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
equation
channel
to e v a l u a t e
; the
;
interconnecting
two
different
states
gain.;
of
set
of i n t e r e o n n e c t i o n s ,
g. T h e r e f o r e ~
x Ccontro I + m x Ccontrol
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
be
i.e.,
process
va-
chosen the
output
e.g.,
i Jaccuracy
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
controller Cinterconnectio n
subcosts,
of s u b c o s t s
cost
energy
of the m o d e l
]
ai(Yi-Ymi)2
Z b li i
; Yi-Ymi
l
for the input
same
sequence
.
(5)
[etc Although function
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
its
JG
to a p p l y
application
A generalization
is a n a l y t i c a l l y
second
Js.c.
only
Js.c.
the
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
S ~:
[~T~]
is a s i n g u l a r
the
integer
entries
are
the
~
the
results
of
JM
treat-
is o p t i of the
is d i s c u s s e d e.g.
[4].
In the
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.
In this since
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
one w i s h e s
modeling
control
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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J.P.,
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Co,
1967
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H.R,1970
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Statis-
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(27)
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C.R.
Acad.
Multivariate
Data
Analysis
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view
Inf.
Theoretical Gestalt C-20,
Freeman
and L o n d o n . of
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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
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les m o d i f i c a t i o n s
Computer
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Congress
entre
off-line
Joint
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Control methods
Clusters.
a l g o r i t h m s . IFIP
TA-2
8,pp.
338-353
for
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I.E.E.E.
n ° I, J a n u a r y .
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and
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on C o m p u -
254
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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,
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of Convex
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linearen
verw.
Gebiete
IOl-ll2.
Differential ~]
for a Class
(1972).
Programmierung",
[3]
Conditions
of Systems
"Fonctionelles
aux deriv~es
Governed
Springer-Verlag,
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Convexes",
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by Partial (]971).
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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 p