Acta Numerica 2004: Volume 13 (Acta Numerica)

ij,ij(xj)} in (3.8). Again, as in our introductory example of piecewise linear hierarchical bases of Section 3.1, we use these ID functions as the input of the tensor product construction, which provides a suitable piecewise d-dimensional basis function i,i(x) := n 7 _ i
4-4- Prewavelets and wavelets Note again that the presented hierarchical multiscale bases with higherorder polynomials or interpolets, after the tensor product construction, allow for a relatively cheap evaluation of differential operators and discretization schemes on sparse grids in a Galerkin approach owing to their interpolating properties. Also, upper error estimates can be easily derived for various sparse grid spaces following the arguments in Sections 3.2 and 4.2. However, they all exhibit a main drawback: there is no lower error estimate with constants independent of the number of levels involved, that is, they form no stable multiscale splitting (Oswald 1994). The consequences are twofold. First, the (absolute) value of the hierarchical coefficient is just a local error indicator and no true error estimator. We obtain sufficiently refined sparse grids (compare the associated numerical experiments in Section 5) on which the error is properly reduced, but it may happen that too many grid points are employed for a prescribed error tolerance in an adaptive procedure. Efficiency is thus not guaranteed. Second, the condition number of the linear system which results from a symmetric elliptic partial differential operator in multiscale basis representation is, after diagonal scaling, in general not independent of the finest mesh size involved.3 To obtain a fast mesh independent solver, additional lifting tricks (Sweldens 1997, Daubechies and Sweldens 1998, Koster 2002) or multigrid-type extensions (Oswald 1994, Griebel 1994a, Griebel and Oswald 1994, Griebel and Oswald 1995a, Griebel and Oswald 19956) are necessary. These difficulties are avoided if we employ stable multiscale splittings (Oswald 1994, Dahmen and Kunoth 1992, Carnicer, Dahrnen and Pena 1996, Cohen 2003) and the respective Li- and ifi-stable multiscale bases, for which two-sided estimates exist.

3

This is the case for stable splittings with wavelets. Then a simple fast solver results from the diagonal scaling preconditioner: see Dahrnen and Kunoth (1992) and Oswald (1994).

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-1/2

209

—1/2

Figure 4.10. Non-orthogonal linear spline wavelet (left: interior; right: boundary). Since Daubechies' fundamental discovery of orthogonal multiscale bases with local support, i.e., the classical wavelets (Daubechies 1988), an enormous literature has arisen. At present, there exists a whole zoo of such bases from the wavelet family. Here, we stick to simple hierarchical spline-like function systems. Within this class, we have orthonormal wavelets (Lemarie 1988), biorthogonal spline wavelets (Cohen, Daubechies and Feauveau 1992), semi-orthogonal spline wavelets (Chui and Wang 1992), fine grid correction spline wavelets (Cohen and Daubechies 1996, Lorentz and Oswald 1998) and multiwavelets derived from splines. The construction principles of such functions are highly developed, and there is abundant literature on them. Most theory can be found in Daubechies (1992), Chui (1992) and Meyer (1992), and the references cited therein. Further reading is Daubechies (1988, 1993), and Cohen et al. (1992, 2001). A nice introduction, similar in spirit to this paper, is given in Cohen (2003). In the following, let us briefly mention the simplest but also cheapest mother wavelets which are made up from piecewise linears by stable completion procedures (Carnicer et al. 1996). They are sufficient to be used for a second-order PDE within the Galerkin method. These are the so-called linear pre-prewavelets, which are a special case of non-orthogonal spline wavelets (Cohen and Daubechies 1996, Lorentz and Oswald 1998, Stevenson 1996) of linear order. The corresponding mother function (f> is shown in Figure 4.10 (left). Again, by translation and dilation in an analogous way to (3.8), we get a ID multilevel basis from this (f>. Note that a modification is necessary near the boundary. For homogeneous Dirichlet conditions, the scaled and dilated function in Figure 4.10 (right) can be used in points next to the boundary. Another, still reasonably cheap approach involves linear prewavelets, a special case of semi-orthogonal spline wavelets (Chui and Wang 1992, Griebel

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9/10

-6/10

-6/10

-6/10

Figure 4.11. Semi-orthogonal linear spline wavelet (left: interior; right: boundary). and Oswald 19956). The corresponding mother function is shown in Figure 4.11 (left). Again, a modification is necessary near the boundary. For homogeneous Dirichlet conditions, this is shown in Figure 4.11 (right). A construction for Neumann conditions can be found in Griebel and Oswald (19956). Finally, let us mention the so-called lifting wavelets. They are the result of the application of Sweldens' lifting scheme (Carnicer et al. 1996, Sweldens 1997) to the interpolet basis {i-,i-(xj)} or other interpolatory multiscale bases, for example. Here, the hierarchical (i.e., if we consider odd indices only; see (3.11)) lifting wavelets are defined by

+ on finer levels lj > 0. The basic idea is to choose the weights Q/'j. in this linear combination in such a way that 0/.^. has more vanishing moments than c/iij^, and thus to obtain a stabilization effect. If we apply this approach to the hierarchical interpolet basis of Section 4.3 so that we achieve two vanishing moments in the lifting wavelet basis, we end up with 1 for the odd indices 2ij + 1. The corresponding mother function (j> is shown in Figure 4.12. Again, as in our introductory example of piecewise linear hierarchical bases of Section 3.1, we can use these ID multilevel basis functions as the input of the tensor product construction which provides a suitable piecewise d-dimensional basis function 0i,i(x) in each grid point xi j. As in Sections 3

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-1/4

Figure 4.12. Lifting interpolet with N = 2 and two vanishing moments.

and 4.2, we can derive sparse grids based on wavelets. Also, most other considerations regarding the estimates for the cost and error complexities carry over accordingly. Further information on wavelets and sparse grids can be found in Griebel and Oswald (19956), Hochmuth (1999), DeVore, Konyagin and Temlyakov (1998), Koster (2002), Knapek (2000a), Griebel and Knapek (2000) and the references cited therein. At the end of this subsection, let us dwell a bit into theory, and let us give an argument why there is a difference between hierarchical polynomials and interpolets on the one hand and prewavelets and pre-prewavelets on the other hand. For all mother functions and the resulting product multiscale bases and sparse grid subspaces, we can, in principle, follow the arguments and proofs of Sections 3 and 4.2, respectively, to obtain cost complexity estimates and upper error bounds with relatively sharp estimates for the order constants. The main tools in the proofs are the simple triangle inequality and geometric series arguments. However, as already mentioned, a lower bound for the error can not be obtained so easily. An alternative approach is that of Griebel and Oswald (19956) and Knapek (2000a), where we developed a technique for which two-sided error norm estimates for the ID situation can be carried over to the higher-dimensional case. The approach is based on the representation of Sobolev spaces Hs([0, l]d), s > 0, as Hs([Q,l]d) = f | L2([0,1]) g> • • • ® L2([0,1]) ®Hs([0,1])®

(4.39)

(i—1) times

(d—i) times

The Sobolev space # m i x ( [ 0 , l]d),s

> 0, on the other hand is defined as the

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simple tensor product

d times

Now, for the different components of the intersection, we obtain twosided norm estimates, if the univariate multiscale functions 0 / ^ of (3.8) allow two-sided norm estimates for both Hs and LiTheorem 4.10. Let the univariate multiscale basis {4>i-,i} satisfy the norm equivalence (4.40)

u H»

where U(XJ) = Y.i^ij uij,ij 0. Then, the multivariate basis functions { ( ^ ( x ) } fulfil the norm equivalences

\\u\\l* = u Hi 1,1

where Here, ~ denotes a two-sided equivalence, i.e., a ~ b means that there exist positive constants C\, C2 such that c\ • b < a < C2 • b. Note the distinct difference in the quality of the two estimates. It is exactly the difference between l l ^ and |l|i which leads to a significant reduction in e-complexity, i.e., it allows us to use substantially fewer degrees of freedom to reach the same truncation error in the if* lix -norm than in the i? s -norm. For the proof and further details, see Griebel and Oswald (19956), Knapek (2000a) and Koster (2002). Here, the bounds 71,72 for the range of the regularity parameter s depend on the specific choice of the mother function (p. The value 72 is determined by the Sobolev regularity of 0, i.e., 72 = sup{s : 6 Hs}. The theory here works with biorthogonality arguments and dual spaces: our univariate multiscale basis {<&-,i-} possesses a dual basis {z-,i-}, in the sense that (for our hierarchical functions4) the 4

In the general setting, there is a multiresolution analysis with spaces spanned by scaling functions and difference spaces spanned by wavelets. Then, biorthogonality conditions must be fulfilled between all these function systems. In the hierarchical approach, certain scaling functions are simply wavelets, and the biorthogonality conditions are reduced.

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biorthogonality relations (Cohen et al. 1992)

If the primal and dual functions are at least in L2, then (.,.) is the usual 1,2-inner product. The value of 71 is here just the Sobolev regularity of the dual mother function 0. Note that, with 71,72 > 0, the ID norm equivalence for s = 0, and thus the Instability of the ID hierarchical basis {fii^ij}, is a prerequisite in Theorem 4.10. However, the linear hierarchical bases of Section 3, the higher-order Lagrangian interpolation functions of Section 4.2 and the interpolets of Section 4.3 do not fulfil this prerequisite, so they are not L2-stable.5 Then we obtain simply one-sided estimates. For these function systems, we get the same upper estimates with our simple tools as the triangle inequality and the geometric series arguments, but we can pinpoint, additionally, quite sharp bounds for the constants. The one-sided estimates still allow us to use the hierarchical coefficients lu^il {e.g., after suitable weighting) as error indicators in a refinement procedure, but give us no true error estimator. To this end, grids are constructed in an adaptation process steered by the indicator such that a prescribed global error tolerance is reached. This aim, however, might be reached by using more points than necessary. For the function systems built from the wavelet-type mother functions, the prerequisite of instability is fulfilled, and we obtain two-sided estimates. Thus, the wavelet coefficients \u\ti\ can serve as local error estimators, and the lower estimate part then gives us efficiency of the corresponding adaptive scheme, which means that only the necessary number of points (up to a constant) to reach a prescribed error tolerance is employed in a grid adaptation process. Furthermore, a fast solver can now be gained with levelindependent convergence rate by simple diagonal scaling as a preconditioner in the rnultiscale system. Note that from relation (4.39) it also becomes clear that the constants in the multivariate norm equivalence are more or less given by the dth power of the constants in the univariate equivalence. This causes rather large constants for higher dimensionality d. 4-5. Sparse grid applications PDE discretization techniques Since its introduction, the sparse grid concept has been applied to most of the relevant discretization schemes for PDEs. These are finite element 5

The reason is that the dual 'functions' are Dirac functionals. Then 71 = — i and the norm equivalency (4.40) is only valid for s €]|,72[, i-e., it does not hold for s = 0.

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O.S

1.0

Figure 4.13. Laplace equation with a singularity on the boundary: adaptive sparse grid (left) and corresponding solution (right). methods and Galerkin techniques, finite differences, finite volumes, spectral methods, and splitting extrapolation, which leads to the combination technique. The first main focus of the development of sparse grid discretization methods was on piecewise linear finite elements. In the pioneering work of Zenger (1991) and Griebel (19916), the foundations for adaptive refinement, multilevel solvers, and parallel algorithms for sparse grids were laid. Subsequent studies included the solution of the 3D Poisson equation Bungartz (1992 a, 19926), the generalization to arbitrary dimensionality d (Balder 1994) and to more general equations (the Helmholtz equation (Balder and Zenger 1996), parabolic problems using a time-space discretization (Balder, Rude, Schneider and Zenger 1994), the biharmonic equation (Stortkuhl 1995), and general linear elliptic operators of second order in 2D (Pflaum 1996, Dornseifer and Pflaum 1996). As a next step, the solution of general linear elliptic differential equations and, via mapping techniques, the treatment of more general geometries was implemented (Bungartz and Dornseifer 1998, Dornseifer 1997) (see Figure 4.13). Since then, algorithmic improvements for the general linear elliptic operator of second order have been studied in detail (Schneider 2000, Achatz 2003a, Achatz 20036). In the first experiments with polynomial bases of higher-order, bicubic Hermite bases for the biharmonic equation (Stortkuhl 1995) and hierarchical Lagrange bases of degree two (Bungartz 1996) as well as of arbitrary degree (Bungartz and Dornseifer 1998, Bungartz 1997) were studied. The general concept of the hierarchical Lagrangian interpolation on sparse grids was introduced in Bungartz (1998). Afterwards, the higher-order concept was combined with a duality-based adaptivity approach (Schneider 2000) and with operator extension (Achatz 2003a). In addition to that, aspects of complexity, especially for the solution of the Poisson equation (Bungartz

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and Griebel 1999), were discussed with regard to Werschulz (1995). Furthermore, besides the classical hierarchical bases according to Section 3, prewavelet bases (Oswald 1994, Griebel and Oswald 19956) and wavelet decompositions (Sprengel 1997a) were considered. Also, the parallelization of the sparse grid algorithms was pursued: see Pfaffinger (1997), Hahn (1990), Griebel (1991a, 19916). Another development in the discretization of PDEs on sparse grids using second-order finite differences was derived and implemented in Griebel (1998), Griebel and Schiekofer (1999), Schiekofer (1998), and Schiekofer and Zumbusch (1998). There, based on Taylor expansions and sufficient regularity assumptions, the same orders of consistency can be shown as they are known from the comparable full grid operators. This means, in particular, that the typical log-term does not occur. Generalizations to higher-order finite difference schemes using conventional Taylor expansion (Schiekofer 1998) and interpolets (Koster 2002) as discussed in Section 4.3 were also developed. The conceptual simplicity of finite difference schemes allowed for straightforward adaptive refinement strategies based on hashing (Griebel 1998, Schiekofer 1998), for the application of sparse grids to elliptic problems with general coefficient functions, and for the handling of general domains and nonlinear systems of parabolic PDEs of reaction-diffusion type (Schiekofer 1998). This opened the way to a first solution of the NavierStokes equations on sparse grids: see Schiekofer (1998), Griebel and Koster (2000), Koster (2002) and Griebel and Koster (2003). Furthermore, the so-called combination technique (Griebel, Schneider and Zenger 1992), an extrapolation-type sparse grid variant, has been discussed in several papers (Bungartz, Griebel, Roschke and Zenger 19946, 1994c, Bungartz, Griebel and Rude 1994a, Pflaum and Zhou 1999). Because a sparse grid can be represented as a superposition of several (much coarser) full grids Cl\ (see Figure 4.14 for the 2D case), a sparse grid solution for some PDEs can be obtained from the linear combination of solutions u\ computed on the respective coarse grids. Since the latter ones are regular full grids, existing solvers can be used without any need for an explicit discretization on a sparse grid. For the 2D case, we get

4c)(x):= Yl u^~ E u'(x)' |l|l=n+l

|l|i=n

and the 3D combination formula is given by

|l=n+2

Note that, in contrast to (3.19), where u\ € W\, u\ denotes the approximate solution of the underlying PDE on the regular full grid Qi. Compared with the direct discretization on sparse grids, the combination technique

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r

7

r

•

/ •

/

I •r i u '3

/

T

^(2,2) •

t

/

^

/

- I

^(1,3)

-/ 7

/

J

/

^(3,1)

/

1 + 1• /

/

/ /

/

^(2,1)

(1-2)

Figure 4.14. Combination technique for d = 2: combine coarse full grids Q\, |l|i € {n,n + 1}, with mesh widths 2"' 1 and 2~'2 to get a sparse grid Qn corresponding to V,l '.

turns out to be advantageous in two respects. First, the possibility of using existing codes allows the straightforward application of the combination technique to complicated problems. Second, since the different subproblems can be solved fully in parallel, there is a very elegant and efficient inherent coarse-grain parallelism that makes the combination technique perfectly suited to modern high-performance computers (Griebel 1992, 1993). This has been shown in a series of papers dealing with problems from computational fluid dynamics, including turbulence simulation (Bungartz and Huber 1995, Huber 19966, Griebel and Thurner 1993, Griebel, Huber and Zenger 1993a, Griebel 1993, Griebel and Thurner 1995, Griebel and Huber 1995, Griebel, Huber and Zenger 1996, Huber 1996a). Furthermore, note that there are close relations to the above-mentioned Boolean and blending methods (Delvos 1982, Delvos and Schempp 1989, Gordon 1969, Gordon and Hall 1973, Baszenski et al. 1992, Hennart and Mund 1988) as well as to the so-called splitting extrapolation method (Liem, Lu and Shih 1995). Concerning alternative discretization schemes, there are ongoing investigations on sparse grid finite volume methods for the Euler equations (Hemker 1995, Hemker and de Zeeuw 1996, Koren, Hemker and de Zeeuw 1996). In this context, a variant - the so-called sets of grids or grids of grids - has been developed (Hemker and Pflaum 1997, Hemker, Koren and Noordmans 1998). This approach goes back to the notion of semi-coarsening (Mulder 1989, Naik and van Rosendale 1993). Furthermore, spectral methods have been

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implemented on sparse grids and studied for PDEs with periodic boundary conditions (Kupka 1997). Adaptivity and fast solvers Especially in the context of PDE, two ingredients are essential for each kind of discretization scheme: adaptive mesh refinement and fast linear solvers. Concerning the first, grid adaptation based on local error estimators (typically closely related to the hierarchical surplus, i.e., the hierarchical basis coefficients) is a very widely used approach (Zenger 1991, Bungartz 1992a, 19926, Griebel 1998, Koster 2002, Griebel and Koster 2000). However, in Schneider (2000), the dual problem approach (Becker and Rannacher 1996, Eriksson, Estep, Hansbo and Johnson 1996) has been successfully applied to sparse grids, allowing for global error estimation too. As a result, the error control can be governed by some appropriate linear error functional, and the respective refinement strategies include ^-driven as well as point-driven adaptation. Concerning the efficient multilevel solution of the resulting linear systems, a number of contributions using additive and multiplicative subspace correction schemes have been made (Griebel 1991a, 19916, Pflaum 1992, Griebel, Zenger and Zimmer 19936, Griebel 19946, Griebel and Oswald 1994, 1995a, 19956, Bungartz 1997, Pflaum 1998), showing the availability of fast algorithms for the solution of the linear systems resulting from sparse grid discretizations. Applications Flow problems were the first focus for the use of sparse grid PDE solvers. Now, however, sparse grids are also used for problems from quantum mechanics (Garcke 1998, Garcke and Griebel 2000, Yserentant 2004, Hackbusch 2001), for problems in the context of stochastic differential equations (Schwab and Todor 2002, 2003), or for the discretization of differential forms arising from Maxwell's equations (Hiptmair and Gradinaru 2003). Aside from the field of PDEs, sparse grids are being applied to a variety of problems that will not be forgotten here. Among these problems are integral equations (Frank, Heinrich and Pereverzev 1996, Griebel, Oswald and Schiekofer 1999, Knapek and Koster 2002, Knapek 20006), general operator equations (Griebel and Knapek 2000, Knapek 2000a, Hochmuth, Knapek and Zumbusch 2000), eigenvalue problems (Garcke 1998, Garcke and Griebel 2000), periodic interpolation (Poplau and Sprengel 1997), interpolation on Gauss-Chebyshev grids (Sprengel 19976), Fourier transforms (Hallatschek 1992), tabulation of reduced chemical systems (Heroth 1997), digital elevation models and terrain representation Gerstner (1995, 1999), audio and image compression (Frank 1995, Paul 1995) (see Figure 4.15), and possibly others not listed here.

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Figure 4.15. Data compression with sparse grids: Mozart's autograph of the fourth movement of KV 525 'Eine kleine Nachtmusik'. Top left: original image. Top right: compressed version. Right: corresponding adaptive sparse grid.

