n m e t h o d s h a v e b e e n p u b l i s h e d b y A r m o u r a n d B u f f a 1 2 G i l m o r e 1 3 14 15 Hillier and Connors , G r a v e s and W h i n s t o n . Heuristic algorithms are feasible fop problems

say~ a n d t h e s o l u t i o n is g e n e r a l l y 16 about these methods can be found elsewhere

formation Finally,

the implicit

zero-one

variables

applied to u s .

up to f i f t y v a r i a b l e s ,

enumeration

algorithm

under quadratic

t o the q u a d r a t i c in t h e s o l u t i o n

for minimizing

constraints

assignement i however

of the nonlinear

continuous

a q u a d r a t i c f u n c t i o n in 1? due to Hansen m i g h t be u s e f u l l y experimental

results

programming

problem

deal with two main points, once a method for unconstrained T h e y a r e h o w to t r e a t t h e c o n s t r a i n t s Conditions matrix

(g}, ( 3 ) , ( 4 ) e r ' e r e a d i l y

X

spect to

a n d l e t t i n g an ( n - l ) x.. Ij

(18) Conditions

by (n-l)

optimization

and how to choose the initial

eliminated,

9 o o d . M o p e in

deleting

unknown matrix

a r e not k n o w n w e h a v e to is a v a i l a b l e .

point.

the last row and column from X. Derivatives

d.. with reI.I

a r e g i v e n b y the f o r m u l a

d.. = d.. + d - d . - d . ~j ~j nn fn nj (1)

a r e d e a l t w i t h in o u r a p p r o a c h

by building

the penalty function

560

(19)

'~= f +

where

the " l o s s "

coefficients function

h,.

ij

~ k [×.,(x..-1)+hij] ~.~_| i j U U

g

coefficients

k.. are positive and both k.. and the "cor'rection u U U a p e c o n s t a n t for" e v e r y u n c o n s t r a i n e d m i n i m i z a t i o n . A L a g r a n g i a n

approach

has been discarded

as t h e n u m b e r of e q u a l i t y

i:er t h a n the n u m b e r of i n d e p e n d e n t

variables.

k.)s

conver'gence

U

a n d t h e h . . l s in o r d e r ' t o f o r c e Ij

methods.

In t h e f i p s t ,

k ~

where

e

vergence

is grea

i s h o w to c h a n g e t h e

to a feasible

point.

We u s e d t w o

M i e l e et a l . : ~ w e h a v e t h e h.~.s i d e n t i c a l l y z e r o w h i l e LI e q u a l to a v a l u e k ; i n i t i a l l y k = 1 0 0 a n d t h e n it i s m o d i f i e d a c c o r d i n g

t h e k..~s a p e aJt U to t h e f o r ' m u l a (20)

A question

constraints

1Ok ]

following

(T-f)/ke

is a c o n v e r g e n c e to a f e a s i b l e

ed m i n i m i z a t i o n .

parameter.

As Mieie~s formula

point we limited

gives fast

to f o u r t h e n u m b e r o f c y c l e s

In t h e s e c o n d m e t h o d w e f o l l o w e d

Powel119

Pate of c o n -

of u n c o n s t r a i n -

. In the f i r s t

cycle

the

h..1s a r e i d e n t i c a l l y z e r o a n d t h e k. ~s a p e a l l e q u a l t o k = 100. T h e n , a c c o r d i n g IJ U to s o m e t e s t s , t h e k . . I s a n d the h..Ts a r e m o d i f i e d a t e v e r y c y c l e b y f o r m u l a s l i k e U U multiplication b y a c o n s t a n t or- t h e m a p p i n g (21)

h..4--ij

Convergence

h..+x.,(x..-l) U

Ij

ij

is supposed w h e n m a x

Ix..(x..-1) I /_ .0/4 and no ij

are

m o p e than /4 cycles

U

allowed.