Of course, owing to the historic background described in Section 4.1, numerical quadrature has always been a hot topic in sparse grid research. Starting from the explicit use of the piecewise linear basis functions to calculate integrals Bonk (19946, 1994a), quadrature formulas based on the midpoint rule (Baszenski and Delvos 1993), the rectangle rule (Paskov 1993), the trapezoidal rule (Bonk 1994a), the Clenshaw-Curtis rule (Cools and Maerten 1997, Novak and Ritter 1998), the Gauss rules (Novak and Ritter 1997), and the Gauss-Patterson rules (Gerstner and Griebel 1998, Petras 2000) were used as ID input for Smolyak's principle according to (4.1). Furthermore, the higher-order basis polynomials from Section 4.2 were used for an adaptive quadrature algorithm (Bungartz and Dirnstorfer 2003). These techniques can be used advantageously for the computation of stiffness matrix entries in conventional finite element methods and especially in partition of unity methods (Griebel and Schweitzer 2002, Schweitzer 2003). Gerstner

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and Griebel (2003) developed a generalization of the conventional sparse grid approach which is able to adaptively assess the dimensions according to their importance. For further information on numerical quadrature on sparse grids and on data mining, another interesting field of application, we refer to the respective numerical examples in Section 5.3. 5. Numerical experiments In this section, we present a collection of numerical results for different problems solved on sparse grids. We start with the discussion of the basic features of sparse grid methods for PDEs, applied to simpler 2D and 3D model problems. Then we turn to the solution of the Navier-Stokes equations on sparse grids. Finally, we illustrate the potential of sparse grids for problems of a higher dimensionality, here in the context of numerical quadrature and data mining. 5.1. PDE model problems

We start with some PDE model problems to demonstrate the interpolation properties of the Z/2-based sparse grid spaces presented as well as their behaviour concerning the approximation of a PDE's solution. We omit examples with energy-based sparse grids, since the situation with respect to the energy norm in a finite element context is completely clear owing to Cea's lemma, whereas the L2-quality of the finite element solution is still an open question. Hence we choose a standard Ritz-Galerkin finite element setting and use the hierarchical Lagrangian basis polynomials from Section 4.2 in the pregular way, i.e., choosing the local degree as high as possible according to (4.16) up to some given maximum value for p. Note that a Petrov-Galerkin approach with the higher-order basis functions appearing in the approximation space only leads to the same approximation properties. We will not discuss the algorithmic scheme which is applied to ensure the linear complexity of the matrix-vector product, the so-called unidirectional principle (Bungartz 1998), allowing us to treat the single dimensions separately. However, note that, especially in the case of more general operators, this is actually the challenging part of a sparse grid finite element implementation; see Dornseifer (1997), Bungartz and Dornseifer (1998), Schneider (2000), and Achatz (2003a, 20036). Concerning the iterative solution of the resulting linear systems, several multilevel schemes are available, most of them based upon the semi-coarsening approach of Mulder (1989) and Naik and van Rosendale (1993) and its sparse grid extension (Griebel 19916, 1991a, Griebel et al. 19936, Griebel and Oswald 19956, Pflaum 1992, 1998, Bungartz 1997). In the context of solvers, it is important that the influence

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100

0

2

4 6 level n of regulor sparse grid

8

1

Figure 5.1. Condition numbers of the (diagonally preconditioned) stiffness matrix for the sparse grid spaces Vn'

of the polynomial degree starting from p = 2 be only moderate, which is in contrast to the behaviour known from hierarchical polynomials in a por h-p-version context (c/. Zumbusch (1996), for example). Figure 5.1 hints this, showing spectral condition numbers of the stiffness matrix. Our model problems cover regular and adaptively refined sparse grids in two and three dimensions, mainly using the Laplacian as the operator, but also the second-order linear operator with more general coefficients. For measuring the error, we consider the error's discrete maximum norm, the discrete /2-norm, and the error in the centre point P = (0.5,..., 0.5). Where grid adaptation is used, it is based on a sparse grid extension of the dual problem approach (Becker and Rannacher 1996, Eriksson et al. 1996); see Schneider (2000).

The 2D Laplace equation O n O = [0,1]2, let in Q,

Att(x) = 0 sinh(7r)

(5.1)

on

Figure 5.2 shows the smooth solution u(x) of (5.1). We study the accuracy of the hierarchical Lagrangian approach for the regular sparse grid spaces V^r'1' with polynomial degrees p 6 { 1 , . . . , 6 } . Figure 5.3 illustrates the maximum and the /2-error. The effect of the higher-order approximation can be seen clearly. For p = 6, we already get troubles with machine accuracy because of the use of standard double precision floating point numbers.

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Figure 5.2. Solution u(x) = sin(?rxi) • s's"n[T^?' of (5.1).

10'

icr ioJ degrees of freedom N

10*

10

10' 10 J degrees of freedom N

10*

Figure 5.3. Example (5.1): maximum and ^-error for regular sparse grid spaces V\f , p € { 1 , . . . , 6}. In Figure 5.4, a summary of the convergence behaviour with respect to all error norms regarded and for p € { 1 , . . . , 6} is provided. In addition to the error plots, we show the curves of expected sparse grid convergence (sgc) due to the results (3.88) and (4.32), stating £oo (jV)

= O{N ~2

(5.2)

These curves indicate the (asymptotic) behaviour of the e-complexity of sparse grids with respect to the problem of interpolating a given function. For all polynomial degrees presented, we observe a rather striking correspondence of theoretical results concerning the mere quality of interpolation and experimental results for the accuracy of calculating approximate solutions of PDEs. Note that this correspondence is still an open question with respect to the L2- or maximum norm, in contrast to the energy norm for which the answer is provided by Cea's lemma.

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p=4

n-2

§ 10- 10

*

.

»

,

^_

o

I2 Imidpointl

12

Imoximuml

Imidpointl 0

0

101

Imoximuml

102 10 3 degrees of freedom N

10^ 10 J degrees of freedom N

10

10

p=5

§ io-10

10 ,5

10^ 10 J degrees of freedom N

_ ,n0 10

10

O

10

10^ 10 degrees of freedom N

10

10 u

10^ 10 J degrees of freedom N

10

10 J

102 10 3 degrees of freedom N

10'

10=

O

10'

Figure 5.4. Example (5.1): convergence on regular sparse grid spaces Vjf'1] for p G {1,2,3} (left) and p 6 {4,5,6} (right); the solid lines indicate the respective expected sparse grid convergence (sgc) due to the interpolation accuracy.

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Figure 5.5. Solution u(x) = ,

+Q Av?/^.

_0 5\2 of example (5.3).

p=1

sgc(l)

-•

12 Imidpointl Imaximuml

101

10* 10 J degrees of freedom N

10 2 10 3 degrees of freedom N

to4

10 J

Figure 5.6. Example (5.3): convergence on Vn' for p = 1 (left) and p = 2 (right); the solid lines indicate the respective expected sparse grid convergence (sgc; position of curve chosen for clarity). Next, we turn to adaptively refined sparse grids. On 0, = [0,1]2, let Ait(x) = 0 u(x) =

in n, :

—-^—-— :

r^

(5.3)

on dVt.

In x = (—0.1,0.5), •u(x) has a singularity which is outside Cl, but close to the boundary dQ. Figure 5.5 shows the solution u(x). Figure 5.6 illustrates the case of the regular sparse grid spaces Vn ' (left) (2 1) and (right). Obviously, things are not too convincing for reasonable N. However, decisive progress can be made if we apply our /2-adaptivity.

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10

lO* 10 J degrees of freedom N

10

10 J degrees of freedom N

10*

Figure 5.7. Example (5.3): maximum and ^-evvov for adaptive refinement, p € {1,2,6}.

Figure 5.8. Example (5.3): error on regular (left) and adaptive sparse grid (right). Figure 5.7 shows a gain in accuracy with higher p that is comparable to the smooth situation of (5.1). In Figure 5.8, we compare the error on a regular sparse grid and on an adaptively refined one. As expected, the ^-adaptation process reduces the error equally over the whole domain. In contrast to that, regular sparse grids show large errors near the singularity. As for (5.1), the achieved accuracy will be compared to the theoretical results concerning the £-complexity of interpolation on sparse grids. In Figure 5.9, again for p = 1, the correspondence is striking. For p = 6, it seems that the asymptotic behaviour needs bigger values of N to appear. This was to be expected, since our hierarchical Lagrangian basis polynomials of degree p = 6 need at least level 5 to enter the game. However, the adaptation process causes a delayed creation of new grid points in the

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10 J degrees of freedom N

Figure 5.9. Example (5.3): convergence with adaptive mesh refinement for p — 1 (left) and p = 6 (right); the solid lines indicate the respective expected sparse grid convergence (sgc; position of curve chosen for clarity).

!

;

•

*

!

!

•

•

;

;

•

•

•

•

;

:

•

•

!

•

!

•

•

:

•

•

•

!

.1.;.;.;.;.;.;

-

!

•

•

•

•

; _

.: . 1.: |

-

S-

m m

i[i

|

-±±

m-

•H:

7"".'\".'\

—; • i • ;

i

Figure 5.10. Example (5.3): adaptive grids for p = 1 (left) and p = 6 (right). smooth areas. Nevertheless, with the adaptive refinement advancing, the higher-order accuracy comes to fruition. Finally, to get an impression of the adaptation process, Figure 5.10 shows two adaptively refined grids with 7641 grid points (p = 1, left), and 10965 grid points (p = 6, right). With these results supporting the efficiency of the combination of higherorder approximation and adaptive mesh refinement on sparse grids, we close the discussion of the 2D Laplacian and turn to 3D problems.

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The 3D Laplace equation For the 3D case, we restrict ourselves to a smooth model problem. On Q = [0,1]3, let Ait(x) = 0

in

(5.4)

/

ti(x) =

sinh(v 27ra;i) . . -= • sin(7ra;2J • sm(7ra;3) sinh(V27r)

on as I.

For the polynomial degrees p S {1,... ,4}, Figure 5.11 compares the accuracy with respect to the error's maximum norm or its /2-norm, respectively. Again, the effects of the improved approximation properties of our hierarchical polynomial bases are evident. Figure 5.12 shows that we do already come quite close to the asymptotic behaviour predicted for the quality of mere interpolation. This is true in spite of the fact that up to 100000 degrees of freedom are not excessive for a 3D problem. Remember that, although the curse of dimensionality is lessened significantly by the L2- or Loo-based sparse grid spaces Vn and Vn , there is still some d-dependence. Evaluating the respective order terms of the e-complexity derived before and given once more in (5.2) for the 3D case, we observe an exponent of (p + 2) • {d — 1) = 2p + 4 in the log-factor, i.e., an exponent of 12 for polynomial degree p = 4, for example. Nevertheless, as in the 2D case of (5.1) and (5.3), the benefit caused by higher polynomial degrees in combination with sparse grid discretization is again rather impressive.

10 J 10 degrees of freedom N

10 J 1(T degrees of freedom N

Figure 5.11. Example (5.4): maximum and t e r r o r for regular sparse grid spaces Vn' , p £ {1,2,3,4}.

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10 3

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degrees of freedom N

degrees of freedom N

degrees of freedom N

degrees of freedom N

Figure 5.12. Example (5.4): convergence on Ki ' , V G {1,2} (left) and p £ {3,4} (right); the solid lines indicate the respective expected sparse grid convergence (sgc; position of curve chosen for clarity).

Towards greater generality

Figure 5.13 illustrates the effect of using the point functional with respect to the midpoint P of Q for the dual problem-based control of adaptive refinement instead of the ^-adaptation regarded up to now. It can be shown that the resulting error estimator is strongly related to the energy of the basis function living in P. Since we have an O(/ip)-behaviour of the energy norm, the energy decreases of the order O(h2p). Consequently, a convergence behaviour of O(h2p) for the error in the midpoint can be expected. Actually, for the quadratic case, this can be seen from Figure 5.13 for both example (5.1) and even for the root singularity Re {z1/2) on Q := ]0, l[- x ] — 5, ^[. Note that the solid line now indicates an iV~4-behaviour. In addition to (5.3), the results for the root singularity show that we can tackle really singular problems too.

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H . - J . BUNGARTZ AND M . GRIEBEL 10" p=2

Sir;--.

m-2

order 4 »

10"

12 Imidpointl

*

Imoximuml

o

a

n-«5 10°

101

10 2 10 3 degrees of freedom N

10

100 1000 degrees of freedom N

Figure 5.13. Reducing the error in the midpoint for p = 2: example (5.1) (left) and root singularity Re(z 1 / 2 ) (right); the solid lines indicate the expected convergence N~4 (position of curve chosen for clarity).

In Figure 5.14, we present the error resulting from adaptive refinement based on the point functional for example (5.1). Obviously, the error is primarily reduced near the midpoint. The right-hand side of Figure 5.14 shows the underlying adaptive grid consisting of 12 767 grid points. Since all examples have so far been treated with a p-regular approach, we want to present at least one result for a p-asymptotic proceeding, i.e., for the scenario of increasing p for each new level I of grid points occurring. On n= ]0,l[ 2 , let -Ait(x) = - 2

(x\ x\

3x1 (x\ + I)

2x\ 3

in Q,

(5.5)

on

According to the p-asymptotic strategy, in Figure 5.15, the polynomial degree is chosen as p := 1 + 1 up to p = 12. In order to avoid trouble with machine accuracy, we switch to quadruple precision. Of course, striving for twenty-five or more correct digits is not particularly useful. Nevertheless, we obtain a convincing diagram that shows the sub-exponential behaviour of the p-asymptotic strategy which, by the way, performs better than theory had suggested. The last finite element example of this section demonstrates that our approach is not limited to simple operators such as the Laplacian. Here we turn to a variable diffusion coefficient. For a detailed discussion of the general linear elliptic operator of second order, see Achatz (20036), for instance.

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:::.p

•

!:

•

;

!

"

•

Figure 5.14. Example (5.1): reducing the error in the midpoint for p = 2; error (left) and adaptively refined grid (right).

p<=12

io°

IO1

icr

icr

io'

icr

degrees of freedom N

Figure 5.15. Example (5.5): regular sparse grid spaces V-jf' , p-asymptotic proceeding up to p = 12 with quadruple precision. On the unit square, let - V • (A(x)Vw(x)) = /(x)

in n,

(5.6)

where ., s / 4 + sin(27ra;i) + sin(2-7TX2) 1 + 4^2 • (1 — £2) • sin(27ra;i) ' '~ \1 + 4x2 • (1 — X2) • sin(27ra;i) 4 + sin and where /(x) and the Dirichlet boundary conditions are chosen such that u(x) := si

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10 degrees of freedom N

10

10 degrees of freedom N

10

Figure 5.16. Example (5.6): maximum and ^-error for the regular sparse grid spaces Vn , p G {2,4,6}.

P=2 10

10"

10"°

10

-10

0 o

10,-12

10 10

10z 10 J degrees of freedom N

10

,5

,n0

10'

lO* 10 J degrees of freedom N

10

Figure 5.17. Example (5.6): convergence on Vn' for p = 2 (left) and p = 6 (right); the solid lines indicate the respective expected sparse grid convergence (sgc; position of curve chosen for clarity).

is the solution. We present results for p € {2,4,6}. Note that, owing to the smoothness of «(x), we restrict ourselves to the regular sparse grid spaces Vn . Figure 5.16 shows the maximum and the ^-error. Obviously, the higher-order approximation of our hierarchical Lagrangian basis polynomials comes to fruition in the more general situation of example (5.6) too. Figure 5.17 illustrates that we come close to the expected asymptotic behaviour already for moderate values of N.

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So far, all problems have been tackled with a finite element approach. However, finite differences have been successfully used for the numerical solution of PDEs too (see Griebel and Schiekofer (1999), Schiekofer (1998), Schiekofer and Zumbusch (1998), for example). Hence we want to present one model problem that has been solved with finite differences on sparse grids, i.e., on Vn , to be precise. On the unit cube, define the matrix r-3

TX

sin(n)

0.1 • [x\ + 2x2),

the vector S(x) :=

:=

cos(exp(a;i + x2x3)) xxx\

and the scalar function c(x) := 7T • (xi + x 2 + X3). Furthermore, choose the right-hand side /(x) such that it(x) = arctan 100

X\ + X2 +X3

(5.8)

solves - V•

Vu(x) + c(x)w(x) = /(x)

(5.9)

on the unit cube with Dirichlet boundary conditions. Figure 5.19 (overleaf) illustrates the solution u(x) due to (5.8) for the two X3-values 0.5 and 0.875. Numerical results are presented in Figure 5.18.

.1e-3 1e-05.1e3

.1e4

.1e5

1e+05

Figure 5.18. Example (5.9), finite differences, regular sparse grid spaces Vd : maximum ^-error as well as errors in Pi = (0.5,0.5,0.5) and P 2 = (1/TT, 1/TT, 1/TT).

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o.o

0.2

0.0

0.2

Figure 5.19. Example (5.9), finite differences, regular sparse grid spaces Vn • solution M(X) and its contour lines for 2:3 = 0.5 (top) and £3 = 0.875 (bottom).

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233

5.2. Flow problems and the Navier-Stokes

equations

We now consider the application of an adaptive sparse grid method to the solution of incompressible flow problems. The governing relations are given by the Navier-Stokes equations dtu + V - ( u O u ) = f - Vp + ^Au,

(x,i) EQ, X [0,T],

(5.10)

V • u = 0, u(x,0) = u°(x), here in primitive variables, with velocity u = (u,v,w)T and pressure p. The parameter v denotes the kinematic viscosity; f is a prescribed volume force. The system has to be completed by proper initial and boundary conditions. We apply the the pressure correction scheme (Chorin 1968, Temam 1969, Bell, Colella and Glaz 1989). To this end, the discretization of the NavierStokes equations is split into two subproblems. Transport step. Calculate the auxiliary velocity u " + 1 by f.n+1 _

nn

— + C ( u n , u n - \ ...) = fn+1 - Vpn + i/Au n+1 .

(5.11)

k Here, C(u n , u™" 1 ,...) resembles a stabilized, space-adaptive discretization of V • (u (g> u) by means of a Petrov-Galerkin/collocation approach using sparse grid interpolets. Adaptive refinement or coarsening takes place in every time-step steered by the size of the actual coefficients of the multiscale representation of the current solution approximation; see Griebel (1998), Koster (2002) and Griebel and Koster (2000) for further details. Projection step. Calculate u n + 1 , pn+1 as the solution of ,,n+l _ f,n+l

= - V ( p n + 1 - pn),

(5.12)

V • u n + 1 = 0. Instead of the simple Euler scheme, higher-order time discretization techniques can be applied analogously. Of course, both subproblems have to be augmented by boundary conditions - not only for u and u, but also for the pressure p. The (in general non-physical) pressure boundary conditions especially are the subject of controversy (c/. Gresho and Sani (1987), Karniadakis and Sherwin (1999), Prohl (1997)), which is beyond the scope of this paper. In the following, we will assume periodic boundary conditions for u and p. The saddle point problem (5.12) is treated by solving the Schur complement equation vu, k

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(5.13)

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H . - J . BUNGARTZ AND M . GRIEBEL

followed by the correction of the velocity u n+l =

,n+l

fi

(5.14)

To this end, a weakly divergence-free adaptive discretization of V • V is applied together with sparse grid interpolets. The solver involves a transform to the lifting interpolets of Section 4.4, a diagonal scaling, and a backtransform; see Koster (2002) and Griebel and Koster (2000) for further details. Merging of modons Now, we apply the adaptive version (Griebel and Koster 2000, Koster 2002, Griebel and Koster 2003) of this sparse grid interpolet solver to the model problem of the interaction of three vortices in a 2D flow. Here we use the interpolets of Section 4.3 with N = 6. The initial velocity is induced by three vortices, each with a Gaussian vorticity profile w(x, 0) = UIQ + ^

Ui exp

-

X-Xj

x

6

[0,1]2

The first two vortices have the same positive sign u\, UJ2 > 0, and the third has a negative sign; see Figure 5.20. The different parameters UQ, uJi, and at are chosen such that the mean value of u(. ,0) vanishes and that u(., 0)L fo 112 is almost UQ to allow for periodic boundary conditions. Owing to the induced velocity field, the three vortices start to rotate around each other. In a later stage, the two same-sign vortices merge, which leads to a configuration of two counter-rotating vortices. This process is the basic mechanism in 2D turbulence, and it takes place, e.g., in the shear layer problem of the next subsection or during the convergence of OJ to the solution of the Joyce-Montgomery equation (Chorin 1998) for random initial vorticity fields.

Figure 5.20. Initial configuration for the three vortices' interaction.

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Ill

Figure 5.21. Isolines of u) at t = 5, 10, 15, 35 (from top to bottom). Left: adaptive wavelet solver (e = 4 • 10~ 4 ). Right: adaptive sparse grid.

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In our numerical experiments, the viscosity was set to v = 3.8 • 10 6 . The maximum velocity of the initial condition is 7 • 10~2, which corresponds to a Reynolds number of ~ 55200. For the time discretization in the Chorin projection scheme, we used the third-order Adams-Bashforth scheme. The time-step used was dt — 10~2. The values of the threshold parameter e were set to {8, 4, 1} • 10~4, and the finest level 1 was limited to (10,10) to avoid very fine time-steps due to the CFL-condition. In Figure 5.21 (page 235), the contour lines of the vorticity of the adaptive solutions are shown. The contour levels are equally spaced from —1.5 to 3, which is approximately the minimum/maximum value of w(., 0). Obviously, our method recognizes the arising complicated flow patterns in space and time (see Figure 5.21) and devotes many more degrees of freedom to these than to smooth regions. Note that the left-hand pictures of Figure 5.21 are enhanced by 20%. Further comparisons of these results with a conventional Fourier spectral solution showed that the results of the adaptive wavelet solver are quite accurate. But to achieve the same accuracy, only a small number of degrees of freedom was needed by our adaptive sparse grid approach - less than 1% compared with the Fourier spectral technique. 2D shear layer flow

Now we apply the adaptive wavelet solver to a shear layer model problem. The initial configuration of the temporally developing shear layer is a velocity field with a hyperbolic-tangent profile u(y) = Ut&nh(2y/5), v = 0, where 5 is the vorticity shear layer thickness S = 2t//max(du/dy); see Figure 5.22. This initial condition is an exact stationary solution of the Navier-Stokes equations for the case v = 0, i.e., for inviscid fluids. However, from linear stability analysis this initial configuration is known to be

u(y) = U tanh y/S

-1/2 Figure 5.22. Velocity profile of the initial condition for the shear flow model problem. The thickness of the boundary layer is defined as 5 = 2(7/max^ \dyu(y)\.

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SPARSE GRIDS

Figure 5.23. Isolines of the rotation for the values {0.1, 0.2, 0.3, 0.5, 0.75, 1.0,..., 2.25, 2.5} at time t = 4, 8, 16, 36. Left: isolines of computed rotation. Right: adaptive sparse grid.

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H . - J . BUNGARTZ AND M. GRIEBEL

inviscidly unstable. Slight perturbations are amplified by Kelvin-Helmholtz instabilities and vortex roll-up occurs (Michalke 1964). In our numerical experiments, the vorticity thickness 6 was chosen so that ten vortices should develop in the computational domain [0,1]2. The maximum velocity was U = 1.67 • 10~2, and the viscosity was v = 3.8 • 10~6. The instabilities were triggered by a superimposed weak white noise in the shear layer. The numerical simulations were applied to the periodized version of the problem with two shear layers on [0,1] x [0,2]. We used the same adaptive sparse grid solver as in the previous experiment, based on interpolets with N = 6, a stabilized Petrov-Galerkin collocation, and Chorin's projection method. The values of the threshold parameter e were set to {12, 8, 4}-10~4 and the finest level 1 was limited to (10,10). The time-step was 5 • 10~3. On the left-hand side of Figure 5.23 (page 237), the resulting rotation w = V x u for e = 8 • 10~4 is given. The evolution of the sparse grid points is shown on the right-hand side of Figure 5.23. The initial velocity is not very smooth owing to the white noise added for triggering the instability. Therefore, a large number of degrees of freedom (DOF) is required to resolve the initial velocity in the shear layer sufficiently well. Then, in a first phase, the diffusion makes the velocity smooth very fast, which leads to the strong decay of the number of DOF. This process is stopped by the development of the Kelvin-Helmholtz instabilities leading to an increase of the number of DOF (4 < t < 10). In the last phase (t > 10), the number of coherent vortices constantly decreases by successive merging. The final state comprises two counter-rotating vortices which dissipate. Besides the generation and the roll-up of Kelvin-Helmholtz vortices, we also see that the vortices merge with time. This process, in which a few large vortices are created from many small ones, is typical for 2D flows; cf. Chapter 4.6 of Chorin (1998). It comes with an energy transfer from fine to coarse structures, i.e., from fine to coarse levels in the sense of an isotropic multiscale representation (Novikov 1976). Thus 2D turbulence behaves fundamentally differently to 3D turbulence. Here, predominantly an energy transfer from coarse to fine levels occurs, that is, coarse structures decay to fine structures. The maximal level was limited to 10 to avoid overly small time-steps resulting from the CFL condition. 3D shear layer flow

As seen in the previous section, vortices with the same rotation direction merge successively in 2D flows. In 3Dflows,however, this effect is no longer present. Here the additional third dimension allows for vortex tubes. They can be bent, folded, and stretched over time until they break apart. Their diameter reduces during stretching, and therefore smaller-scale structures

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SPARSE GRIDS

1

Figure 5.24. Iso-surfaces of the rotation for the 3D turbulent shear layer problem at time 0,15,30,60,90,150.

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are created. In this way energy is transported to finer scales. This energy cascade proceeds recursively until the structures are so tiny and the associated relative Reynolds number is so small that energy is annihilated by dissipation. The different phases of a developing 3D shear layer are shown in Figure 5.24 (page 239), from which this increase of complexity in the flow becomes clear. The initial conditions for this calculation are the 3D analogues of the initial conditions for the previous 2D example. However, viscosity was increased, i.e., we have set v = 5.0410"6. The remaining parameters were the same as for the 2D calculation. 5.3. Problems of high dimensionality We now turn to higher-dimensional problems. To this end, a few general remarks are helpful. First, the sparse grid approach is limited to (topologically) quadrilateral domains due to its tensor product nature. Complicated domains must be built up by gluing together quadrilateral patches on which locally sparse grids are employed, or by appropriate mapping techniques. Now, in higher dimensionalities, this question of the shape of the domain is not as important as in the 2D and 3D case, since complicated domains typically do not appear in applications. Conceptually, besides Md itself, we use mainly hypercubes [—a,a]d, a > 0, and their straightforward generalizations using different values of a for each coordinate direction, as well as the corresponding structures in polar coordinates. These domains are of tensor product structure and cause no difficulties to the sparse grid approach. Second, complexity issues are still crucial in higher-dimensional applications. In the simplest case of a sufficiently smooth function with bounded mixed derivatives, the cost and error estimates for the Lp-norms still possess (log Ar)d~1-terms which depend exponentially on the dimensionality d. This limits the practical applications of the sparse grid method for PDEs to at most 18 dimensions at present. Furthermore, in the nonsmooth case, adaptivity towards a singularity is a difficult task in higher dimensions. In principle, it can be done, as demonstrated in the examples of the previous sections. However, an implementation of refinement strategies, error indicators, and solvers with optimal complexity is more difficult than in the low-dimensional case. Hence, utmost care has to be taken in the implementation not to obtain order constants in the work count that depend exponentially on d. For a simple example of this problem, just consider the stiffness matrix for d-linear finite elements on a uniform grid. There, the number of nonzero entries grows like 3 d for a second-order elliptic PDE with general coefficient functions. Also the work count for the computation of each entry by a conventional quadrature scheme involves terms that grow exponentially with d. Here, at least in certain simple cases like the Laplacian, an algorithmic scheme can be applied, which is based on the

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241

so-called unidirectional principle (Bungartz 1998). It allows us to treat the single dimensions separately and results in linear complexity in both the degrees of freedom and d, for the matrix-vector product in a Galerkin sparse grid approach. However, note that, especially in the case of more general nonseparable operators, this is the challenging part of a sparse grid finite element implementation. The next question concerns the sources of high-dimensional problems and PDEs. Here, besides pure integration problems stemming from physics and finance, typically models from the stochastic and data analysis world show up. For example, high-dimensional Laplace/diffusion problems and high-dimensional convection-diffusion problems result from diffusion approximation techniques and the Fokker-Planck equation. Examples are the description of queueing networks (Mitzlaff 1997, Shen, Chen, Dai and Dai 2002), reaction mechanisms in molecular biology (Sjoberg 2002, Elf, Lotstedt and Sjoberg 2001), the viscoelasticity in polymer fluids (Rouse 1953, Prakash and Ottinger 1999, Prakash 2000), or various models for the pricing of financial derivatives (Reisinger 2003). Furthermore, homogenization with multiple scales (Allaire 1992, Cioranescu, Damlamian and Griso 2002, Matache 2001, Matache and Schwab 2001, Hoang and Schwab 2003) as well as stochastic elliptic equations (Schwab and Todor 2002, 2003) result in high-dimensional PDEs. Next, we find quite highdimensional problems in quantum mechanics. Here, the dimensionality of the Schrodinger equation grows with the number of considered atoms. Sparse grids have been used in this context by Garcke (1998), Garcke and Griebel (2000), Yserentant (2004) and Hackbusch (2001). In the following, we illustrate the potential of sparse grids for problems of a higher dimensionality in the context of numerical quadrature and data mining.

Classification and regression in data mining Data mining is the process to find hidden patterns, relations, and trends in large data sets. It plays an increasing role in commerce and science. Typical scientific applications are the post-processing of data in medicine (e.g., CT data), the evaluation of data in astrophysics (e.g., telescope and observatory data), and the grouping of seismic data, or the evaluation of satellite pictures (e.g., NASA earth observing system). Financial and commercial applications are perhaps of greater importance. With the development of the internet and e-commerce, there are huge data sets collected, more or less automatically, which can be used for business decisions and further strategic planning. Here, applications range from contract management to risk assessment, from the segmentation of customers for marketing to fraud detection, stock analysis and turnover prediction.

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Usually, the process of data mining (or knowledge discovery) can be separated into the planning step, the preparation phase, the mining phase (i.e., machine learning) and the evaluation. To this end, association-analysis classification, clustering, and prognostics are to be performed. For a thorough overview of the various tasks arising in the data mining process, see Berry and Linoff (2000) and Cios, Pedrycz and Swiniarski (1998). In the following, we consider the classification problem in detail. Here, a set of data points in d-dimensional feature space is given together with a class label in { — 1,1}, for example. From these data, a classifier must be constructed which allows us to predict the class of any newly given data point for future decision making. Widely used approaches are nearest neighbour methods, decision tree induction, rule learning, and memory-based reasoning. There are also classification algorithms based on adaptive multivariate regression splines, neural networks, support vector machines, and regularization networks. Interestingly, these latter techniques can be interpreted in the framework of regularization networks (Girosi, Jones and Poggio 1995). This approach allows a direct description of the most important neural networks, and it also allows for an equivalent description of support vector machines and n-term approximation schemes (Girosi 1998). We follow Garcke, Griebel and Thess (2001) and consider a given set of already classified data (the training set):

S={(xi,yi)GRdxR}fL1. We now assume that these data have been obtained by sampling of an unknown function / which belongs to some function space V defined over Rd. The sampling process was disturbed by noise. The aim is now to recover the function / from the given data as well as possible. This is clearly an ill-posed problem since infinitely many solutions are possible. To get a well-posed, uniquely solvable problem, we have to restrict / . To this end, regularization theory (Tikhonov and Arsenin 1977, Wahba 1990) imposes an additional smoothness constraint on the solution of the approximation problem, and the regularization network approach considers the variational problem

mm R(f) fev w '

with m m

E

i), W)

Here, C(.,.) denotes an error cost function which measures the interpolation error and <&(/) is a smoothness functional which must be well defined for / € V. The first term enforces closeness of / to the data, the second term enforces smoothness of / , and the regularization parameter A balances these

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two terms. We consider the case C(x, y) = (x- yf

and

$(/) = ||P/|||

with Pf = V / .

The value of A can be chosen according to cross-validation techniques (Allen 1972, Golub, Heath and Wahba 1979, Utreras 1979, Wahba 1985) or to some other principle, such as structural risk minimization (Vapnik 1982). We find exactly this type of formulation in the case d = 2,3 in many scattered data approximation methods (see Arge, Daehlen and Tveito (1995) and Hoschek and Lasser (1992)), where the regularization term is usually physically motivated. We now restrict the problem to a finite-dimensional subspace Vjv C V. The function / is then replaced by N

).

(5.16)

Here {tpj}^ should span V/v and preferably should form a basis for V/v. The coefficients {aj}jL1 denote the degrees of freedom. Note that the restriction to a suitably chosen finite-dimensional subspace involves some additional regularization (regularization by discretization) which depends on the choice of V/v- In this way we obtain from the minimization problem a feasible linear system. We thus have to minimize 1

M

{

^

\l2,

fNevN

(5.17)

in the finite-dimensional space V/v. We plug (5.16) into (5.17) and obtain M / N

\

2

N

(5.18)

Differentiation with respect to a^, k = 1 , . . . , N, gives nt*c\

o=

r*ri

v

-—- i _—.

\

N

dak (5.20)

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This is equivalent to (k = 1 , . . . , N) N

N 1 v—r

M ^—^

M ,

N

.

.

1

(5.21) and we obtain (k = 1,..., N

N) M

M

_

(5.22)

i=i

In matrix notation, we end up with the linear system (XC + B-BT)a = By.

(5.23)

Here, C is a square N x N matrix with entries Cj^ — M • j , k = 1 , . . . , N, and B is a rectangular N x M matrix with entries Bjti =
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SPARSE GRIDS 100 98 96

5

94

s

92

j* 90

Level 7 Train Level 7 Test ....... Level 8 Train •a Level 8 Test „„.— Level 9 Train .*.... Level 9 Test - • • - • Level 10 Train - • Level 10 Test

\

V| \

"i

V *^ \ \

"""*•""""

88 0.0

0.25

1e-05

0.0001 0.001 Lambda

0.01

Figure 5.25. Left: chessboard data set, combination technique with level 10, A = 4.53999 • 10" 5 . Right: plot of the dependence on A (in logscale) and level.

To evaluate our method we give the correctness rates on testing data sets if available and the 10-fold cross-validation results on training and testing data sets. For the 10-fold cross-validation we proceed as follows. We divide the training data into 10 equally sized disjoint subsets. For % = 1 to 10, we pick the ith of these subsets as a further testing set and build the sparse grid combination classifier with the data from the remaining nine subsets. We then evaluate the correctness rates of the current training and testing set. In this way we obtain ten different training and testing correctness rates. The 10-fold cross-validation result is just the average of these ten correctness rates. For further details, see Stone (1974). For a critical discussion on the evaluation of the quality of classifier algorithms, see Salzberg (1997). We first consider 2D problems with small sets of data that correspond to certain structures. Then we treat problems with huge sets of synthetic data with up to 5 million points. The first example is taken from Ho and Kleinberg (1996) and Kaufman (1999). Here, 1000 training data points were given which are more or less uniformly distributed in Q = [0,1]2. The associated data values are plus one or minus one depending on their location in Q such that a 4 x 4 chessboard structure appears: see Figure 5.25 (left). We computed the 10-fold crossvalidated training and testing correctness with the sparse grid combination method for different values of the regularization parameter A and different levels n. The results are shown in Figure 5.25 (right). We see that the 10-fold testing correctness is well around 95% for values of A between 3 • 10~5 and 5 • 10~3. Our best 10-fold testing correctness was 96.20% on level 10 with A = 4.54 • 10~5. The chessboard structure is thus reconstructed with less than 4% error.

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Figure 5.26. Spiral data set, sparse grid with level 6 (left) and 8 (right), A = 0.001. Another 2D example with structure is the spiral data set, first proposed by Wieland; see also Fahlmann and Lebiere (1990). Here, 194 data points describe two intertwined spirals: see Figure 5.26. This is surely an artificial problem, which does not appear in practical applications. However it serves as a hard test case for new data mining algorithms. It is known that neural networks can have severe problems with this data set, and some neural networks can not separate the two spirals at all. In Figure 5.26 we give the results obtained with our sparse grid combination method with A = 0.001 for n = 6 and n — 8. Already for level 6, the two spirals are clearly detected and resolved. Note that here only 577 grid points are contained in the sparse grid. For level 8 (2817 sparse grid points), the shape of the two reconstructed spirals gets smoother and the reconstruction gets more precise. The BUPA Liver Disorders data set from the Irvine Machine Learning Database Repository (Blake and Merz 1998) consists of 345 data points with six features plus a selector field used to split the data into two sets with 145 instances and 200 instances, respectively. Here we only have training data and can therefore only report our 10-fold cross-validation results. No comparison with unused test data is possible. We compare with the two best results from Lee and Mangasarian (2001), the smoothed support vector machine (SSVM) introduced therein, and the feature selection concave minimization (FSV) algorithm due to Bradley and Mangasarian (1998). Table 5.1 gives the results for the 10-fold correctness. Our sparse grid combination approach performs on level 3 with A = 0.0625 at 69.23% 10-fold testing correctness. But our other results were also in this range. Our method performs only slightly worse here than the SSVM but

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Table 5.1. Results for the BUPA liver disorders data set.

train. % test. %

sparse grid combination method level 1 level 2 level 3 level 4 A = 0.0001 A = 0.1 A = 0.0625 A= 0.625

SSVM

FSV

70.37 70.33

68.18 65.20

83.28 66.89

79.54 66.38

90.20 69.23

88.66 68.74

Table 5.2. Results for a 6D synthetic massive data set, A — 0.01.

# of points

training corr.

testing corr.

total time (sec)

data matrix time (sec)

#of iterat.

level 1

50 000 500 000 5 million

87.9 88.0 88.0

88.1 88.1 88.0

158 1570 15933

152 1528 15514

41 44 46

level 2

50 000 500 000 5 million

89.2 89.4 89.3

89.3 89.2 89.2

1155 11219 112656

1126 11022 110772

438 466 490

clearly better than FSV. Note that the results for the robust linear program (RLP) algorithm (Bennett and Mangasarian 1992), the support vector machine using the 1-norm approach (SVMn.^), and the classical support vector machine (SVMji.p) (Bradley and Mangasarian 1998, Cherkassky and Mulier 1998, Vapnik 1995) were reported to be somewhat worse in Lee and Mangasarian (2001). Next, we produced a 6D data set with 5 million training points and 20 000 points with DatGen (Melli 2003) for testing. We used the call datgen - r l -X0/100,R,0 :0/100,R,0:0/100,R,0:0/100,11,0:0/200, R,0:0/200,R.O -R2 -C2/4 -D2/5 -T10/60 -0502 0000 -p -e0.15. The results are given in Table 5.2. On level one, a testing correctness of 88% was achieved already, which is quite satisfying for this data. We see that really huge data sets of 5 million points could be handled. We also give the CPU time which is needed for the computation of the matrices G\ = B\- Bf. Here, more than 96% of the computing time is spent on the matrix assembly. Again, the execution times scale linearly with the number of data points.

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Analogously, we can at present deal with up to 18-dimensional problems before storage limitations on our available computers stop us. This moderate number of dimensions, however, is still sufficient for many practical applications. In very high-dimensional data, there exist mostly strong correlations and dependencies between the dimensions. Then, in a preprocessing step, the effective dimensions can be determined, for instance, by means of principal component analysis, and the dimension can be reduced in many cases to 18 or less. To reach 18 dimensions, a generalization of the sparse grid combination technique to simplicial basis functions (Garcke and Griebel 2001a, 2002) is needed. Note that the sparse grid combination technique can be parallelized in a straightforward way (Garcke and Griebel 20016, Garcke, Hegland and Nielsen 2003). Finally, the sparse grid technique can be used in a dimensionadaptive fashion (Hegland 2002, 2003), which further enhances the method's capabilities. This approach will be discussed for the example of numerical integration now.

Integration The computation of high-dimensional integrals is a central part of computer simulations in many application areas such as statistical mechanics, financial mathematics, and computational physics. Here, the arising integrals usually cannot be solved analytically, and thus numerical approaches are required. Furthermore, often a high-accuracy solution is needed, and thus such problems can be computationally quite challenging even for parallel supercomputers. Conventional algorithms for the numerical computation of such integrals are usually limited by the curse of dimensionality. However, for special function classes, such as spaces of functions which have bounded mixed derivatives, Smolyak's construction (Smolyak 1963) (see (4.1)) can overcome this curse to a certain extent. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of appropriate ID formulas. In this way, the number of function evaluations and the numerical accuracy become independent of the dimension of the problem up to logarithmic factors. Smolyak's construction is simply our sparse grid approach. It has been applied to numerical integration by several authors, using the midpoint rule (Baszenski and Delvos 1993), the rectangle rule (Paskov 1993), the trapezoidal rule (Bonk 1994a), the Clenshaw-Curtis rule (Cools and Maerten 1997, Novak and Ritter 1998), the Gauss rules (Novak and Ritter 1997), and the Gauss-Patterson rules (Gerstner and Griebel 1998, Petras 2000) as the ID basis integration procedure. The latter approach, in particular, achieves the highest polynomial exactness of all nested quadrature formulas and shows very good results for sufficiently smooth multivariate integrands. Further studies have been

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made concerning extrapolation methods (Bonk 1994 a), discrepancy measures (Frank and Heinrich 1996), and complexity questions (Wasilkovski and Wozniakowski 1999). There is also a large variety of other methods for the numerical integration of multivariate functions such as Monte Carlo and quasi-Monte Carlo methods (Niederreiter 1992), lattice rules (Sloan and Joe 1994), adaptive subdivision methods (Genz and Malik 1980, Dooren and Ridder 1976), and approximation methods based on neural networks (Barron 1994, Mhaskar 1996). Each of these methods is particularly suitable for functions from a certain function class and has a complexity which is then also independent or nearly independent of the problem's dimensionality. Despite the large improvements of the quasi-Monte Carlo and sparse grid methods over the Monte Carlo method, their convergence rates will suffer more and more with rising dimension owing to their respective dependence on the dimension in the logarithmic terms. Therefore, one aim of recent numerical approaches has been to reduce the dimension of the integration problem without affecting the accuracy unduly. In some applications, the different dimensions of the integration problem are not equally important. For example, in path integrals the number of dimensions corresponds to the number of time-steps in the time discretization. Typically, the first steps in the discretization are more important than the last steps since they determine the outcome more substantially. In other applications, although the dimensions seem to be of the same importance at first sight, the problem can be transformed into an equivalent one where the dimensions are not. Examples are the Brownian bridge discretization or the Karhunen-Loeve decomposition of stochastic processes. Intuitively, problems where the different dimensions are not of equal importance might be easier to solve: numerical methods could concentrate on the more important dimensions. Interestingly, complexity theory also reveals that integration problems with weighted dimensions can become tractable even if the unweighted problem is not (Wasilkovski and Wozniakowski 1999). Unfortunately, classical adaptive numerical integration algorithms (Genz and Malik 1980, Dooren and Ridder 1976) cannot be applied to high-dimensional problems, since the work overhead in order to find and adaptively refine in important dimensions would be too large. To this end, a variety of algorithms have been developed that try to find and quantify important dimensions. Often, the starting point of these algorithms is Kolmogorov's superposition theorem: see Kolmogorov (1956, 1957). Here, a high-dimensional function is approximated by sums of lowerdimensional functions. A survey of this approach from the perspective of approximation theory is given in Khavinson (1997). Further results can be found in Rassias and Simsa (1995) and Simsa (1992). Analogous ideas are followed in statistics for regression problems and density estimation.

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ex +ey

10 • e"* 2 + 10 old active

1

2

3

4

5

6

t x

2 2 e-Wx -10y

old active

+ X

old active

7

+

+ x

+

+

+

+ x

+

+

+

+

+ x

3

4

5

1

'

' I' *

.