TabJe 2 s h o w s h o w

a matrix

X

is generated Ca case with

P4). With both strategies the m a x i m u m the nonlinear p r o g r a m m i n g

n=6

and starting point

n u m b e r of computer` operations required by

problem is easily bounded. N o mor,e than four cycles ape

allowed; no m o r e than I00(n-I) function evaluations are allowed in each cycle, co~ responding to about I0(n-I) iterations and ten function evaluations per linear search. N o w t h e n u m b e r of o p e r a t i o n s auxiliary calculate

matrices D

a p e u s e d to s t o r e

this number grows

We s u p p o s e t h a t a d d i t i o n s is n e g l i g i b l e . for

n

efficient

required

large

of G i t m o r e l s

g r , o w s as ? n 3 + 3 0 n 2, if t h r ' e e otherwise

if t w o a u x i l i a P y

it is

matrices

O (n4); to ape used.

t a k e the s a m e t i m e a n d that s h i f t t i m e

gradient

the n u m b e r of o p e r a t i o n s requires

f

products,

as 7 n 3 + 3 b n 2,

by c o n j u g a t e

aigorithms

n e e d s O (n 3) o p e r a t i o n s

matrix

and multiplications

Time required

sufficiently

to calculate

algorithm

(n2)~ t h e r , e f o r , e 4 is b o u n d e d b y 2b01 n . T h e m o s t

O (n 5) c o m p u t e r

is

O

operations;

CRAFT

per` ~ t e r , a t i o n ; f o p s m a l l n t h e n u m b e r o f i t e r a t i o n s

r,alty tow; its probabitistic

d e p e n d e n c e on

n

is not known

to us.

is g e n e -

561 TABLE

2 -

G e n e r a t i o n of a p e r m u t a t i o n m a t r i x

CYCLE

ITERATIONS

MATRIX

X

I

28

•631,.065,.549,.037,-. 183 .228, . 376, . 024, .401 , - . 0?5 .019, . 0 5 3 , .022, . 202, . 6 7 .09 ~.43 , . 4 3 , . 1 4 , - . 0 8 .05 , . 0 6 , . 0 0 , . 1 2 , . 7 3

2

23

1.16,-.1,-, 18,-.03,-.09 - . 1 6 , - . 17, . 1 7 , 1.21,-./4.4 .11 , . 1 5 , . 9 6 , - . 0 3 , - . 1 8 - . 0 6 , 1.2 , - . 1 6 , - . 0 5 , - . 0 1 -.06,-.09,.1 , - . 0 ? , 1.1

20

1.00, 0 , 0 , 0 , 0 ,

O u r bound is r e a l i s t i c ; 2801 is a h i g h c o e f f i c i e n t ,

0,0,0, 0 0,0, I,-.02 O, . g g , o , o 1,0, 0 , - . 0 1 0 , 0 , O, .99

so o n l y f o r l a r g e n, s a y n•

100,

the time r e q u i r e d by the c o n t i n u u m a p p r o a c h is e s t i m a t e d to be l e s s than that requi_ r e d by h e u r i s t i c p r o c e d u r e s . te o n l y v a r i a t i o n s of

f

An a d v a n t a g e of h e u r i s t i c methods is that they c a l c u l a

due to e x c h a n g e of e l e m e n t s w h i c h r e q u i r e

O(n) c o m p u t e r

operations. T h e s e c o n d main p o i n t is how to c h o o s e the i n i t i a l m a t r i x

X.

It is w e l l known that

if the o b j e c t f u n c t i o n is not c o n v e x , then d i f f e r e n t i n i t i a l p o i n t s may l e a d to d i f f e r e n t minima. In o u r p r o b l e m e v e r y f e a s i b l e p o i n t is a local minimum and a l s o t h e r e may be many g l o b a l m i n i m a . FoP q u a d r a t i c a s s i g n e m e n t p r o b l e m s a r i s i n g f r o m p l a n t l a y out m u l t i p l i c i t y of 91obal minima is often a c o n s e q u e n c e of g e o m e t r i c a l s y m m e t r y . In m o r e g e n e r a l c a s e s a r o u g h e s t i m a t e of the n u m b e r of 91obal minima can be done as follows.

L e t us assume that A and B a r e i n t e g e r v a l u e d and max

maxlbij I Zp,

p

b e i n g an i n t e g e r . Then

laij I L p

on the s p a c e of p e r m u t a t i o n m a t r i c e s it

holds

(22)

0 L. f

~

~

1

p

2

n

2

A s s u m i n g that the i n t e g e r v a l u e s of

f

e r e d i s t r i b u t e d u n i f o r m l y on the s p a c e of per"

m u t a t i o n m a t r i c e s then the a v e r a g e n u m b e r of 91obal minima is

22.3.4 ....

( n - 2 ) / p 2.