.

2

+ x

.

OJ 07

• • •

06 Oi m 03 Hi

. I • i, • I. . U

00

07

01

M

Figure 5.27. The resulting index sets and corresponding sparse grids for TOL = 10~ 15 for some isotropic test functions.

Here, examples are so-called additive models (Hastie and Tibshirani 1990), multivariate adaptive regression splines (MARS) (Friedman 1991), and the ANOVA decomposition (Wahba 1990, Yue and Hickernell 2002); see also Hegland and Pestov (1999). Other interesting techniques for dimension reduction are presented in He (2001). If the importance of the dimensions is known a priori, techniques such as importance sampling can be applied in Monte Carlo methods (Kalos and Whitlock 1986). For the quasi-Monte Carlo method, a sorting of the dimensions according to their importance leads to a better convergence rate (yielding a reduction of the effective dimension). The reason for this is the better distributional behaviour of low-discrepancy sequences in lower dimensions than in higher ones (Caflisch, Morokoff and Owen 1997). The sparse grid method, however, a priori treats all dimensions equally and thus gains no immediate advantage for problems where dimensions are of different importance. In Gerstner and Griebel (2003), we developed a generalization of the conventional sparse grid approach which is able to adaptively assess the dimensions according to their importance, and thus reduces the dependence of the computational complexity on the dimension. This is quite in the spirit of Hegland (2002, 2003). The dimension-adaptive algorithm tries to find

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*-**+10.e-v> old + active x

1 2

3

4

5

6

7

1

- • • • ' I

• • ' • ! ' •

•'

w

•

.

i

.

-

t

.

v

.

; •

.

. •

.

j

.

•.

.

•

•

•

•

*

.

v

.

i

.

i

.

M

OS

04

03

..

•

:

•

. . . . . . . . . 1...., 0-3

0.1

U

0.8

0.7

OJ

Figure 5.28. The resulting index sets and corresponding sparse grids for TOL = 10~ 15 for some anisotropic test functions.

important dimensions automatically and adapts (places more integration points) in those dimensions. To achieve this efficiently, a data structure for a fast bookkeeping and searching of generalized sparse grid index sets is necessary. We will now show the performance of the dimension-adaptive algorithm in numerical examples. First, we consider some 2D test functions, which allows us to show the resulting grids and level index sets. In these cases, the exact value of the integral is known (or can be computed quickly). The second example is a path integral of 32 dimensions in which the integral value is also known beforehand. The third example is a 256-dimensional application problem from finance where the exact value is unknown. We use the wellknown Patterson formulas (Patterson 1986) for univariate quadrature in all examples. These were shown to be a good choice for the sparse grid construction by Gerstner and Griebel (1998). Let us first consider simple combinations of exponential functions defined over [0,1]2. In Figures 5.27 and 5.28 we depict the level index sets used by the algorithm as well as the resulting dimension-adapted sparse grids for some isotropic and some anisotropic functions, respectively. In these examples, the selected error threshold is TOL = 10~15.

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The first example is a sum of ID functions. The dimension-adaptive algorithm correctly selects no indices in joint dimensions. Also, more points are placed in the z-direction than in the y-direction in the anisotropic case. Clearly, here the conventional sparse grid would use too many points in joint directions. The second example is not separable, nor does it have product structure. The resulting level index set is almost triangular, like the conventional sparse grid. However, the dimension-adaptive algorithm chooses to select more points on the axes, while the conventional sparse grid would have used too many points in the interior. In our experience, many application problems fall in this category, which we would call nearly additive. The third example is the well-known Gaussian hat function, and has product structure. In this example, many points in joint dimensions are required, and the conventional sparse grid would have placed too few points there. At first sight, this is a surprising result, since product functions should be more easily integrable by a tensor product approach. However, the mixed derivatives of the Gaussian can become large even if they are bounded, which reduces the efficiency of both the conventional sparse grid and the dimension-adaptive approaches. Let us now approach some higher-dimensional problems. We will first consider an initial value problem given by the linear PDE du

1 d2u

,

,

with initial condition u(x,0) = f(x). The solution of this problem can be obtained with the Feynman-Kac formula as

where £ represents a Wiener path starting at £(0) = x. The expectation EXio can be approximated by a discretization of time using a finite number of time-steps ti = i • At with At = t/d. The integral in the exponent is approximated by a ID quadrature formula such as a sufficiently accurate trapezoidal rule. The most natural way to discretize the Wiener path is by a random walk, i.e., by the recursive formula & = &-i + >/At zfc, where £o = % and Zk are normally distributed random variables with mean zero and variance one. The dimensions in the random walk discretization are all of the same importance since all the variances are identical to At.

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SPARSE GRIDS 0.1

Sparse Grid Random Walk — • — Sparse Grid Brownian Bridge —*-— Dimension-Adaptive Random Walk - • * • • • , Dimension-Adaptive Brownian Bridge — o —

0.01 ^ • s .

Maximum Level with Random Walk Maximum Level with Brownian Bridge

I

0.001

5

0.0001 le-05 10-06

"A

1e-07

^

10-08

_

4

J

3

I I :

:

2 1 0

1

10

100

1000 100001000001e+06 1e+07 1e+08 1e+09 Function Evaluations

15

20

Dimension

Figure 5.29. Computational results for the path integral (d = 32): integration error vs. number of function evaluations (left) and maximum level over all dimensions (sorted) for the dimensionadaptive algorithm with Brownian bridge discretization (right). In the Brownian bridge discretization (Caflisch et al. 1997), however, the path is discretized using a future and a past value

6s = ^feStarting with £0 := x and £d : = x + \fi Zd, the subsequent values to be computed are £d/2, fd/4, 6d/4. £d/8> £,3d/8, ^5d/8, &d/8, £id/i6> ^3d/i6> • • • with corresponding h = 1/2, 1/4, 1/4, 1/8, 1/8, 1/8, 1/8, 1/16, 1/16,.... The Brownian bridge leads to a concentration of the total variance in the first few steps of the discretization and thus to a weighting of the dimensions. Let us now consider the concrete example (Morokoff and Caflisch 1995) v(x,t) = with initial condition u(x,0) = $, 1 . The exact solution is then

The results for d = 32, t = 0.02 and x = 0 are shown on the left-hand side of Figure 5.29. We see the integration error plotted against the number of function evaluations in a log-log scale. Here, the conventional sparse grid method is compared with the dimension-adaptive algorithm for the random walk and Brownian bridge discretizations. In this example, the conventional sparse grid is for the random walk discretization obviously close to the optimum, since the dimension-adaptive method cannot improve on the performance. The conventional sparse grid gains no advantage from the Brownian bridge discretization, but the convergence rate of the dimension-adaptive algorithm

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is dramatically improved. Note that the convergence rate of the quasi-Monte Carlo method (with Brownian bridge) is comparable to that of the conventional sparse grid approach (Morokoff and Caflisch 1995, Gerstner and Griebel 1998). On the right-hand side of Figure 5.29, we plot the maximum level per dimension of the final level index set of the dimension-adaptive method with and without the Brownian bridge discretization. Here the dimensions are sorted according to this quantity. For the Brownian bridge discretization, the maximum level decays with the dimension. This shows that only a few dimensions are important and thus contribute substantially to the total integral while the other dimensions add significantly less. Let us now consider a typical collateralized mortgage obligation problem, which involves several tranches, which in turn derive their cash flows from an underlying pool of mortgages (Caflisch et al. 1997, Paskov and Traub 1995). The problem is to estimate the expected value of the sum of present values of future cash flows for each tranche. Let us assume that the pool of mortgages has a 211 year maturity and cash flows are obtained monthly. Then the expected value requires the evaluation of an integral of dimension d = 256 for each tranche,

Juc with Gaussian weights g(£i) = {2na2) 1//2e ^/2
ik :=

Koeil+-+^io

with a certain normalizing constant KQ and an initial interest rate IQ (for details see the first example in Caflisch et al. (1997) and compare Gerstner and Griebel (1998) and Paskov and Traub (1995)). Again the interest rates can be discretized using either a random walk or the Brownian bridge construction. For the numerical computation, the integral over Rd is transformed into an unweighted integral on [0, l]d with the help of the inverse normal distribution. In Figure 5.30, we again compare the conventional sparse grid method with the dimension-adaptive method for the random walk and the Brownian bridge discretization. The error is computed against an independent quasiMonte Carlo calculation. Note also that in this example the convergence rate of the conventional sparse grid approach is comparable to the quasiMonte Carlo method (Gerstner and Griebel 1998). We see again that a weighting of the dimensions does not influence the convergence of the conventional sparse grid method. But for the dimensionadaptive method the amount of work is again substantially reduced (by several orders of magnitude) for the same accuracy when the Brownian bridge discretization is used, and thus higher accuracies can be obtained.

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Sparse Grid Random Walk Sparse Grid Browntan Bridge Dimension-Adaptive Random Walk Dimension-Adaptive Brownian Bridge

0.001

• • -

Maximum Level with Random Walk Maximum Level with Brownian Bridge

0.0001

I

_

4

J

3

1

10-05

LU

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2

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1

"

I ; i

0 1e-08 1

10

100

1000 10000 100000 1e+06 Function Evaluations

1e+07

1e+08

0

100

150

200

300

Dimension

Figure 5.30. Computational results for the CMO problem (d = 256): integration error vs. number of function evaluations (left) and maximum level over all dimensions (sorted) for the dimension-adaptive algorithm with and without Brownian bridge discretization (right). In this example the dimension-adaptive method also gives better results than the conventional sparse grid method for the random walk discretization. This implies that the conventional sparse grid uses too many points in mixed dimensions for this problem. The problem seems to be intrinsically lower-dimensional and nearly additive (Caflisch et al. 1997). At present, we are working to carry this dimension-adaptive approach over to PDEs.

6. Concluding remarks In this contribution we have given an overview of the basic principles and properties of sparse grids as well as a report on the state of the art concerning sparse grid applications. Starting from the dominant motivation breaking the curse of dimensionality - we discussed the underlying tensor product approach, based upon different ID multiscale bases such as the classical piecewise linear hierarchical basis, general hierarchical polynomial bases, interpolets, or wavelets. We then presented various resulting sparse grid constructions and discussed their properties with respect to computational complexity, discretization error, and smoothness requirements. The approach can be extended to nonsmooth solutions by adaptive refinement methods. We demonstrated the effectiveness of sparse grids in a series of applications. The presented numerical results include 2D and 3D PDE model problems, flow problems, and even two non-PDE applications in higher dimensions, namely numerical quadrature and data mining. Since their introduction slightly more than a decade ago, sparse grids have seen a very successful development and a variety of different applications.

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Especially for higher-dimensional scenarios, we are convinced that sparse grids, together with dimension adaptivity, will also have a thriving future. For readers who want to stay up-to-date on sparse grid research, we refer to the sparse grid bibliography at www. ins. uni-bonn. de/inf o/sgbib, which gives roughly 300 articles from the past 40 years.

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Ada Nurnerica (2004), pp. 271-369 DOI: 10.1017/S0962492904000194

© Cambridge University Press, 2004 Printed in the United Kingdom

Complete search in continuous global optimization and constraint satisfaction Arnold Neumaier Institut fiir Mathematik, Universitdt Wien, Nordbergstrafle 15, A-1090 Wien, Austria E-mail: Arnold. NeumaierOunivie. ac. at www. mat. univie. ac. at/~neum/

This survey covers the state of the art of techniques for solving generalpurpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably find all solutions (if there are finitely many). The core of the material is presented in sufficient detail that the survey may serve as a text for teaching constrained global optimization. After giving motivations for and important examples of applications of global optimization, a precise problem definition is given, and a general form of the traditional first-order necessary conditions for a solution. Then more than a dozen software packages for complete global search are described. A quick review of incomplete methods for bound-constrained problems and recipes for their use in the constrained case follows; an explicit example is discussed, introducing the main techniques used within branch and bound techniques. Sections on interval arithmetic, constrained propagation and local optimization are followed by a discussion of how to avoid the cluster problem. Then a discussion of important problem transformations follows, in particular of linear, convex, and semilinear (= mixed integer linear) relaxations that are important for handling larger problems. Next, reliability issues - centring on rounding error handling and testing methodologies - are discussed, and the COCONUT framework for the integration of the different techniques is introduced. A list of challenges facing the field in the near future concludes the survey.

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CONTENTS 1 Introduction 272 2 Why global optimization? 277 3 Basic ideas 279 4 Problem formulation 282 5 First-order optimality conditions 284 6 Software for complete global optimization 287 7 Incomplete methods for simple constraints 291 8 Bound-constrained approximation 295 9 Pure branching methods 298 10 Box reduction: an example 300 11 Interval arithmetic 302 12 The branch and bound principle 307 13 The role of local optimization 310 14 Constraint propagation 313 15 The cluster problem and second-order information 318 16 Linear and convex relaxations 323 17 Semilinear constraints and MILP 329 18 Semilinear relaxations 335 19 Other problem transformations 339 20 Rigorous verification and certificates 341 21 Test problems and testing 344 22 The COCONUT environment 346 23 Challenges for the near future 348 References 351

1. Introduction Consider everything. Keep what is good. Avoid evil whenever you recognize it. St. Paul, ca. 50 A.D. (The Bible, 1 Thess. 5:21-22)

1.1. Early history As the above quote shows, continuous global optimization or constraint satisfaction and the associated global search methods have been part of the art of successful living since antiquity. In the mathematical literature published before 1975, there are occasional references related to the topic, few and important enough to mention them individually. (Please inform me about other significant work on continuous global optimization published before 1975 not mentioned here!) Several independent strands of work (probably done in ignorance of each other) are discernible.

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(1) Markowitz and Manne (1957) and Dantzig et al. (Dantzig, Johnson and White 1958, Dantzig 1960) used piecewise linear approximations for the approximate global minimization of separable nonconvex programs, formulating them as mixed integer linear programs. Land and Doig (1960) and Little, Murty, Sweeney and Karel (1963) introduced the branch and bound technique for discrete optimization, applicable to mixed integer linear programs. Motzkin and Strauss (1965) showed that solving the (discrete) maximum clique problem is equivalent to finding the global minimum (or maximum) of a special nonconvex quadratic program. Falk and Soland (1969) gave the first piecewise linear relaxations of nonconvex problems, thus making them available for obtaining bounds in a branch and bound scheme. Beale and Tomlin (1970) and Tomlin (1970) introduced special ordered sets, defining the way piecewise linear functions are still handled today in mixed integer linear programming solvers. McCormick (1972) introduced the now frequently used linear relaxations for products and quotients, which made the solution of general factorable global optimization problems accessible to the branch and bound technique. (2) Moore (1962) showed in Part 4 of his PhD thesis, which introduced interval arithmetic to numerical analysis (following an unpublished technical report by Moore and Yang (1959)), that by repeated subdivision and simple interval evaluation, the range - hence in particular the global minimum of a rational function over a box can be determined in principle to arbitrary accuracy. Skelboe (1974) improved this basic but excessively slow method by embedding it into (what would today be called) a branch and bound scheme for continuous variables, giving (what is now called) the MooreSkelboe algorithm. Moore's thesis (Moore 1962) also showed that interval methods can be used to prove the nonexistence of solutions of nonlinear systems in a box (which nowadays is used to discard boxes in a branch and bound scheme) and to reduce the region where a solution can possibly lie (which is now used to avoid excessive splitting). Kahan (1968) discovered that interval techniques can also be used to prove the existence of solutions of nonlinear systems in a box (and hence to verify feasibility). Krawczyk (1969) simplified Moore's methods for systems of equations; the Krawczyk operator based on his paper is used in several state-of-the-art global solvers. (See also the historical remarks in Hansen (2001) and Madsen and Skelboe (2002).) Piyavskii (1972) introduced complete global optimization methods based on Lipschitz constants, which are similar in flavour to interval methods. (3) Tsuda and Kiono (1964) introduced the Monte Carlo method for finding solutions of systems of equations; and the thesis by Mockus (1966) applied it to global optimization. Becker and Lago (1970) were the first to use clustering methods, and Torn (1972, 1974) suggested combining these with local optimization, a combination which currently defines the most efficient

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class of stochastic global optimization algorithms (Janka 2000). Holland (1973) introduced genetic algorithms, still a popular (although usually slow) stochastic heuristics for global optimization. As a recognizable mathematical discipline with diverse solution methods for precisely formulated problems involving continuous variables, the field essentially dates back to 1975 when the first book containing exclusively global optimization papers appeared: the volume Towards Global Optimization, edited by Dixon and Szego (1975). In the almost 30 years since the publication of this landmark volume, tremendous progress has been made, and many signs indicate that the field is now ripe for manifold applications in science and engineering. 1.2. Scope

Global optimization is the task of finding the absolutely best set of admissible conditions to achieve an objective under given constraints, assuming that both are formulated in mathematical terms. It is much more difficult than convex programming or finding local minimizers of nonlinear programs, since the gap between the necessary KKT (Karush-Kuhn-Tucker) conditions for optimality and known sufficient conditions for global optimality is tremendous. Many famous hard optimization problems, such as the travelling salesman problem or the protein folding problem, are global optimization problems. The truth of the famous unresolved conjecture P ^ NP (Garey and Johnson 1979) would imply (Murty and Kabadi 1987, Pardalos and Schnitger 1988) that there are no general algorithms that solve a given global optimization problem in time that is polynomial in the problem description length. However, some large-scale global optimization problems have been solved by current methods, and a number of software packages are available that reliably solve most global optimization problems in small (and sometimes larger) dimensions. The author maintains a web site on Global (and Local) Optimization (1995) that contains many links to online information about the subject. The different algorithms can be classified according to the degree of rigour with which they approach the goal. • An incomplete method uses clever intuitive heuristics for searching but has no safeguards if the search gets stuck in a local minimum. • An asymptotically complete method reaches a global minimum with certainty, or at least with probability one, if allowed to run indefinitely long, but has no means of knowing when a global minimizer has been found.

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• A complete method reaches a global minimum with certainty, assuming exact computations and indefinitely long run-time, and knows after a finite time that an approximate global minimizer has been found (to within prescribed tolerances). • A rigorous method reaches a global minimum with certainty and within given tolerances even in the presence of rounding errors, except in neardegenerate cases, where the tolerances may be exceeded. (Often, the label deterministic is used to characterize the last two categories of algorithms; however, this label is slightly confusing since many incomplete and asymptotically complete methods are deterministic, too.) 1.3. Complete search Complete methods (and a fortiori rigorous ones) are (in exact arithmetic) guaranteed to find the global minimizer (within some tolerances) with a predictable amount of work. Here predictable only means relative to known problem characteristics such as Lipschitz constants or other global information (needed for the convergence proof, but usually not for the algorithm itself). The bound on the amount of work is usually very pessimistic - exponential in the problem characteristics. It is only a weak guarantee that does not ensure that the algorithm is efficient in any sense, but it guarantees the absence of systematic deficiencies that prevent finding (ultimately) a global minimizer. The simplest complete method for bound-constrained problems is grid search, where all points on finer and finer grids are tested, and the best point on each grid is used as a starting point for a local optimization. Since the number of points on a grid grows exponentially with the dimension, grid search is efficient only in one and two dimensions. More efficient complete methods generally combine branching techniques with one or several techniques from local optimization, convex analysis, interval analysis and constraint programming. Generally, complete methods (including approximation methods that reduce the problem to one treated by complete methods) are more reliable than incomplete methods since, to the extent they work (which depends on the difficulty of the problem), they have built-in guarantees. Complete methods with finite termination require more or less detailed access to global information about the problem. In most complete codes, this is obtained using interval arithmetic (which provides global control of nonlinearities) in an automatic differentiation-like manner (c/. Section 16), traversing a computational graph either explicitly, or implicitly by operator overloading. If only black box function (and sometimes gradient) evaluation routines are available, complete methods will find the global minimizer with

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certainty after a finite time, but will know when this is the case only after an exponentially expensive dense search (cf. Theorem 9.1 below). Thus, for complete black box algorithms, stopping must be based on heuristic recipes. Good heuristics and probabilistic choices (similar to but usually simpler than those for incomplete methods) also play a role in complete methods, mainly to provide good feasible points cheaply that benefit the complete search.

1.4- About the contents In this survey, the reader will be introduced to theory and techniques that form the backbone of the packages implementing complete or even rigorous algorithms. The core of the material is presented in sufficient detail that the survey may serve as a text for teaching constrained global optimization. We deliberately exclude methods specific to special problem classes (such as distance geometry or protein folding (Neumaier 1997)), and methods specific to combinatorial optimization (Nemhauser and Wolsey 1988, 1989, Wolsey 1998). Moreover, the discussion of incomplete methods is limited to a short overview, and to techniques that remain useful for complete methods. No attempt has been made to be objective in selection and evaluation of the material; even for the topics I discuss, there is often much more in the references quoted. Instead I have tried to give personal value judgements whenever I found it appropriate. At the present state of the art, where so many methods compete and reliable comparative information is only just beginning to become available, this seems justified. Thus I discuss the methods that I find most interesting, most useful, and most promising. I hope that my selection bias will be justified by the future. Also, while I try to give accurate references, I do not always refer to the first paper discussing a concept or method but rather quote convenient books or articles summarizing the relevant information, where available. As one can see from the list of current codes for complete global optimization given in Section 6, none of these codes makes use of all available state-of-the-art techniques. Indeed, in the past, many research groups on global optimization worked with little knowledge of or consideration for related areas. It is hoped that this survey will help to change this lack of communication across the borders of the various traditions in global optimization. Reviews from other perspectives, with less emphasis on the complete search aspect, are given in Gray et al. (1997), Pinter (19966), Torn (2000), and Torn, Ali and Viitanen (1999). For recent books and other basic references, see Section 3.