T h i s i m p l i e s that e v e n if d i f f e r e n t s t a r t i n g p o i n t s may g i v e 91obal minima, g e n e r a l ly we can e x p e c t o n l y l o c a l m i n i m a . T h e c h o i c e of the s t a r t i n g p o i n t s c o u l d be made

562

using values possibility

g i v e n by c o m b i n a t o r i a l

('I~) a n d w e c h o s e a r b i t r a r i l y

considered

3

Starting

x.

P1

I0

if

P2

0

if

P3

Change inequality

P4

1/in- 1 ~2

P5

Stochastic

j + i(n-1)

points.

The five

this

choices

-t0

I

s i g n in c a s e P 2

for

n=3,5,6,7.

They were

the same for MieieTs and

except a few cases which can be explained

coefficients.

(36 v a r i a b l e s

otherwise

s e q u e n c e 0, 1

are given in Table 4 methods,

U

is even,

sin[j+itn-1)] Z .5, otherwise

there are many minima; penalty

initia~ unfeasible

we did not explore

points

CASE

Powell~s

however

a r e g i v e n in T a b l e 3.

TABLE

Results

procedures;

by t h e f a c t t h a t w h e n

the actual one calculated

Matrices

m a y d e p e n d on t h e s e q u e n c e o f t h e 20 A a n d B a r e f o u n d in N u g e n t et a l . . For n=7

and 49 constraints)

the time of execution

i s a b o u t t w e n t y m i n u t e s on

the IBM 1800. Notation

n.m.

means that the case was not considered;

ce to a permutation some identicaI

matrix

means that convergen

In t h i s c a s e t h e f i n a l m a t r i x

showed

rows.

TABLE

/4.

Resuits N. w;

N,b, n.m°

n, mo

Pt

P2

P3

P4

P5

15

17

f5

25

17

25

31

n.c.

34

35

n.c.

30

/43

46

49

62

52

49

43

74

84

n.m.

96

n.c.

84

n.m.

We can observe

that under

njugate gradient

algorithms

(~)

was qot obtained.

n.c.

the knowledge

certain

conditions

cannot work.

of derivatives

of

a c h c o u l d b e u s e d in t h e c o m b i n a t o r i a l be ~nterchanged

f

For

which

a penalty function instance,

if

is a byproduct

heuristics

to suggest

approach

using co-

a..=b..= 1 - d.. (d.. the U

I.l

iJ

of the continuum which

U

appro-

elements should

563

Kronecker

symbol) and initially

x i j = r~ then

every variable

is c h a n g e d by the s a

me a m o u n t at e v e r y i t e r a t i o n and a p e r m u t a t i o n m a t r i x is not g e n e r a t e d . Column headed N.b.

and N. w. c o n t a i n the b e s t and t h e w o r s t r e s u l t s q u o t e d b y

augent fop some heuristic

procedures.

T h e f o l l o w i n g c o n c l u s i o n s can be made: 1) the n o n l i n e a r p r o g r a m m i n g a p p r o a c h g e n e r a l l y g i v e s a p e r m u t a t i o n m a t r i x in two or three cycles 2)

the f i n a l m a t r i c e s do not u s u a l l y c o r r e s p o n d # g l o b a l

m i n i m a and v a l u e s of

f are

rather scattered 3)

the heuristic

combinatorial

procedures are therefore superior

both for quality

of s o l u t i o n a n d t i m e of e x e c u t i o n in t h e r a n g e c o n s i d e r e d f o r n. F o r l a r g e r n~ w h e r e the c o n t i n u u m a p p r o a c h m i g h t b e c o m e c o m p e t i t i v e , unfortunately

the c , o m p u t e r t i m e is

t o o d e m a n d i n g to m a k e e x p e r i m e n t s p o s s i b l e .

CONCLUSION Combinatorial

o p t i m i z a t i o n p r o b l e m s h a v e been e x p r e s s e d as n o n l i n e a r p r o g r a m m -

ing p r o b l e m s and e f f i c i e n t t e c h n i q u e s f o r h a n d l i n g the l a r g e u n c o n s t r a i n e d m i n i m i z a t i o n p r o b l e m a r i s i n g h a v e been c o n s i d e r e d .

T h e c h o i c e of the i n i t i a l p o i n t s is a c r i

t i c a l p r o b l e m w h i c h has n o t been s o l v e d s a t i s f a c t o r i l y . assignement problem are inferior fact,

S o l u t i o n s f o r the q u a d r a t i c

to t h o s e g i v e n by c o m b i n a t o r i a l

t e c h n i q u e s i in

the n e c e s s i t y of e x p l o r i n g a s u c c e s s i o n of l o c a l m i n i m a r e p r o d u c e s

tain sense the original combinatorial

in a c e r -

problem.