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2. Why global optimization? Superficially, global optimization is just a stronger version of local optimization, whose great usefulness in practice is undisputed. Instead of searching for a locally unimprovable feasible point we want the globally best point in the feasible region. In many practical applications, finding the globally best point is desirable but not essential, since any sufficiently good feasible point is useful and usually an improvement over what is available without optimization. For such problems, there is little harm in doing an incomplete search; and indeed, this is all that can be achieved for many large-scale problems or for problems where function values (and perhaps derivatives) are available only through a black box routine that does not provide global information. However, there are many problem classes where it is indispensable to do a complete search, in particular the following. • Hard feasibility problems (e.g., robot arm design, cf. Lee and Mavroidis (2002) and Lee, Mavroidis and Merlet (2002)), where local methods do not return useful information since they generally get stuck in local minimizers of the merit function, not providing feasible points (though continuation methods are applicable for polynomial systems in low dimensions). • Computer-assisted proofs (e.g., the proof of the Kepler conjecture by Hales (1998)), where inequalities must be established with mathematical guarantees. • Safety verification problems, where treating nonglobal extrema as worst cases may severely underestimate the true risk (emphasized in the context of robust control by Balakrishnan and Boyd (1992)). • Many problems in chemistry (cf. below), where often only the global minimizer (of the free energy) corresponds to the situation matching reality. • Semi-infinite programming, where the optimal configurations usually involve global minimizers of auxiliary problems. These problems, as well as the fact that algorithms doing a complete search are significantly more reliable and give rise to more interesting mathematics, justify our focus on complete solution techniques. To show the relevance of global optimization for both pure and applied mathematics, we sketch here a number of typical applications. Of course, this is only the tip of an iceberg (i) Graph structure. Many problems in graph theory are global optimization problems. For example, the maximum clique problem asks for the maximal number of mutually adjacent vertices in a given graph. By a wellknown theorem of Motzkin and Strauss (1965), an equivalent formulation

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is the indefinite quadratic program max xTAx s.t. eTx = 1, x > 0, where A is the adjacency matrix of the graph and e is the vector of all ones. Since the maximum clique problem is TVP-hard, the same holds for all classes of global optimization problems that contain indefinite quadratic programming. (ii) Packing problems. The problem is to place a number of ^-dimensional (k < 4) objects of known shape within a number of larger regions of /c-space of known shape in such a way that there is no overlap and a measure of waste is minimized. The simplest packing problem is the knapsack problem where a maximal number of objects of given weights is to be placed into a container with given maximum weight capacity. Many packing problems arise in industry; but there are also a number of famous packing problems in geometry, of which the 300 year-old Kepler problem of finding the densest packing of equal spheres in Euclidean 3-space was only solved recently by Hales (1998), reducing the problem to several thousand linear programs and some interval calculations to ensure rigorous handling of rounding errors. (The proof is still disputed because of the difficulty to check it for correctness; cf. Lagarias (2002). A proof based on rigorous global optimization algorithms would probably be more transparent.) (iii) Scheduling problems. The problem is to match tasks (or people) and slots (time intervals, machines, rooms, airplanes, etc.) such that every task is handled in exactly one slot and additional constraints are satisfied. If there are several feasible matchings, one that minimizes some cost or dissatisfaction measure is wanted. Simple scheduling problems such as the linear assignment problem can be formulated as linear programs and are solved very efficiently, but already the related quadratic assignment problem is one of the hardest global optimization problems, where already most instances with about 30 variables are at the present limit of tractability: cf. Anstreicher (2003). (iv) Nonlinear least squares problems. In many applications, we need to fit data to functional expressions. This leads to optimization problems with an objective function of a form such as

where xi,yi are given data vectors and 6 is a parameter vector. Under certain assumptions, the most likely value of 9 is the global minimizer; it generally must have a small objective function value at noise level if the

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model is to be deemed adequate. If the Fi are nonlinear in 9, a nonconvex optimization problem results that frequently has spurious local minima far above the noise level. A particularly obnoxious case is obtained for data fitting problems in training neural networks. (v) Protein folding. The protein folding problem (Neumaier 1997) consists in finding the equilibrium configuration of the N atoms in a protein molecule with given amino acid sequence, assuming the forces between the atoms are known. These forces are given by the gradient of the 3Ndimensional potential energy function V(x\,... , XJV), where xi denotes the coordinate vector of the ith atom, and the equilibrium configuration is given by the global minimizer of V. Because short-range repulsive forces act like packing constraints, there are numerous local minima. (vi) Chemical equilibrium problems. (Floudas 1997, McDonald and Floudas 1995.) The task here is to find the number and composition of the phases of a mixture of chemical substances allowed to relax to equilibrium. Local optimization of the associated Gibbs free energy is notorious for giving wrong (nonglobal) solutions. The need to solve such problems was one of the main driving forces for the development of constrained global optimization packages in the chemical engineering community, which is still among the leaders in the field. (vii) Robotics. For applications in robotics, see Neumaier (20036). (viii) Other applications. Many more applications can be found in the books by Pinter (1996a), Floudas and Pardalos (1990), Floudas et al. (1999) and Jaulin et al. (2001).

3. Basic ideas In the following, we discuss complete methods for finding the global minimizer(s) of an objective function subject to constraints. Such problems are typically much more difficult than local optimization problems, since it is often hard to decide whether a local minimizer found is global, and since nonlocal space-covering techniques are needed to avoid being trapped in a region with only nonglobal local minimizers. Basic to almost all complete global optimization algorithms is the branching principle (Section 9). This technique consists in splitting (branching) the original problem recursively into subproblems which are sooner or later easy to solve. In pure branching methods, the more prospective branches are split more frequently, while in branch and bound methods we compute bounds on the objective function for each subproblem, in the hope of being able to eliminate many subproblems at an early stage.

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The very useful technique of constraint propagation, discussed in Section 14, allows us to reduce the feasible region in many cases by exploiting properties of separable constraints of the form

keK

with simple, often linear or quadratic functions q^ of a single variable only. This technique may save a lot of branching steps and thus speeds up the branch and bound procedure. This is a reason why special care should be taken in presenting (or transforming) the problem in a form which has as much separability as possible, and we introduce the notion of a semiseparable program adapted to this feature. Section 18 addresses ways to transform general problems into semiseparable form by introducing appropriate extra variables. Semiseparable programs are also amenable to approximation by a mixed integer linear program (MILP), the only class of global optimization problems that has long been successfully solvable, even for large problems. We shall not discuss techniques for solving MILPs (see, e.g., Bixby et al. (2000), Nemhauser and Wolsey (1988, 1989), Wolsey (1998)), but we show how to approximate (and indeed rigorously relax) general global optimization problems by MILPs in Sections 17 and 18. In order to be able to eliminate subproblems quickly, it is important that one can easily locate good feasible points. This is usually done by local optimization (often in a somewhat rudimentary form); see Section 13. However, especially for problems with many local extrema, it is important to use heuristics which (hopefully) prevent a local method being trapped in a high-lying local minimum. A suitable tunnelling technique is discussed in Section 13. Another basic principle, discussed in Section 16, is that of outer approximation of the feasible domain and underestimation of the objective function, in order to obtain relaxed problems which are convex and hence solvable by local methods. Indeed, this is the traditional way to obtain the bounds on the subproblem. In particular, we consider the use of cutting planes and more general cutting surfaces. Nonconvex relaxations are also of interest if they can be solved efficiently. A useful tool for the automatic construction of tight bound constraints, outer approximations and underestimating functions in nonlinear problems is interval arithmetic. Though little known in the optimization community, interval arithmetic is an elegant way of calculating with bound constraints, intervals, and simple higher-dimensional geometric shapes like boxes and parallelepipeds. Its most prominent feature is that it allows strict estimates of the approximation error in linear and quadratic approximations of nonlinear functions over a box, thereby providing non-local information even in large boxes. In Section 11, we shall give a very short introduction to this

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subject (just sufficient for writing programs); a more leisurely introduction embedded into a standard numerical analysis course can be found in Neumaier (20016), and a much more extensive treatment in Neumaier (1990). Interval arithmetic can also be used to certify rigorously the validity of calculations with finite precision arithmetic, and some such applications to optimization are briefly treated in Section 20. The state of the art in 1996 of certified global optimization with interval methods may be found in Kearfott (19966). 3.1. Basic references A basic reference on most aspects of global optimization is the Handbook of Global Optimization by Horst and Pardalos (1995). It contains chapters written by the experts in the respective subfields, on global optimality conditions, complexity issues, concave minimization, dc methods, indefinite quadratic programming, complementarity problems, minimax problems, multiplicative programming, Lipschitz optimization, fractional programming, network problems, continuation methods, interval methods, and stochastic methods (including simulated annealing). Some more recent books present the state of the art in deterministic global optimization from different perspectives: The interval point of view is in Kearfott's book Rigorous Global Search (Kearfott 19966). The constraint propagation perspective is in the book Numerica by Van Hentenryck, Michel and Deville (19976); see also the tutorial by Lustig and Puget (2001). The convex analysis perspective is in the books Deterministic Global Optimization by Floudas (1999) and Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming by Tawarmalani and Sahinidis (20026). An attempt to give a synthetic view of the field (but mostly restricted to discrete optimization) is in the book Logic-Based Methods for Optimization by Hooker (2000); see also his survey (Hooker 2002). A comprehensive background on local optimization (needed as part of most good global optimization algorithms) can be found in the book Numerical Optimization by Nocedal and Wright (1999). For interior point methods, this should be complemented by Wright (1997). Other books on global optimization methods include those of Forgo (1988), Hansen (1992a), Horst, Pardalos and Thoai (1995), Horst and Tuy (1990), Mockus (1989), Pardalos and Rosen (1987), Pinter (1996a), Ratschek and Rokne (1984, 1988), Torn and Zilinskas (1989), Van Laarhoven and Aarts (1987), Zhigljavsky (1991), and proceedings of conferences on global optimization include Bliek, Jermann and Neumaier (2003), Bomze, Csendes, Horst and Pardalos (1996), Dixon and Szego (1975), Floudas and Pardalos (1992, 1996, 2000) and Grossmann (1996). The Journal of Global Optimization is devoted exclusively to papers on global optimization and its applications.

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4. Problem formulation In the present context, a global optimization problem is specified in the form

s.t.

/(*) x € x, F(x) 6 F,

xi integral.

Here x

= [x, x] — {x e Rn | x < x < x}

is a bounded or unbounded 602; in Rn (with i € ( K U {-oo})1l,x € (M U {oo}) n ,x < x), and xi denotes the subvector (x^,... ,Xit)T of x when / = [i\,... ,ii) is a list of indices. Inequalities between vectors are interpreted componentwise. Here / : x —> R is a continuous objective function, F : x —> Rm is a vector of m continuous constraint functions Fi(x),..., Fm(x), and F is a box in R m denning the constraints on F(x). The feasible domain is denned by C = {a; E x I xj integral, F(x) € F } .

(4.2)

Points in C are called feasible, and a solution of (4.1) is a feasible point x G C such that f(x) = minf(x).

(4.3)

A local minimizer only satisfies f{x) < f(x) for all x € C in some neighbourhood of £, and the solutions are precisely the global minimizers, i.e., the local minimizers with smallest objective function value. A local (global) solver is an algorithm or programming package designed for finding a local (global) minimizer. (We avoid the ambiguous term optimizer which may denote either a minimizer or a solver.) The difficulties in global optimization stem mainly from the fact that there are generally many local minimizers but only one of them is the global minimizer (or just a few), and that the feasible region may be disconnected. (Consider, e.g., the set of positions in the Alps above a certain altitude.) Already a linear objective function has one minimizer in each connected component of the feasible domain, and local descent methods usually fail if they start in the wrong component. Even the constraint satisfaction problem, i.e., the problem of deciding whether the feasible set is nonempty (and finding a feasible point) is frequently highly nontrivial, and may be essentially as difficult as the optimization problem itself (cf. Section 13). The usual device of minimizing a suitable measure of infeasibility does not work when the constraints are sufficiently nonlinear, since this measure has itself local minima in which descent methods often get stuck.

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Usually, it is possible to reformulate a global optimization problem such that / and F are smooth, i.e., twice continuously differentiable. Note that (4.1) is sufficiently flexible to take care of: • free variables x^. take x^ = — oo, Xi = oo; • nonnegative variables xf. take Xj = 0, Xi — oo; • binary variables xf. take Xj = 0, a7j = 1, i € I; • equality constraints Fi(x) = 0: take F^ = Fi = 0; • inequality constraints Fi(x) < 0: take Fj = —oo, Fi = 0. If / is not empty then, if / and F are linear then (4.1) is called a mixed integer linear program (MILP); and if / and F are convex, and F^ = —oo for all nonlinear Fi, (4.1) is called a mixed integer nonlinear program (MINLP). Strictly speaking, this term should apply for all problems (4.1); however, the current techniques for MINLP use the convexity in an essential way, so that it is appropriate to reserve the term for the convex case. Nonconvex mixed integer global optimization problems have received little attention, but see, e.g., Floudas (1995), Grossmann (2002), Sahinidis (1996), and Tawarmalani and Sahinidis (20026). The only class of global optimization problems that can be reliably solved for many large problem instances (say, ~ 105 variables and |/| w 103) is the class of MILPs. This is due to the fact that after fixing the integer variables one is left with a linear program, which can be solved efficiently. Instead of trying all integer combinations separately, branching techniques (branch and bound, branch and cut), combined with preprocessing the resulting linear programs, drastically cut down the number of cases to be looked at. MINLP shares with MILP the feature that fixing all integer variables leads to a tractable problem, in this case a convex nonlinear program, for which every local minimizer is a solution; however, the dimensions are here more limited since nonlinear programming codes are significantly slower than their linear counterparts. Most constrained global optimization is nowadays best viewed as an adaptation of mixed integer programming technology to nonlinear problems. Historically, however, many of the techniques were devised independently by groups working in integer programming, combinatorial optimization, unconstrained optimization, interval analysis, and constraint logic programming. Other important classes of global optimization problems are as follows: • simply constrained: if d i m F = 0, • continuous: if / = 0, • bound-constrained: if simply constrained and continuous,

. separable: if f(x) = £ £ = 1 fk{xk) and F(x) = £ £ = 1 Fk(xk),

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• factorable: if / and F are obtained by applying a finite sequence of arithmetic operations and unary elementary functions to constants and the Xk, • reverse convex: if / , F are concave, and F^ = —oo for all nonlinear Fi, • DC: if / , F are differences of convex functions.

5. First-order optimality conditions Nonlinear programming provides the following necessary (Karush-John) optimality conditions for local minimizers. We assume that / , F are continuously differentiate, and denote by f'{x) and F'(x) the derivatives at x. Note that f'{x) is a row vector and F'{x) a matrix, the Jacobian. Theorem 5.1. (Karush (1939), John (1948)) For every local minimizer x of (4.1) (which defines the notation) there are a number K > 0 and a vector y, not both zero, such that the row vector gT = Kf'(x) + yTF'(x)

(5.1)

satisfies 0 9% ' < 0

if if

o

if

u

> 0

' • %i = X "I

<^ ^

Xi < X •i = %i < X '"

i d T

'Ftij ] %

(5.2)

Xi.

,


i&I,

if Fi < Fi(x) = Tu

< 0 if ^

= Fi{x) < Fi,

(5.3)

!

= 0 if Zi < Fi{x) < Fi. Note that there is no restriction on gi if i € / or x_t = Xi, and no restriction on yi if F_i — Fi(x) = Fi. Buried implicitly in results of Mangasarian (1969), and spelled out explicitly in Neumaier and Schichl (2003), is the observation that one may in fact assume that either K or the subvector yj of y is nonzero, where J is the set of indices i such that either Fi(x) is nonconvex and Fi(x) = F_i or Fi(x) is nonconcave and Fi(x) = F{. (In particular, J does not contain any index i such that Fi(x) is linear.) In view of the homogeneity of the statement of the theorem, one can therefore scale the multipliers such that K + yjDyj = l, (5.4) where D is an arbitrary diagonal matrix with positive entries. This condition is relevant for the application of exclusion box techniques (cf. Section 15).

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We say that x satisfies a constraint qualification (CQ) if (5.1)-(5.3) hold for some K > 0. In this case, one can scale g, K, y to enforce K = 1 and obtains the more frequently used Kuhn-Tucker conditions (Kuhn and Tucker 1951, Kuhn 1991). A sufficient condition for the constraint qualification is that the rows of F'(x) are linearly independent; various weaker conditions guaranteeing CQ are known. If K — 1 then y is called an optimal Lagrange multiplier corresponding to x (it need not be unique). In this case, g is the gradient of the associated Lagrangian (Lagrange 1797)

L(x,y) = f(x) + yTF(x) at x = x. Note that minimizers with huge Lagrange multipliers are best considered as points nearly violating the constraint qualification, so that (5.1) holds with y = 0(1) and tiny K. If there are only nonnegativity constraints and equality constraints,

C = {x > 0 | F(x) = b}, corresponding to x^ = 0, Xi = oo, F^ = F{ = b{ then the conditions (5.3) are vacuous, and (5.2) reduces to the traditional complementarity condition min(gi,Xi) = 0 for all i. Example 5.2. min s.t.

We consider the problem

f(x) = -xi - 2x2 2

. 2

F{x) = ( n - I) + (x2 - I) = 1,

.

x1,x2£[-l,l].

The feasible region is a quarter circle, and the contour lines of the objective function are linear, decreasing in the direction indicated in Figure 5.1. This implies that there is a unique maximizer at P , a local minimizer at Q and a global minimizer at R. The solution is therefore x = (0,1). Since there are only two variables, we analysed the problem graphically, but we could as well have proceeded symbolically as follows. Assuming for simplicity the validity of the CQ, we find for the gradient of the Lagrangian

The Kuhn-Tucker conditions require that gi > 0 if Xi = — 1, gi < 0 if Xi = 1, and gi = 0 otherwise. This leaves three cases for each component, and a total of 3 • 3 = 9 cases. If we assume \x\\, \x2\ < 1 we must have g = 0, hence x\ = 1 + l/2y, x2 = 1 + 1/y- Since x must be feasible, y < 0, and since F(x) = 1, y = — \\f§, x = (1 — 1/VE, 1 - 2/v / 5) r , which is the local maximizer P. If we assume Xi = — 1 or x2 = — 1, or x\ = x2 = 1,

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Figure 5.1. Feasible region and contour lines for Example 5.2.

we find a contradiction with F(x) — 1. (These are 6 cases!) If we assume |^21 < 1 = X\ we find Q, and for \x\\ < 1 = %2 we find R. Thus we have three feasible points satisfying the Kuhn-Tucker conditions, and a comparison of their function values shows that R is the global minimizer. In general, we have three cases for each two-sided inequality and two for each one-sided inequality; since the number of independent choices must be multiplied, the total number of cases grows exponentially with the number of inequalities in the problem formulation. Hence this symbolic approach is limited to problems with few inequality constraints. Even then it only works if the resulting nonlinear equations are symbolically solvable and have few solutions only. Thus, in general, we need to resort to numerical methods. We draw several conclusions from the example. First, there is a combinatorial aspect to the continuous global optimization problem, so that it resembles a mixed integer problem. Second, several cases can often be excluded by a single argument, which is the basis for the branch and bound approach to global optimization. Third, the Karush-John or Kuhn-Tucker conditions do not distinguish between maxima and minima (and other 'stationary' points); all these would have to be enumerated in a naive approach. Since there may be an exponential number of Kuhn-Tucker points, additional techniques are needed to reduce the search space. Lagrange multiplier techniques involving second-order conditions will address this last point; cf. Theorem 15.1.

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6. Software for complete global optimization Here we list some of the better complete global optimization codes available on the Web, with short comments on scope and method. Several of the codes (LGO, BARON, SBB, DICOPT) can be called from the GAMS modelling system (GAMS World 2003), allowing for very convenient input. Input from the AMPL modelling system (Fourer, Gay and Kernighan 1993) will be possible through an AMPL to GAMS translator available within the COCONUT environment; cf. Section 22. Only two of the codes (GlobSol and Numerica) are rigorous solvers. 6.1. Some branching codes using function values only The codes listed use black box function evaluation routines, and have heuristic stopping rules, so that the actual implementation yields an incomplete search only. (i) DIRECT: Divide Rectangles (in Fortran) Gablonsky and Kelley (2001) ftp://ftp.math.ncsu.edu/FTP/kelley/iffco/DIRECTv204.tar.gz Direct.m, a MATLAB implementation of DIRECT www4.ncsu.edu/~definkel/research/ Implementations of a simple and efficient global optimization method by Jones, Perttunen and Stuckman (1993) for bound-constrained problems. DIRECT is based on branching and a Pareto principle for box selection. (ii) MCS: Multilevel Coordinate Search Huyer and Neumaier (1999) www.mat.univie.ac.at/~neum/software/mcs/ A MATLAB program for bound-constrained global optimization using function values only. MCS is based on branching and sequential quadratic programming. (iii) LGO: Lipschitz Global Optimization (commercial) Pinter (1996a, 1999) i s . dal. ca/~ jdpinter/l_s_d. htm An integrated development environment for global optimization problems with Lipschitz-continuous objective and constraints. LGO is based on branching and stochastic estimation of Lipschitz constants; constraints other than simple bounds are handled by L\ penalty terms, but interior convex constraints by projection penalties. (LGO also has options for incomplete search methods; in many cases, these give better results than the branching option.)