A P P E N D IX The definitionofapermutationmatrixX[s

e q u i v a l e n t to s a y that the e l e m e n t s of

X

must s a t i s f y e q u a H o n s

(1.1) (1.2)

x..(x..-l) =0 ij 1j

i,j= 1,2,. .... n

.~" x i j = 1

j = 1,2 ......

n

i=

n

t=l

(1.3) Summing

E

xij = 1

1,2, .....

e q u a t i o n s ( 1 . 2 ) and ( 1 . 3 ) r e s p e c t i v e l y

ties. Therefore

j

and

i

we o b t a i n i d e n t i -

s y s t e m ( I - 4 ) is r e a d i l y g e n e r a t e d . A l s o the f o l l o w i n g

holds: Theorem:

over

C o n d i t i o n s ( I - 4 ) a r e not r e d u n d a n t .

Theorem

564

Proof. ed. Put

Suppose firstty

t h a t o n e of r e l a t i o n s

Xli=1 for i ~ m,

Xjm=1

¢,I ,j, s a y

for j ~Z I , Xlm = -n+2

X l m ( X l m _ t ) = 0 can be d e l e t -

and all remair~ing xij~s e-

qual to zero. T h e n all conditions are satisfied but the matrix is not a permutation matrix. W i t h o u t t o s s of g e n e r a l i t y ,

s u p p o s e n o w t h a t one of e q u a t i o n s (2), s a y

xil=l,

can be d e l e t e d . S y s t e m ( 2 - 4 ) can be w r i t t e n in t h i s c a s e

(1.4)

~: × i j = l

i=1,2

......

n

J--I (1.5)

~_ x.. = 1 ~=l Ij

N o w put the d i a g o n a l

j= !,2,.o..n , j~/l,l+l w i s e j ~/ 1,1-1.

e l e m e n t s of X e q u a l to one e x c e p t

w i s e Xl_ 1,I_1=1 if I = n. P u t x. Js e q u a l to z e r o . U T h e n c o n d i t i o n s (1. 1 - 1 . 4 - 1 . 5 )

Xl,l+l

= i if

if

ILn,

other-

x l + l , I+1 ' if ! ~ n, other'

1 ~ n, o t h e r w i s e

x 1,1_1 = 1. P u t a l l o -

ther

a r e s a t i s f i e d but the r e s u l t i n g m a t r i x is not a p e r m u

ration matrix. Finally, xTx

o b s e r v e t h a t a p e r m u t a t i o n m a t r i x s a t i s f i e s the u n i t a r i t y

condition

= X X T= 1. T h i s e q u a t i o n is e a s i l y d e r i v e d f r o m e q u a t i o n s ( 1 - 3 } ,

r i t y b e i n g c o n n e c t e d w i t h the n o n l i n e a r i t y

its n o n l i n e a -

of e q u a t i o n { I . 1).

REFERENC ES 1.

Nelder~d.A. and N e a d , R . : A s i m p l e x m e t h o d f o r f u n c t i o n m i n i m i z a t i o n , d.~7, 308-313~ 1965

CompuL

2.

P o w e l I , M . d. D. : An e f f i c i e n t m e t h o d of f i n d i n g the m i n i m u m of a f u n c t i o n of s e v e Pal v a r i a b l e s w i t h o u t c a l c u l a t i n g d e r i v a t i v e s , C o m p u t . d . , 7, 1 5 5 - 1 6 2 , 1964

3.

F i e t c h e r , R. and R e e v e s , C . M . : F u n c t i o n m i n i m i z a t i o n by c o n j u g a t e g r a d i e n t s ~ C o m p u t e r J. ~ 7, 1 4 9 - 1 5 4 , 1964

4.

P o l a k , E. a n d R i b i e r e , G . : N o t e sup te c o n v e r g e n c e de m e t h o d e s des d i r e c t i o n s c o n j u g e e s , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , D e p t . of E l e c t r i c a l E n g i n e e r i n g and C o m p u t e r S c i e n c e s , w o r k i n g p a p e r , 1969

5.

Sorenson, m.w.: Conjugate Direction Procedures for Function Minimization~ J o u r n a l of t h e F r a n k l i n Institute, 288, 4 2 1 - 4 4 1 , 1969

6o ~ r i e d , I. : N - s t e p C o n j u g a t e G r a d i e n t M i n i m i z a t i o n S c h e m e f o r N o n q u a d r a t i c F u n c t i o n s , A I A A J o u r n a l , 9, 2286-2287~ 1971 7.

S p e d i c a t o , E. : Un p o l i a l g o r i t m o p e r la m i n i r n i z z a z i o n e di u n a f u n z i o n e di pi~ v a r i a b i l i ~ A t t i del C o n v e g n o A I C A su T e c n i c h e di S i m u l a z i o n e e A l g o r i t m i , M i l a no, ' n f o r m a t i c a ~ N u m e r o s p e c i a l e ~ 1972

8.

F i e l d i n g , K . : F u n c t i o n m i n i m i z a t i o n and l i n e a r s e a r c h ~ A l g o r i t h m of ACM) 13~8) 1970

387~ C o m m u n .

565

9.

S p e d i c a t o , E. : Un p o l i a l g o r i t m o a 9 r a d i e n t e c o n i u g a t o pep la m i n i m i z z a z i o n e di f u n z i o n i n o n l i n e a r i in m o l t e v a r i a b i l i , Nota t e c n i c a C 1 S E - 7 3 . 0 1 2 , M i l a n o , 1973

10.Spedicato~E.: CISE-Report !l. Lawler, E.L.: 599, 1963

to a p p e a r

T h e Q u a d r a t i c A s s i g n e m e n t P r o b l e m , Management S c i , 9 , 5 8 6 -

19-. A r m o u r , G . C . and B u f f a , E . S . : A H e u r i s t i c A l g o r i t h m and S i m u l a t i o n A p p r o a c h to R e l a t i v e L o c a t i o n of F a c i l i t i e s , Management S c i . , 9~ 2 9 4 - 3 0 9 , 1963 if 3. G i l m o r e , P. C. : O p t i m a l and S u b o p t i m a l A l g o r i t h m s ment, S I A M d.~ 10~ 3 0 5 - 3 1 3 , 1962

f o r the Q u a d r a t i c A s s i g n e -

14. Hi II i e r , F . S . and C o n n e r s , M. M. : Q u a d r a t i c A s s i g n e m e n t P r o b l e m AI 9 o r i thms and the L o c a t i o n of I n d i v i s i b l e F a c i l i t i e s , Management S c i . , 13, 4 2 - 5 7 , 1966 15. G r a v e s , G . W . and W h i n s t o n , A . B . : An A l g o r i t h m f o r the Q u a d r a t i c A s s i g n e m e n t Problem~ Management S c i . , 16, 4 5 3 - 4 7 1 , 1970 1 6 . C a s a n o v a ~ M . and T a g l i a b u e , G . " C I S E - R e p o r t

to a p p e a r

1 7 . H a n s e n , P . : Q u a d r a t i c Z e r o - O n e P r o 9 r a m m i n 9 by l i m p l i c i t E n u m e r a t i o n , in N u m e r i c a l Methods fop n o n - l i n e a r O p t i m i z a t i o n ~ ( F . A . Lootsma, e d . ) ~ A c a d e m i c P r e s s , 1972 1 8 . M i e l e , A . , C o g g i n s ~ G . M . and L e v y , A . V . : U p d a t i n g r u l e s f o p the p e n a l t y c o n s t a n t used in the p e n a l t y f u n c t i o n method f o p m a t h e m a t i c p r o g r a m m i n g p r o b l e m s , A e r o - A s t r o n a u t i c s R e p o r t n. 90, R i c e U n i v e r s i t y , H o u s t o n , 1972, 19. Povvel l, M. J. D. : A M e t h o d f o r N o n l i n e a r C o n s t r a i n t s in M i n i m i z a t i o n P r o b l e m § in O p t i m i z a t i o n ~ (R. F l e t c h e r , ed.)~ A c a d e m i c Press~ 1969 20. Nugent, C. E. ~ V o l l m a n n , T . E. and Ruml J. : An e x p e r i m e n t a l c o m p a r i s o n of t e c h niques f o r the a s s i g n e m e n t of f a c i l i t i e s to Iocations~ O p e r a t i o n s R e s e a r c h , 16, 150-173, 1968.

T h i s w o r k was s u p p o r t e d by the C o n s i g l i o N a z i o n a l e d e l l e R i c e r c h e , w o r k of the r e s e a r c h c o n t r a c t C I S E / C N R

n. 7 1 . 0 2 2 0 7 . 7 5 -

115. 2946

in the f r a m e -

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