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6.2. Some branch and bound codes

The codes listed use global information (generally from required symbolic problem input). They have finite termination with guarantee that the global minimizer is found within certain tolerances; in difficult cases storage or time limits may be exceeded, however, leading to appropriate error messages. All codes use at least basic constraint propagation, but differ considerably in the other techniques implemented. Not listed are the many MILP codes available (see the Global Optimization web page mentioned in the Introduction). (i) BARON: Branch-And-Reduce Optimization Navigator (commercial) Sahinidis et al. (Ryoo and Sahinidis 1996, Sahinidis 1996, 2000, 2003, Tawarmalani and Sahinidis 20026, 2004) archimedes.scs.uiuc.edu/baron/baron.html A general-purpose solver for optimization problems with nonlinear constraints and/or integer variables. Fast specialized solvers for many linearly constrained problems. BARON is based on branching and box reduction using convex and polyhedral relaxation and Lagrange multiplier techniques. (ii) GlobSol: Global Solver (in Fortran 90) Kearfott (19966, 2003) www.mscs.mu.edu/~globsol/ Branch and bound code for global optimization with general factorable constraints, with rigorously guaranteed results (even round-off is accounted for correctly). GlobSol is based on branching and box reduction using interval analysis to verify that a global minimizer cannot be lost. (iii) LINGO (commercial) Gau and Schrage (2004) www.lindo.com/cgi/frameset.cgi?leftlingo.html;lingof.html Branch and bound code for global optimization with general factorable constraints, including nondifferentiable expressions. LINGO is based on linear relaxations and mixed integer reformulations. C and Excel interfaces are available. (iv) Frontline Interval Global Solver (commercial) Nenov and Fylstra (2003) www.solver.com/technology5.htm This solver is based on interval methods and linear relaxations. Visual Basic and Excel interfaces are available.

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(v) ALIAS (in C) Merlet (ALIAS-C++ 2003, Merlet 2001) www-sop.inria.fr/coprin/logiciels/ALIAS/ALIAS-C++/ALIAS-C++. html Branch and bound environment for solving constraint satisfaction problems (and rudimentary global optimization). A toolkit of interval analysis and constraint programming techniques with a Maple interface for symbolic preprocessing. (vi) Numerica Van Hentenryck et al. (19976) Branch and bound code for constrained optimization (with mathematically rigorous results). This code (no longer available) was based on branching and box reduction using interval analysis and deeper constraint propagation techniques. The box reduction and interval analysis algorithms of Numerica are now available in ILOG Solver (commercial) at www.ilog.com/products/solver/ (vii) aBB Adjiman, Floudas and others (Adjiman et al. 1996, 1998a, 19986, Androulakis et al. 1995) titan.princeton.edu/soft.html#abb Branch and bound code for nonlinear programs. The site currently has the description only, no code. aBB is based on branch and bound by convex underestimation, using interval analysis to write nonlinearities in DC (difference of convex functions) form. (viii) LaGO Nowak, Alperin and Vigerske (2003) www-iam.mathematik.hu-berlin.de/~eopt/#Software Branch and bound code for mixed integer nonconvex nonlinear programming, using block-separable structure and convex underestimation. The site currently has the description only, no code. (ix) GloptiPoly (in MATLAB) Henrion and Lasserre (2003a, 20036, 2003c) www.laas.fr/~henrion/software/gloptipoly/ Global optimization of polynomial nonlinear programs, using semidefinite relaxations. Currently limited to problems with fewer than 20 variables. (x) SOSTOOLS (in MATLAB) Prajna, Papachristodoulou and Parrilo (2002) control.ee.ethz.ch/~parrilo/sostools/index.html

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toolbox for solving sums of squares (SOS) optimization programs. Allows the solution of polynomial global optimization problems. MATLAB

(xi) cGOP Visweswaran and Floudas (1996a, 19966) titan.princeton.edu/soft.html

Branch and bound code for linearly constrained global optimization problems with an objective containing linear, bilinear, and convex terms, using convex relaxations. (xii) M I N L P B B (commercial) Fletcher and Leyffer (1994, 1998) www-unix.mcs.anl.gov/~leyffer/solvers.html Branch and bound code for mixed integer nonlinear programming; finding the global optimum is guaranteed only if all constraints become convex when all integer variables are fixed. Problems with AMPL input can be solved online via NEOS at www-neos.mcs.anl.gov/neos/solvers/MINCO:MINLP-AMPL/ MINLP uses standard mixed integer programming techniques and filter methods for the local subproblems. (xiii) D I C O P T (commercial) Duran and Grossmann (1986), Kocis and Grossmann (1989) www.gams.com/solvers/dicopt/main.htm

Solver for mixed integer monlinear programming (MINLP) problems. Finding the global optimum is guaranteed only if all constraints become convex when all integer variables are fixed. (xiv) SBB (commercial) Bussiek and Drud (2001) www.gams.com/solvers/solvers.htm#SBB Branch and bound code for mixed integer nonlinear programming; finding the global optimum is guaranteed only if all constraints become convex when all integer variables are fixed. Problems with GAMS input can be solved online via NEOS at www-neos.mcs.anl.gov/neos/solvers/MINCO:SBB-GAMS/ SBB uses standard mixed integer programming techniques and sequential quadratic programming methods for the local subproblems. MINLPBB, DICOPT and SBB are borderline cases in our list since they do not use truly global techniques for the continuous variables, and will not be discussed further.

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7. Incomplete methods for simple constraints As mentioned in the Introduction, numerical methods for global optimization can be classified into four categories according to the available guarantees. We shall be mainly concerned with complete methods; however, since incomplete and asymptotically complete methods are frequently successful and, for many difficult problems, are the only feasible choice, we give an overview of the main possibilities. In these categories are several deterministic and most stochastic methods. For some of the latter, it is possible to prove convergence with probability arbitrarily close to 1 (if running arbitrarily long), but this does not yet guarantee convergence. (Moreover, the assumptions underlying the convergence proofs are frequently not verifiable for particular examples.) The simplest incomplete method is multiple random start, consisting of picking random starting points and performing local optimizations from these points, in the hope that one of them is in the basin of attraction of the global minimizer. Most stochastic techniques can be regarded as devices for speeding up this basic method, by picking the points more carefully and by doing only rudimentary local optimization, or optimizing only selectively. Most of the research on incomplete search has been concentrated on global optimization methods for simply constrained problems only. Many different incomplete methods are known for simply constrained problems, and we sort them into four categories: local descent techniques, including among others multiple random start, clustering (Boender, Rinnooy Kan, Timnier and Stougie 1982), tunnelling (Levy and Montalvo 1985), and smoothing methods (Guddat, Guerra Vasquez and Jongen 1990, Kostrowicki and Scheraga 1996, More and Wu 1997, Stillinger 1985); response surface techniques, including Bayesian stochastic techniques Mockus (1989, 1994) and related techniques (Bjorkman and Holmstrom 2000, Jones 2001, Jones, Schonlau and Welch 1998); nonmonotonic search techniques, including among others tabu search (Glover 1989/90, Glover and Laguna 1997), simulated annealing (Kirkpatrick, Geddat, Jr. and Vecchi 1983, Ingber 1993, Van Laarhoven and Aarts 1987), and deterministic variants such as threshold accepting (Dueck and Scheuer 1990); ensemble methods, including genetic algorithms (Holland 1973, Forrest 1993, Michalewicz 1996) and variants such as ant colony minimization (Dorigo, Maniezzo and Colorni 1991). No attempt is made to be representative or complete on referencing or describing the large literature on incomplete techniques; we only mention the 1975 book by Dixon and Szego (1975), which marks the start of a tradition of comparing different global optimization methods, an excellent exposition of stochastic global optimization methods for bound-constrained problems on the Web by Torn (2000), and another web survey of (mainly incomplete) methods by Gray et al. (1997). For incomplete search in combinatorial

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optimization (where the underlying ideas are also called metaheuristics), see, e.g., Aarts and Lenstra (1997), Yagiura and Ibaraki (2001). Instead of describing technical details of the various methods (these vary from author to author and even from paper to paper), we give an informal view of the ideas, strengths and weaknesses of one method from every category, each based on analogies to natural processes where more or less global optima are reached. While these techniques are motivated by nature it is important to remember that processes in nature need not be the most efficient ones; at best they can be assumed to be efficient given the conditions under which they have to operate (namely an uncertain and changing environment that is potentially hazardous to those operating in it). Indeed, much of our present technology has vastly surpassed natural efficiency by unnatural means, and it would be surprising if it were different in global optimization. Even assuming that nature solves truly global optimization problems (a disputable assumption), simple lower estimates for the number of elementary steps - roughly corresponding to function evaluations - available to natural processes to converge are (in chemistry and in biology) in the range of 1015 or even more. This many function evaluations are unacceptable for present day computers, and will be so in the near future. With a limited number of function evaluations, the quality of incomplete methods depends a lot on details of the implementation; comparisons on relative efficiency are virtually missing. Indeed, the techniques must generally be tuned to special classes of applications in order to be fast and competitive, which makes general-purpose comparisons difficult and inconclusive. Smoothing (= homotopy = continuation) methods are based on the intuition that, in nature, macroscopic features are usually an average effect of microscopic details; averaging smooths out the details in such a way as to reveal the global picture. A huge valley seen from far away has a well-defined and simple shape; only by looking more closely, the many local minima are visible, more and more at smaller and smaller scales. The hope is that by smoothing a rugged objective function surface, most or all local minima disappear, and the remaining major features of the surface only show a single minimizer. By adding more and more details, the approximations made by the smoothing are undone, and finally one ends up at the global minimizer of the original surface. In mathematical terms, one has to define a homotopy by introducing an additional parameter t into the problem in such a way that t = 0 gives the original problem, while t — 1 gives either a related convex problem or a related problem with a unique and known global minimizer. (There are various ways of doing this; homotopies whose parameter has a natural interpretation in the context of the original problem usually perform better.) Then a sequence of local problems is solved for t = t\, £2 • • •, *w, where the U

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form a decreasing sequence starting at 1 and ending at 0. Each time, the solution of the previous problem is taken as the starting point for the current problem. The quality of the final local minimizer depends on the homotopy, and is frequently the global or at least a good local minimizer. There is no theoretical work on conditions that would ensure convergence to the global minimum. In particular, it is quite possible for such a method to miss the global minimum. However, for properly chosen homotopies, smoothing methods at least give good local minima with a small number of function evaluations. (For more theory on homotopy methods, see, e.g., Guddat et al. (1990, Section 6.3).) Response surface techniques are designed specifically for the global optimization of functions that are very expensive to evaluate. They construct in each iteration an interpolation or approximation surrogate function of known analytic form. The surrogate function is then subjected to global optimization, e.g., by some form of multiple random start (started at a selection of the current points). The resulting optimizers (or some points where the feasible region has only been sparsely explored) are taken as new evaluation points. Since this is sequential global optimization, each step is much more expensive than the others, but the reduction of the number of function values needed gives (for sufficiently expensive function evaluations) a net gain in speed. In principle, these methods may have convergence guarantees if the point selection strategy is well chosen; but this is irrelevant in view of the fact that for expensive functions, only few (perhaps up to 1000) function evaluations are admissible. Simulated annealing takes its intuition from the fact that the heating (annealing) and slow cooling of a metal brings it into a more uniformly crystalline state that is believed to be the state where the free energy of bulk matter takes its global minimum. (Incidentally, even for the simplest potential energy functions, it is still an unsolved problem whether this is indeed true with mathematical rigour. Apart from that, even very pure crystals still have defects; i.e., the global minimum is not quite achieved in nature.) The role of temperature is to allow the configurations to reach higher energy states with a probability given by Boltzmann's exponential law, so that they can overcome energy barriers that would otherwise force them into local minima. This is quite unlike line search methods and trust region methods on which good local optimization programs are based. In its original form, the simulated annealing method is provably convergent in a probabilistic sense but exceedingly slow; various ad hoc enhancements make it much faster. In particular, except for simple problems, success depends very much on the implementation used.

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Genetic algorithms make use of analogies to biological evolution by allowing mutations and crossing over between candidates for good local optima in the hope of deriving even better ones. At each stage, a whole population of configurations is stored. Mutations have a similar effect to random steps in simulated annealing, and the equivalent of lowering of the temperature is a rule for more stringent selection of surviving or mating individuals. The ability to leave regions of attraction to local minimizers is, however, drastically enhanced by crossing over. This is an advantage if, with high probability, the crossing rules produce offspring of similar or even better fitness (objective function value); if not, it is a severe disadvantage. Therefore the efficiency of a genetic algorithm (compared with simulated annealingtype methods) depends in a crucial way on the proper selection of crossing rules. The effect of interchanging coordinates is beneficial mainly when these coordinates have a nearly independent influence on the fitness, whereas if their influence is highly correlated (such as for functions with deep and narrow valleys not parallel to the coordinate axes), genetic algorithms have much more difficulties. Thus, unlike simulated annealing, successful tuning of genetic algorithms requires a considerable amount of insight into the nature of the problem at hand. Both simulated annealing methods and genetic algorithms are, in their simpler forms, easy to understand and easy to implement, features that invite potential users of optimization methods to experiment with their own versions. The methods often work, if only slowly, and may be useful tools for applications where function values are not very expensive and the primary interest is to find (near-)solutions now, even when the reliability is uncertain and only subglobal optima are reached. To make simulated annealing methods and genetic algorithms efficient, clever enhancements exploiting expert knowledge about the problem class at hand are essential. Theoretical work on explaining the effectiveness of useful enhancements is completely lacking. Also, I have not seen careful comparisons of the various options available and their comparative evaluation on standard collections of test problems. In general, incomplete methods tend to fail systematically to find the global optimum on the more difficult problems in higher dimensions, but they frequently give relatively good points with a reasonable amount of effort. Beyond a certain number of function evaluations (that depends on the problem), progress slows down drastically if the global optimum has not yet been located already. This is unlikely to change in the future, although new heuristics and variations of old ones are discovered almost every year. For general-purpose global optimization, the most promising incomplete methods appear to be clustering methods (see the recent comparison by Janka (2000)), being fairly robust and fast. In particular, the multilevel

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clustering algorithm by Boender et al. (1982), as implemented by Csendes (1988), can be recommended. Among incomplete algorithms adapted to problem structure, I would favour smoothing methods (if a natural homotopy is available) and tabu-search-like strategies (since these have a kind of memory).

8. Bound-constrained approximation For general constraints, incomplete techniques are much less developed. Only the smoothing techniques extend without difficulties to general constraints. To use the other incomplete techniques, it is customary to rewrite problems with general constraints in an equivalent or approximately equivalent form with either simple constraints only, for which the methods of the previous section apply, or into a mixed integer linear problem (MILP), since highly efficient software is available for solving the latter (Bixby et al. 2000). Both transformations are of great practical importance and widely used. Solving the transformed (equivalent or approximate) problem yields an approximate solution for the original problem, and local optimization from this approximate solution gives the global minimizer of the original problem if the approximation was good enough, and usually a good local minimizer otherwise. In this section we treat the approximation of general constrained problems by bound-constrained problems using penalty and barrier functions. The approximation of nonlinear problems by mixed integer linear programs is treated in Section 18. 8.1. Penalty and barrier formulations Traditionally (see Fiacco and McCormick (1990)), constraints that cannot be handled explicitly are accounted for in the objective function, using simple l\ or 1% penalty terms for constraint violations, or logarithmic barrier terms penalizing the approach to the boundary. In both cases, the reformulation changes the solution, so that this is an instance of an approximation method, and the result should be used as a starting point for a subsequent local optimization of the original problem. There are also so-called exact penalty functions whose optimization gives the exact solution (see, for example, Nocedal and Wright (1999)); however, this only holds if the penalty parameter is large enough, and what is large enough cannot be assessed without having global information. The use of more general transformations gives rise to more precisely quantifiable approximation results. In particular, if it is known in advance that all constraints apart from the simple constraints are soft constraints only (so that some violation is tolerated), one may pick a transformation that

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incorporates prescribed tolerances into the reformulated simply constrained problem, using the following variation of a similar, but less flexible result of Dallwig, Neumaier and Schichl (1997), given in Huyer and Neumaier (2003). Theorem 8.1. (Soft optimality theorem) Given A , ^ , ^ > O,/o € R, let

Then the merit function /merit (x) = q(x) + r(x) has its range bounded by ]—1,3[, and the global minimizer x of / mer it either satisfies Fi(x) € \Fi - zu Fi + ai] for all «, f(x)

< mm{f(x)

| F(x) £F,xE

X},

m

x

(8.1) (8.2)

or one of the following two conditions holds: {x e x | F(x) e F} = 0, /o < min{/(x) | F(x) € F, x € x } .

(8.3) (8.4)

Here (8.1) says that a soft version of the constraints is satisfied. The numbers cij and cfj measure the degree to which the lower and upper bounds in the constraint Fi(x) € Fj may be softened; suitable values are in many practical applications available from the meaning of the constraints. Inequality (8.2) says that /merit has a lower global minimum value (attained at a point satisfying the soft constraints) than the global minimum value of the original problem (on the hard version of the constraints). Thus little is lost from a practical point of view. The degenerate cases (8.3)-(8.4) account for the possibility of an empty feasible set (8.3), and for a choice of /o that was too small. If a feasible point is already known we may choose /o as the function value of the best feasible point known (at the time of posing the problem), thus eliminating the possibility (8.3). If none is known, /o should be chosen as a fairly large value to avoid (8.4); it can be reset (and the optimization restarted) when a feasible point becomes available during the search.

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In spite of the absolute value in the definition of q(x), / mer it is continuously differentiable if / and F have this property. A suitable value for A is the median of the values \f{x) — /o| for an initial set of trial points x (in the context of global optimization often determined by a space-filling design (McKay, Beckman and Conover 1979, Owen 1992, Owen 1994, Sacks, Welch, Mitchell and Wynn 1989, Tang 1993)).

8.2. Projection penalties A little-known result by Pinter (1996a) may be used to get in certain cases (in particular, for linear and convex quadratic constraints) an exact reformulation as a non-smooth but Lipschitz-continuous simply constrained problem. The idea is to project infeasible points to the feasible domain. To accommodate linear constraints (or convex quadratic ones), Pinter assumes that XQ is a known interior point. For arbitrary 7 > 0 we now define the modified objective function

~f{x):=f{x) + 1\\x-x\\\

(8.5)

where x = Xx0 + (1 - X)x

(8.6)

and A = \ x > 0 is minimal such that x satisfies the linear constraints. This is well-defined, since A = 1 always works by the choice of XQ. Each constraint contributes a lower bound € [0,1] for A, and the largest of these bounds is the desired value. In particular, a linear constraint aTx < a contributes a nonzero lower bound A > (aTx — a ) / ( a x — a XQ) if both numerator and denominator of the right-hand side are positive. A convex quadratic constraint similarly yields a quadratic inequality that can easily be solved for A. (Convexity can be weakened to star-shapedness with respect to XQ.) The modified objective function (8.5) is Lipschitz-continuous, but nonsmooth at all points where the ray (8.6) hits a lower-dimensional face of the feasible domain. Note that to evaluate (8.5), function values are needed only at points satisfying the linear (or convex quadratic) constraints. An interior point can be found by solving a linear program or convex second-order cone program. If no interior point exists since the feasible set is in a lower-dimensional subspace, each feasible point has the form x = XQ+CZ with z € z, where xo is in the relative interior of the feasible domain, and z a box with 0 € int z. Both XQ and C can be found by techniques from convex analysis for finding a maximal independent set of points in the affine

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subspace spanned by the feasible set. Reposing the optimization problem in terms of z reduces the dimension and yields a problem in which 0 is an interior point.

9. Pure branching methods We begin our analysis of complete methods for global optimization by looking at the options for methods that can access no global information about a problem. The information is made available via black box routines that provide local information only, i.e., function values and possibly gradients or Hessians at single points. A necessary and sufficient condition for complete methods based on local information only is given by the following important density theorem due to Torn and Zilinskas (1989). It formalizes the simple observation that after finitely many local evaluations there are still many 'holes', i.e., balls not containing an already evaluated point, and there are many functions (Neumaier 2003 a) that have the known function values, gradients and Hessians at the evaluation points but an arbitrarily low function value at the centre of such a ball. Theorem 9.1. Any method based on local information only that converges for every continuous / to a global minimizer of / in a feasible domain C must produce a sequence of points x1 ,x2,... that is dense in C. Conversely, for any such method, liminf f(xl) = min{/(x) | x G C}. A global optimization method based on local information only is called convergent if it satisfies the hypothesis of this density theorem. (Actual implementations of a convergent global optimization method usually are not truly convergent since they must have built-in termination criteria that are necessarily heuristic.) Convergence is a minimal requirement and does not make an algorithm good! For example, exhaustive grid search is convergent but far too slow in dimensions > 2. (Compare with local optimization with line searches along the steepest descent direction, which is globally convergent but frequently very slow.) In a sense, the density theorem says that any convergent method must be ultimately exhaustive, though it may delay the detailed exploration of unpromising regions. Since, in practice, only a limited number of points can be explored, the behaviour of a pure branching method is governed by its ability to find a good ordering of the points to be evaluated for which premature termination has no severe effect. Three good asymptotically complete general-purpose global optimization algorithms based on local information only are currently available: DIRECT (Jones et al. 1993), MCS (Huyer and Neumaier 1999) and LGO (Pinter

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1996a). All work for bound-constrained problems only and need the approximation techniques of Section 8 for more general problems. (Some of these are built into LGO, but must be coded by the user for DIRECT and MCS.) All three algorithms enforce convergence by employing a branching scheme. They differ in how and when to split, and what is done within each box. A branching scheme generates a sequence of rooted trees of boxes whose leaves cover the feasible set. At least one point in each box is evaluated. The first tree just has the original box as root and only leaf. Each other tree is obtained from the previous one by splitting one or several leaves. If the diameters of all boxes at all leaves converge to zero, convergence of the algorithm is straightforward. The convergence to zero of the diameters is ensured by appropriate splitting rules that define when and how a box is split. For example, convergence is guaranteed when in each of a sequence of rounds, • we always split the oldest box along the oldest side, and possibly split finitely many other boxes, or • we always split the longest box along the longest side, and possibly split finitely many other boxes (where length = sum of length of sides), provided that each split of the oldest (or longest) box produces boxes whose volume is at most a fixed fraction < 1 of the unsplit box. The possibility of 'and finitely many other boxes' (but not many if the code is to be robust!) can be used with considerable flexibility without destroying the convergence property. Apart from the convergence requirement, the key to efficiency is a proper balance of global and local search. This is achieved in DIRECT by splitting in each round all boxes for which the pair (v, / ) (where v is the volume and / the midpoint function value) is not dominated by another such pair. Here (v,f) is dominated by (v',f) if both v' < v and / ' > / . In particular, the box of largest volume and the box with the best function value are never dominated and hence always split. MCS instead uses domination of pairs (/,/), where I is a suitably assigned level, and in addition employs local optimization steps (using line searches and sequential bound-constrained quadratic programs) from appropriate candidate points. LGO uses lower bounds L > max \\f(xk) - /(rB/)||/||sfc - xt\\ on Lipschitz constants L obtained from the previous function evaluations to decide on the promising boxes to split first. (Upper bounds on L, and hence bounds on function values, cannot be obtained from local information only.) The combination of a suitable branching strategy with the heuristic methods discussed earlier would make the latter complete, and appears to be a fruitful research direction.

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To improve on the density theorem we must find ways to throw away irrelevant parts of the feasible domain that are guaranteed not to contain a global minimizer. To be able to do this reliably, some kind of global information is necessary. This is utilized by box reduction techniques, discussed in Section 10 using a simple example, and afterwards in more depth.

10. Box reduction: an example Box reduction techniques are based on a more or less sophisticated interplay of several components: logical constraint propagation, interval analysis, convex relaxations and duality arguments involving Lagrange multipliers. Before giving a more formal treatment, we illustrate simple arguments of each of these components by reconsidering Example 5.2. Suppose that a local solver has already produced the local minimizer x = (x) for the problem (5.5) discussed in Example 5.2, perhaps as the best local minimizer found by minimizing from a few random starting points. We use box reduction to check whether there is possibly a better feasible point. In fact, we know already that this is not the case, but we obtained this knowledge in a way that works only for very simple problems. Thus we want to do it again, using only techniques of wide applicability. The idea of box reduction is to use various arguments that allow us to shrink the box without losing any feasible point that is at least as good as the best point found already. Since x is feasible with objective function value - 2 , any such point satisfies f(x) = - x i - 2x2 < - 2 , F(x) = (xi - I) 2 + (x2 - I) 2 = 1, xi G [-1,1], x 2 G [ - l , l ] .

(10.1) (10.2) (10.3)

Constraint propagation (see Section 14) is a very cheap and easily formalizable process that gives important initial range reductions in many otherwise difficult problems. It consists in deducing better bounds for a variable by using the other bounds and one of the constraints. In particular, (10.1) implies x2 > 1 —xi/2 > 0.5 since x\ < 1, and x\ > 2-2x2 > 0 since x2 < 1, reducing the bounds to xi 6 [0,1],

x 2 G [0.5,1].

Similarly, (10.2) implies (xi - I) 2 = 1 - (x2 - I) 2 > 1 - 0.25 = 0.75, hence x\ < 1 — \/0.75 < 0.14, giving the improved bound xx G [0,0.14]. This bound could be used to improve again x2 using (10.1); and by alternating use of (10.1) and (10.2) one would obtain a sequence of boxes shrinking

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towards x. This is a special feature of this simple example. In most cases, this simple substitution process gives no or only very little improvements after the first few good reductions. (Look at a problem with the constraints x\ + X2 = 0, x\ — X2 = 0, xi,X2 G [—1,1] to see why.) Interval analysis (see Section 11) can be applied in a number of different ways. Here we use it to produce linear relaxations of the nonlinear constraint. The Jacobian of F(x) at x G x = ([0,0.14], [0.5,1]) T is F'(x) = (2x! - 2, 2x2 - 2) e ([-2, -1.72], [-1, 0]) = F'(x). The mean value theorem implies that, for any x € x,

F(x) € F(x) + F'(x)(x - x) if s e x. Using x = x we find 1 € 1 + [-2, -1.72]xi + [-1,0](x 2 - 1) = [1 - 2xi, 2 - 1.72xi - x 2 ]; the interval evaluation needs no case distinction since x\ and X2 — 1 happen to have constant sign. The lower bound gives no new information, but the upper bound leads to the new constraint 1.72xi + x 2 < 1. By its derivation, this constraint is weaker than (10.2). But since it is linear, the constraint is quite useful for relaxation techniques (see Section 16). It allows us to create a convex relaxation of the problem. Indeed, we may look at the relaxed linear program min s.t.

-xi-2X2 1.72xi + x 2 < 1,

0 < xi < 0.14,

0.5 < x 2 < 1.

By construction, every feasible point better than the best point is feasible for (10.4), hence the minimum of (10.4) will be a lower bound on the bestpossible objective function value of the original problem. Solving (10.4) gives the solution x = (x) with function value —2. Since this lower bound equals the best function value found so far for the original problem, the original problem has global minimum —2. This is a happy accident due to special circumstances: our problem already had a linear objective function, and the global minimizer was at a corner of the feasible set. (But as we shall see, we can adapt the technique to work much more generally if the box is narrow enough.) Multiplier techniques can be used in a variety of ways. Here we use them to check whether there is a second, undiscovered global minimizer. We use the Lagrange multiplier A = 2 associated with the linear constraint of (10.4) at the solution. The associated linear combination —x\ — 2x2 + 2(1.72xi + X2 — 1) is bounded by the best-known function value —2 of the

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original problem, giving 2.44xi — 2 < —2, hence x\ < 0. Thus we must have x\ = 0, and constraint propagation using (10.1) implies X2 = 1. Thus the box has been reduced to x, showing that it is the only global minimizer. What generalizes? The problem discussed was deliberately kept simple so that the complete solution process could be demonstrated explicitly. In general, constraint propagation only gives limited reduction. Similarly, relaxed linear or convex programs usually only give a lower bound on the smallest-possible objective function value, but the linear combination derived from the Lagrange multipliers frequently contains useful information that can be exploited by constraint propagation to get a further significant box reduction. If the reduction process stalls or becomes slow, the box is split into two or more smaller boxes. On the smaller boxes, the same techniques may prove effective, and we alternate box reduction and box splitting until all box sizes are below some termination threshold. Usually, only very few boxes remain if sufficiently good reduction techniques are used (pathological exceptions include min x — x s.t. x 6 [0,1]). If no box remains, the problem is guaranteed to have no feasible point. The total number of boxes processed is a measure of the difficulty of a problem for the particular algorithm used. Simple problems (like the example discussed above) only need a single box; in the worst case, an exponential number of boxes may be needed. In the latter case, time and storage limitations may force a premature termination; in this case the best point found is not verified to be a global minimizer. 11. Interval arithmetic Interval analysis, the study of theory and algorithms for computing with intervals, is a large subject; see Moore (1979) (introductory), Neumaier (20016) (embedded in a numerical analysis context) and Neumaier (1990) (advanced). Its importance for global optimization stems from several, interrelated facts. • Interval analysis gives easily computable (though sometimes only very crude) bounds on the range expressions. • Interval analysis allows one to control nonlinearities in a simple way (via centred forms). • Interval analysis extends classical analysis in its ability to provide semilocal existence and optimality conditions, valid within a pre-specified local region around some point, while classical analysis generally only asserts the existence of such neighbourhoods without providing a simple way to find them.

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We give here a short introduction to the basics and mention the main techniques useful for global optimization. General references on interval methods in global optimization include Adjiman et al. (1998a), Benhamou, McAllister and Van Hentenryck (1994), Benhamou and Older (1997), Berner (1996), Dallwig et al. (1997), Epperly and Pistikopoulos (1997), Epperly and Swaney (1996), Jansson (1994), Jansson and Kniippel (1992, 1995), Hansen (1980, 1992a), Kearfott (19966), Kearfott, Novoa and Chenyi Hu (1991), Neumaier (1996), Ratschek and Rokne (1984, 1988), and Van Hentenryck et al. (19976). If a and b are two intervals we define for o 6 {+,—,*,/," } the binary operation a o b : = Q { a o & | a€a,6<E b},

(11.1)

provided the right-hand side is denned. Here US= [infS,supS] denotes the interval hull of a set of real numbers, i.e., the tightest interval containing S. A monotonicity argument gives for addition and subtraction a + b = [q + b,a + b), a-b

= [a-b,a-b],

(11.2) (11.3)

and for multiplication and division a* b = U{ab,ab,ab, ah}, a / b = •{a/6, a/b, a/b, a/b}

(11-4) if 0 g b;

(11.5)

in most cases only two of these products or quotients need to be computed. We also define elementary functions ip 6 {sqr, sqrt, exp, log, sin, cos, abs,...} of an interval a (and similarly — a, a+, etc.) by ¥>(a) := UMa)

| a G a}

(11.6)

whenever the right-hand side is defined. Again (p(a) can be computed from the value of (p at the end-points of a and the interior extremal values, depending on the monotonicity properties of ip. Note, however, that |a| is defined as supabs(a), since this expression figures prominently in estimates involving interval techniques. For interval vectors (=boxes) x, analogous definitions apply. We also need the interior int x = {x E Mn | x < x < x} of a box x C Rn. For details and a systematic study of interval operations see Neumaier (1990); we only remark here that some rules familiar from real arithmetic

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fail, and in particular the interval evaluation of different expressions equivalent in real arithmetic may give different results. For example (with - a : = 0 - a = [-a,-a]), a + (—a) = a — a ^ 0 except when a = a. Therefore, we also use the converse inner operations a © b := [a + b,a + b],

(11.7)

aeb

(11.8)

:= [a-b,a-b].

Here, expressions of the form ±00^00 in (11.7) or (11.8) must be interpreted as —00 for the lower bounds and as +00 for the upper bounds. Note that the result of an inner operation is not necessarily an interval since it may happen that the lower bound is larger than the upper bound; giving an empty 'interval'. All these operations are very simple to program. Note that many implementations of interval arithmetic are rather slow since they take care to guarantee correct (and often optimal) outward rounding, needed when interval arithmetic is used for mathematically rigorous certification (see Section 20). For global optimization without certification, unsafe interval arithmetic, which uses the standard rounding for floating point operations, and hence is significantly faster but may lose containment of points that lie too close to the boundary, usually suffices if certain safety measures are taken. But it is significantly harder to ensure robust behaviour with unsafe interval arithmetic since occasionally the solution is lost, too. Important. When using unsafe interval arithmetic, proper safeguards must be taken at places (such as inner operations and intersections) where intervals might become (spuriously) empty owing to accumulation of round-off errors. In place of an empty result, a thin interval formed from the arithmetic mean of the two intersecting bounds should be returned in a robust implementation. As already mentioned, an interval evaluation / ( x ) of some expression / often overestimates the desired range Range(/, x) = {f{x) \ x <E x} of a function. However, under very mild conditions (Neumaier 1990, Section 1.4), the evaluation over small boxes satisfies / ( x ) C Range(/,x) + O(e) if x-x

= O(e);

we refer to this as the linear approximation property of simple interval evaluation. Better enclosures, especially for small e, can be obtained by centred forms; the simplest of these (but not the most efficient: see Chapter 2 of Neumaier

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(1990) for better methods based on slopes) is the mean value form. Owing to the mean value theorem, we have f(x)£f(z)

+ f'(x)(x-z)

if i , z € x .

(11.9)

In particular, Range(/,x) is contained in f(z) + / ' ( x ) ( x — z), and it can be shown that, under mild conditions, f(z) + /'(x)(x -z)C

Range(/, x) + O(e2)

if x - x = O(e);

we say that the mean value form (as other centred forms) has the quadratic approximation property. Recently, centred forms based on higher-order Taylor expansions have found considerable attention; these are able to give significantly sharper bounds in cases where simple interval evaluation suffers from severe dependence. See the survey Neumaier (2002) and the numerical comparisons in Makino and Berz (2003); cf. also Carrizosa, Hansen and Messine (2004). Apart from interval evaluation and centred forms, we need interval Newton methods for verifying solutions of nonlinear systems of equations. The prototype (but again not the most efficient method; see Neumaier (1990, Chapter 5) for better methods based on slopes and Gauss-Seidel iteration) is the Krawczyk (1969) method. To check for solutions of F(x) = 0 with x 6 x, Krawczyk multiplies the vector version of (11.9) by a matrix C and subtracts it from x to find x e K(x, z):=z-

CF(z) + (I - CF'(x))(x - z).

For z € x, the resulting Krawczyk operator K(x,z) (cf. Krawczyk (1969), Kahan (1968)) has the following properties, typical for interval Newton operators: (i) any zero x 6 x of F lies in x n K(K, Z), (ii) if x Pi K(x, z) — 0 then x contains no zero of F, (iii) if K(x, z) C int x then x contains a unique zero of F. Properties (i) and (ii) follow directly from the above derivation, while (iii) is a simple consequence of Banach's fixed point theorem. The most important part is (iii), since, applied to the Karush-John conditions, it allows the elimination of large regions around a local minimizer; cf. Section 15. However, (i) and (ii) are also useful as ways of reducing a box or eliminating it, if it contains no zero. This is implemented in GlobSol (Kearfott 19966) and Numerica (Van Hentenryck et al. 19976). Another useful interval Newton operator with analogous properties is x € JV(x, z):=z-

(CF[x,

z}f(CF(z)),

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where C is an approximate inverse of the interval slope F[x, z] and A ^ b is an enclosure for the set of solutions of Ax = b, A G A, b G b computed, e.g., by the Hansen-Bliek method (Bliek 1992, Hansen 19926, Neumaier 1999). 11.1. Convexity check Interval analysis can be used to check the convexity of a function / : x —»• M. in some box x. Let G be a matrix of intervals (usually simply called an interval matrix), calculated as an enclosure of f"(x) for x G x, then, with r = max{Sfc — xk \ k = 1 , . . . , n}, the linear approximation property implies that \G — G_\ = O(r). Such a statement implies that |G — G\ = O(r) for all individual matrices G G G, with absolute values taken component-wise. In particular, if G is positive definite then, provided the underlying box is not too wide, all matrices in G are definite, too; and if this is the case, / is convex in x. The following constructive criterion for simultaneously checking the definiteness of all members of an interval matrix was given in Neumaier (1996). Theorem 11.1. (Sufficient conditions for convexity) Let / : x —> R be twice continuously differentiable on the compact box x, and suppose that G is a symmetric interval matrix such that f"(x)

<E G

for all x e x.

(11.10)

(i) If some symmetric matrix Go G G is positive definite and all symmetric matrices in G are nonsingular then they are all positive definite, and / is uniformly convex in x. (ii) In particular, this holds if the midpoint matrix G = (sup G + inf G)/2 is positive definite with inverse C, and the preconditioned radius matrix A = |C|radG, where \C\ is the componentwise absolute value of C and r a d G = (supG - inf G)/2, satisfies the condition ||A|| < 1

(11.11)

(in an arbitrary norm). Proof, (i) Since the eigenvalues are continuous functions of the matrix entries and the product of the eigenvalues (the determinant) cannot vanish, no eigenvalue changes sign. Hence the eigenvalues of all matrices in G are positive, since this is the case for the positive definite member. Thus

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all symmetric matrices in G are positive definite. By well-known results, uniform convexity of / now follows from (11.10). (ii) Go = G belongs to G, and condition (11.11) implies strong regularity of the interval matrix G (Neumaier 1990, Section 4.1) and hence nonsingularity of all matrices in G. Thus (i) applies. • In many cases, the Hessian of the augmented Lagrangian can be shown to have the form

with constructively available real matrices Ai and intervals Uj = [ui — ri, Uj + ri\. In this case, the above result can be strengthened (with virtually the same proof) by replacing G and A with G= and

respectively. Indeed, it is not difficult to see that for G = ^ UJJ4J, we always have 0 < A' < A, so that the refined test is easier to satisfy. Other sufficient conditions for convexity based on scaled Gerschgorin theorems and semidefmite programming, form the basis of the aBB method (Adjiman et al. 1996, Androulakis et al. 1995) and are given in Adjiman et al. (1998a) and Adjiman et al. (19986).

12. The branch and bound principle The branch and bound principle is a general label (invented in Land and Doig (1960) and Little et al. (1963)) to denote methods to split a problem recursively into subproblems which are sooner or later eliminated by showing that the subproblem cannot lead to a point better than (or as least as good as) the best point found so far. The latter is often checked by computing lower bounds on the objective function, and the splitting produces new branches in the tree of all subproblems tried, according to so-called branching rules; hence the name 'branch and bound'. But in practice, the subproblems are best treated in a more flexible fashion, allowing us also to eliminate subproblems only partially. General references for branch and bound in global optimization include Beale (1979, 1988), Berner (1996), Epperly and Pistikopoulos (1997), Epperly and Swaney (1996), Floudas (1995), Grossmann (1990), Horst et al. (1995), Jansson and Kniippel (1992), Kearfott (19966), Pinter (1996a) and Van Hentenryck et al. (19976). A thorough discussion of branch and bound

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in discrete optimization, with many algorithmic choices that are of potential interest in general global optimization, is given in Parker and Rardin (1988). For a global optimization problem

f(x)

min s.t.

init x G xinit ,

77i/'™\ *- "n

F{x) E F,

„

. l xi :._J integral,

^

'

'

a natural way to define subproblems is to choose boxes x C x i n i t of the initial box x i n i t , and to consider the subproblems !{X)

, , (12-2) s.t. x 6 x, F(x) G F, xi integral, that is, each subproblem is characterized by (and stored as) the box over which the problem is solved. The branching process then consists in splitting a box x into two or several smaller boxes whose union is x. The most typical branching rule is to select a bisection coordinate j and to split the jth component of the box at a bisection point £. Thus, the current box x is replaced by two sub-boxes x l o w , x u p p with Y low

_ upp _ "~ k ~ x fc Cl V1OW _ f

-c i

k

-yUPP

/ • ' Ji - [ f T'

(12.3)

This branching rule is termed bisection. The bisection point £ is often taken as the midpoint £ = (XJ + xA/2 of the interval XJ; but this fails when there are infinite bounds and is inefficient when the interval ranges over several orders of magnitude. In this case, a more useful bisection point is a safeguarded geometric mean, defined by £ = signXjJxjXj

if 0 < XjXj < oo,

and otherwise £= £= £= £= £=

0 min(/i, qxj) max(—[i, qxj) q~lx_j q~xXj

if if if if if

Xj < 0 < Xj, Xj = 0, Xj = 0, Xj > 0, Xj < 0,

where q € ]0,1[ is a fixed constant (such as q = 0.01) and variables whose initial interval contains 0 are assumed to be most likely of magnitude fi. The branching coordinate is more difficult to choose, but the speed of a branch and bound algorithm may be heavily affected by this choice. For a good algorithm, the choice should be scaling-invariant, but the details depend on how the algorithm treats the individual subproblems.

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Sometimes, a trisection branching rule is used which splits some component of a box into three intervals. Also, multisection branching rules may be employed; only one natural choice is described here. Suppose we know that a sub-box x° of x cannot contain a solution of (12.1). (In practice, x° would be the intersection of an exclusion box with x; see Section 11.) Then we can cover x \ x° by (at most) In sub-boxes, namely, for j = 1 , . . . , n, X

fe - Xfc - Xfc 2 1 v ^ — W r°l 2

x

2J-! _ V J _ V l /c "~ x fc "~ x fc

K< " h 2j v — \T° T 1

(12 A)

if k ^> i K > 3-

n

However, this may yield long and thin slices and is then rather inefficient. For a comparison of some branching rules for bound-constrained problems see Csendes and Ratz (1997), Ratz (1996), Ratz and Csendes (1995). The bounding rule in its classical variant requires the solution of a convex relaxation, i.e., a convex (and often linear) optimization problem whose feasible set contains the feasible set of the subproblem [outer approximation) and whose objective function is at no feasible point larger than the original objective function [underestimation). If the convex problem is infeasible, the subproblem is infeasible, too, and can be discarded. If the convex problem is feasible, its solution provides a lower bound on f(x), and when this lower bound is larger than the value of / b e s t for some feasible point xbest known (stored in a list of best feasible points found so far) we conclude that the subproblem no longer contributes to the solution of the global optimization problem and hence can be discarded. Clearly, this procedure is equivalent to adding the constraint f{x) < / b e s t to the definition of the subproblem and checking infeasibility of the resulting reduced subproblem. This suggests a more general approach to defining subproblems by adding other cuts, i.e., derived inequalities that have to be satisfied at a global minimizer. If these inequalities are linear, the cuts define hyperplanes and are referred to as cutting planes; cf. Section 16. Branch and bound methods using cuts are frequently labelled branch and cut. Another important approach to handling subproblems uses constraint propagation and related techniques that define reduction (also called tightening, narrowing, filtering or pruning) rules which serve to reduce (as much as easily possible) the box defining a subproblem without changing its feasible set. If reduction results in an empty box, the subproblem is eliminated; if not, the subproblem may still have been reduced so much that many branching steps are saved. Fast and simple reduction rules use constraint propagation, discussed in Section 14; more expensive rules are discussed in Section 15. The balancing of work done in reduction versus work saved through less branching is a delicate matter, which at present more or less depends on ad hoc recipes.

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Note that reduction techniques may be applied not only to the original constraints but to all constraints that must be satisfied at the global minimizer. This includes cutting planes (see Section 16) and the equations and inequalities derived from the Karush-John optimality conditions (see Section 5). In particular, software based on interval techniques (GlobSol (Kearfott 19966), Numerica (Van Hentenryck et al. 19976)) make essential use of the latter.

13. The role of local optimization Local optimization routines are an important part of most global solvers. They are used for two different purposes. (i) To find feasible points if the feasible domain has a complicated definition, and tofindbetter local minimizers when (after successful tunnelling) a feasible point better than the previously best local minimizer has been found. (ii) To solve auxiliary optimization problems such as relaxations of the original problem (for generating improved bounds) or bound-constrained approximations (for tunnelling). 13.1. Relaxation The auxiliary local optimization problems that need to be solved are simpler in structure since they 'relax' the problem in some way. A relaxation is a modification of the original problem whose solution is tractable and gives some information about the possible location of the global minimizer. In the past, mainly linear and convex relaxation have been used, since for these, local optimization provides global solutions, which usually implies useful global information about the original problem. We shall discuss various ways of obtaining and using linear and convex relaxations in Section 16. Nonconvex relaxations may be useful, too, if they are reliably solvable to global optimality. We therefore discuss semilinear relaxations - which can be solved by MILP techniques - in Section 18. 13.2. Tunnelling One may consider solving a global optimization problem as a sequential nonlinear programming method (SNLP), where local optimization (NLP) steps that improve a feasible point to local optimality alternate with tunnelling steps that produce better (nearly) feasible points by some tunnelling procedure. For complete methods based on branching, the 'tunnelling' is done by finding nearly feasible points during inspection of the subproblems.

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The success of the tunnelling step depends on the details of looking for such points. One strategy (Dallwig et al. 1997) proceeds by solving on selected sub-boxes nonlinear least squares problems that minimize the sum of squares of the constraint violations, and (if a best feasible point with function value / b e s t is already available) the violation of f(x) < fhest - A, where A > 0 is some measure of minimal gain in function value. Alternatively, one may use the soft optimality theorem (Theorem 8.1) in place of least squares. (See also Guddat et al. (1990) for tunnelling by continuation.) Thus, in a sense, the global optimization of (4.1) consists in solving a sequence of harder and harder feasibility problems find x s.t. x e x, F(x) € F,

fix) < f

best

xi integral,

(13-1)

- A.

Typical global optimization methods spend perhaps 5% of their time on finding a global minimizer, and the remaining 95% on the verification that there is no significantly better feasible point, i.e., showing that the feasibility problem (13.1) has no solution. Also, hard problems need a significant amount of time to find the first feasible point. Thus the initial and final (dominant) stages of a global optimization solution process are essentially identical to that for a feasibility problem. In particular, general feasibility problems, also called constraint satisfaction problems, can be as hard as general global optimization problems, and the techniques needed for solving constraint satisfaction problems are essentially the same as those for solving global optimization problems. 13.3. General considerations Considerations of superlinear convergence of local optimization algorithms imply that one generally uses sequential quadratic programming (SQP) techniques, which solve a sequence of related quadratic programs whose solution converges (under certain conditions, cf. below) to a local minimizer of the original problem; if the starting point is feasible (which, initially, need not be the case), the function value of the local minimizer is at or below that of the starting point. To give the reader a rough idea of times and difficulties, here are some (completely unreliable but catchy) rules of thumb. If the time needed to solve a linear program of a certain size is LP then solving a problem of comparable size and sparsity structure may take perhaps the time QP = 5 * LP for a convex quadratic program, QP' = 10 * LP

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for a local minimizer of a nonconvex quadratic program,

SQP = 30 * QP for a convex nonlinear program,

SQP' > 200 * QP for a local minimizer of a nonconvex nonlinear program,

GLPf > 100 * SQP for finding a global minimizer of a nonconvex nonlinear program, and GLPV > 1000 * SQP for verifying that it is a global minimizer. We now comment on the properties of local optimization software that are important for their use in global optimization. Usually, it is more important that the local solver is fast than that it is very robust (i.e., guaranteed to succeed), since lack of robustness in some of the local optimizations is made up for by the structure of the global solution process. To help control the amount of work done in the local part, it should be possible to force a premature return with a less than optimal point when some limit (of time or number of function values) is exceeded. Nevertheless, the local solver should be good to ensure that solving a problem with a unique minimizer (which is automatically global) by the global solver does not take much longer than a good local solver would need. Modern nonlinear programming codes are usually 'globally convergent' in some sense. The global convergence proofs (to a local minimizer only!) usually make more or less stringent assumptions that imply the absence of difficulties in finding feasible points. Formally, we may say that a local optimization algorithm is globally convergent if there is a continuous function dfeas : R n —>• R (defining a 'distance to feasibility') such that dfeBs(%) > 0,

with equality if and only if x is feasible

and the algorithm produces for arbitrary continuous problems and arbitrary starting points a sequence of xl € Mn satisfying one of the following conditions: (i) xl converges to the set of points satisfying the Karush-John conditions (and, possibly, second-order necessary conditions); (ii) 0, ||x'|| -> oo; (v) d{eas(xl) converges to a nonzero local minimum of dfeas.

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Conditions (i) and (ii) characterize the achievement of the optimization goal, while conditions (iii)-(v) characterize various modes of unavoidable failure. Failures of type (iii) or (iv) are usually attributed to bad modelling or bad choice of the optimization methods. Some methods such as bundle methods can cope with lack of differentiability, hence do not lead to case (iii). A failure of type (v) is unavoidable if there is no feasible point. However, failures of type (v) may happen for problems with nonconvex constraints even though feasible points exist. One could say that from a local point of view, an optimization problem is easy (for an algorithm) if (v) cannot occur whenever a feasible point exists. A local algorithm may be considered good if among its easy problems are all problems with convex constraints only, and all problems satisfying certain strong versions (Burke 1991) of the Mangasarian-Fromovitz constraint qualification (Mangasarian and Fromovitz 1967). Ideally, a good local algorithm would provide in these cases a certificate of infeasibility whenever it detects case (v).

14. Constraint propagation In many cases, general constraints can be used to reduce the size of a box in the branching scheme. The general technique is called constraint propagation and was pioneered in constraint logic (Cleary 1987, Older and Vellino 1993) and interval analysis (Neumaier 1988), but has also forerunners in presolve techniques in mathematical programming (Mangasarian and McLinden 1985, Lodwick 1989, Anderson and Anderson 1995). See Babichev et al. (1993), Benhamou et al. (1994), Benhamou and Older (1997), Chen and van Emden (1995), Hager (1993), Hyvonen and De Pascale (1996), Kearfott (1991), Van Hentenryck (1989), Van Hentenryck, Michel and Benhamou (1997a), Van Hentenryck et al. (19976) for further developments, and the COCONUT report (Bliek et al. 2001) for an extensive recent survey. We follow here the set-up by Dallwig et al. (1997), which handles linear constraints (and more generally block-separable constraints) without the need to decompose the constraints into primitive pieces defined by single operations. (In the following, if J is a list of indices, xj denotes the subvector of x formed by the components with index in J.) Proposition 14.1.

Let the qk be real-valued functions defined on xj fc .

(i) If (for suitable gfc,s) qk>sup{qk(xjk)\xjk€xJk},

s>^qk,

(14.1)

k

then, for arbitrary a, k{xjk)

=> qk(xjk) > a-s + qk for all k. (14.2)

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(ii) If M{qk(xJk)

1 xJk

€ x

(14.3)

J f c }' k

then, for arbitrary a, x € x,

Jk)
Qk(xjk) < a

=>

— s + q.

k

Proof. The assumptions of part (i) imply ^

> a + qk - s,

hence the conclusion in (14.2) holds. Part (ii) is proved in the same way. • The proposition is applied as follows to reduce the size of boxes by tightening bound constraints. Suppose that x E x. For any constraint of the form

Y

(14.5)

we form the quantities (14.1). (This is straightforward if the qk depend on a single variable x^ only, Jk = {&}; in the most important cases, qk is linear or quadratic in xk, and the supremum is very easy to calculate; in more complicated cases, upper (resp. lower) bounds can be calculated with interval arithmetic.) Then one checks the condition a<-s.

(14.6)

If it is violated then (14.5) is clearly inconsistent with x G x (and in the branch and bound application, the corresponding subproblem can be discarded). If (14.6) holds, one can exploit the conclusion in (14.2), provided that one can compute the set of xjk £ xjfc (or a superset) such that qk{xjk)>qk + a-s.

(14.7)

If a is sufficiently close to s then xjk will be forced to be close to the global maximum of qk over the interval xjk, thus reducing the component xjfc and hence the box x. This procedure can be applied for each k in turn to get an optimally reduced box. One can similarly proceed for block-separable constraints of the form J2 qk{xjk) < a. (The reader might wish to reconsider the example in Section 10 in the light of the above result.) In the separable case (Jk = {k}), computing the set of xk with (14.7) is easy, especially for linear or quadratic qk. If qk is nonmonotonic, it may happen that the resulting set is disconnected; then one has to make a choice between taking its convex hull - which is an interval - or of considering splitting the box into sub-boxes corresponding to the connected components.

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In the case of two-sided constraints YlQk(xJk) G a : which includes the equality constraint J2qk(xjk) = qo for a = qo, one can combine (14.2) and (14.4) using interval arithmetic as follows. (See (11.7) for the inner addition ©.)

Proposition 14.2. Suppose that qfc 2 U{Qk(xJk) I xJk Gx J f c },

r D a - ^ .

(14.8)

(i) If 0 ^ r then the conditions r <= v

V f l i f r r 1 Pa

( 1 4 Q")

fc are inconsistent. (ii) Any x satisfying (14.9) also satisfies qk{xjk) &r®qk

for all

fc.

(14.10)

Proof. (14.9) implies 0 G a — J2ik{xjk) C a — J^q^, hence 0 G r. Now suppose that 0 G r. In the notation of the previous proposition we have

qk{xjk) G [a - s + qk,a - s + qk] = [a-s,a-s}

©qfc,

and since r = a - [s, s] = [a - s, a — s], this implies (14.10).

D

Again, condition (14.10) can be used to reduce xjfc whenever Qfc^rffiqfc.

(14.11)

We give details for the most important case of quadratic (and including linear) functions, dropping indices for a moment. Proposition 14.3.

Let c be an interval, o, b G M, and put

d : = (62 + 4ac)+,

w := y/d (if d / 0).

Then

ax2 + bx G c} = <

f =^

U =*££

if d = 0, if a = b = 0 ^ c, if a = b = 0 G c, if a = 0, otherwise.

Proof, ax2 + bx = c G c is equivalent to x = c/b when a = 0, and to x = (—6± Vb2 + 4ac)/2a otherwise; in the latter case, the expression under the square root must be nonnegative and hence lies in d. Since the varying c occurs in these formulas only once, the range over c G c is given by c/b if a = 0 and by (—6 ± y/d)/2a otherwise (use monotonicity!). •

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Note that the differences in Proposition 14.1 and the numerators in Proposition 14.3 may suffer from severe cancellation of leading digits, which requires attention in an actual implementation. In the application to reducing boxes, one must of course intersect these formulae with the original interval. If the empty set results, the subproblem corresponding to the box x can be eliminated. (But remember to be cautious when using unsafe interval arithmetic!) If a disjoint union of two intervals results we either split the box into two boxes corresponding to the two intervals or leave x& unchanged; the first alternative is advisable only when the gap in the interval is quite large. All reduction techniques may be used together with the technique of shaving, which may be seen as an adaptation of the probing technique in mixed integer programming. The idea is to try to remove a fraction of the range [XJ, Xi] of some variable x% by restricting the range to a small subrange [x_j, £] or [£,x"j] at one of the two end-points of that variable, and testing whether reducing the small slab obtained in this way results in an empty intersection. If this is the case, the range of x\ can be restricted to the complementary interval [£,XJ] and [x_j,£], respectively. While more expensive, it reduces the overestimation in the processing of constraints which contain a variable several times. In practice, one would perhaps try to shave away 10% of the length of an interval. 14-1- Consistency concepts In constraint logic programming (see the book by Van Hentenryck et al. (19976) and the references at the beginning of this section), there are many consistency concepts that describe the strength of various reduction techniques. Essentially, a box is consistent with respect to a set of reduction procedures if their application does not reduce the box. A simple recursive argument invoking the finiteness of machine-representable boxes shows that every box can be reduced to a consistent box with finitely many applications of the reduction procedures in an arbitrary order. (Depending on the rules used, the resulting reduced box - called a fixed point of the reduction procedures - may or may not depend on the order of applying the rules.) From a practical point of view, it is not advisable to apply the available rules until the fixed point is reached. The reason is that frequently the first few reductions are substantial, and later ones only reduce the box by tiny fractions; the convergence speed may be arbitrarily slow. For example, for the pair of constraints x\ + x^ = 0, x\ — qx^ = 0 with q e]0,1[, the unique fixed point (with respect to the simple reduction described above) reduces the volume in each step by a factor of q. For q close to one this is very inefficient compared to, say, a linear programming relaxation (which gives the result immediately).

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Thus one has to be selective in practice, using suitable strategic rules for when to use which reduction strategy. The choice is usually done by various ad hoc recipes that balance the likely gain and the amount of work needed. Moreover, fine-grained interaction between different computations to avoid some unnecessary computation, such as that described in Granvilliers (2001) may be decisive in getting optimal performance. 14-2. Semiseparable constraints With some more work, the above techniques can also be utilized for semiseparable constraints. We need the following result (essentially a form of the Cauchy-Schwarz inequality). Lemma 14.4. If A is a rectangular matrix such that ATA is nonsingular then \uk\ < ^J{{A?A)-^)kk\\Au\\2

for all u.

Proof. Let A = QR be an orthogonal factorization of A with QTQ = / and R square nonsingular. Then ATA = RTR and ||Au||2 = ||-R«||2- Since

K| = the assertion follows.

•

Now suppose that we have a semiseparable inequality of the form k(xk) + (x- x°)TH(x - x°) < a,

(14.12)

with possibly nonsymmetric H. Using a modified Cholesky factorization (Gill and Murray 1974, Schnabel and Eskow 1990) H + HT = RTR - D with a (nonnegative) diagonal matrix D, we can rewrite (14.12) as

0 < h\R(x - x°)\\22 < a -J^qkix,) 1

k

+ \{x - x°)TD{x - x°).

(14.13)

l

The right-hand side of (14.13) is a separable quadratic form, hence can be written as a - Y, Qk{xk) with qk{xk) = qk(xk) - \Dkk{xk - x°k)2. Therefore, Proposition 14.1(ii) applies. Moreover, one gets the extra inequality \\R(x-x°)\\22<2(a-s),

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which together with the lemma gives the further inequalities

\xk - x°k\ < ^J2(a - s)((RTR)-^)kk,

(14.14)

which may help to reduce x^. 14.3. Block-separable constraints For only block-separable constraints (\Jk\ > 1), the qk are multivariate, and we need to resort to suboptimal interval techniques. How to exploit the enclosures from Proposition 14.1 and 14.2 to reduce the box depends on the special form of the qk. In many cases, we can in turn solve the conditions directly for each variable involved, substituting an enclosing interval for all other variables. If this is not possible directly we can use the mean value form (or another centred form) to rewrite a constraint Fi(x) € Fj as

this is now a separable expression with interval coefficients that can be processed as above to reduce the box. This way of proceeding, dating back to Neumaier (1988), is called conditioning in Van Hentenryck et al. (19976), and is used in the Numerica package. Similarly, by using a Taylor expansion to second order with an interval Hessian, we get a semiseparable expression with interval coefficients that can in principle be processed as above. (However, the interval coefficients cause here additional complications.) In the context of Taylor models of arbitrary order, a variation of this (with thin coefficients and an interval remainder term) has been used in the linear dominated bounder of Makino (1998); cf. the discussion in Makino and Berz (2003) and Neumaier (2002).

15. The cluster problem and second-order information When programming a simple branch and bound algorithm for global optimization, one quickly notices that it is fairly easy to eliminate boxes far away from the global minimizer, while, especially in higher dimensions, there remains a large cluster of tiny boxes in a neighbourhood of the global minimizer that is difficult to eliminate. The occurrence of this situation is called the cluster problem. Often, algorithms try to avoid the cluster problem by providing only a A-optimal solution; i.e., the program stops when it has shown that there is no feasible point with an objective function value of jbest _ ^ w here / b e s t is the function value of the best feasible point found so far. However, when A is small (as we want) then the cluster problem is still present, although to a less pronounced degree.

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Kearfott and Du (1994) studied the cluster problem for unconstrained global optimization, and discovered that the source of the problem was the limited accuracy with which the function values were bounded. In particular, they showed that the cluster problem disappears if, for x in a box of diameter O(e), one can bound the overestimation of f(xbcst) - f(x) by O(e3). Here we give a simplified version of their result. Let x e Mn be a global minimizer of f(x), and let G be the Hessian at x. Near the global minimizer, we have

f(x) = f(x) + \{x - x)TG(x -x) + O(\\x - xf) since the gradient vanishes at x. Suppose we can bound the objective function value over a box of diameter e with an accuracy of A = Kes+1, s < 2. Then no box of diameter e containing a point x with ^(x — x)TG(x — x) + O(\\x — x|| 3 ) < A can be eliminated. For sufficiently small A, this describes a nearly ellipsoidal region with volume proportional to W(2A)"/det G, and any covering by boxes of diameter e contains at least consty (2A) n /(£ n det G) boxes. The number of uneliminated boxes is therefore proportional to at least (const/e) n / 2 /VdetG const71/ det G const-e)"/det G

if s = Q, if s = 1, if s = 2.

We see that for s — 0, the number grows immensely as e gets small. For s = 1, the number of boxes needed - while (for small e) essentially independent of g - may still grow exponentially with the dimension, and it is especially large for problems where the Hessian at the solution is ill-conditioned. However, the number is guaranteed to be small (for small e) when s = 2. For pure constraint satisfaction problems, a similar cluster effect is present (Schichl and Neumaier 2004), but with order reduced by one; thus the quadratic approximation property available from methods exploiting firstorder information only (such as centred forms) already avoids the cluster effect, except in degenerate cases. However, especially near poorly conditioned solutions, the size of boxes that can be eliminated is significantly larger if second-order information is used (Schichl and Neumaier 2004). In the case of nonisolated solution sets, some clustering seems unavoidable, but Lyapunov-Schmidt reduction techniques (Neumaier 2001a) might prove useful. The problem of covering nonisolated solution sets efficiently with a small number of boxes is discussed in considerable algorithmic detail by

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Xuan-Ha Vu, Sam-Haroud and Silaghi (2002, 2003) for the case of pure constraint satisfaction problems; see also Chapter 7 of the COCONUT report (Bliek et al. 2001). For constrained global optimization, similar arguments as for the unconstrained case apply in a reduced manifold with the result that, in the formulas, n must be replaced by n — a, where a is the maximal number of constraints active at the solution, with linearly independent constraint gradients. Clearly, to bound the overestimation over a box of diameter O(e) by O(e 3 ) requires that we know the Hessian up to O(e), and that we are able to bound the deviation from a quadratic model. (Actually, the above argument shows that o(e2) is sufficient, but this still requires the knowledge of the Hessian up to o(l).) Thus it is necessary to have access to second-order information. Unfortunately, in higher dimensions, no cheap method is known that bounds function values over an arbitrary narrow box of diameter O(e) close to a minimizer by O(e3). In a single dimension, cheap methods are known; see Cornelius and Lohner (1984) and Neumaier (1990, Section 2.4). In dimension > 1, peeling methods together with Taylor expansions work with an effort that grows like O{v? • 3 n ); see the discussion in Neumaier (2002, Section 5). Fortunately, however, it turns out that by using interval Hessian matrices (which, for 3-times differentiable functions have the required O(e) accuracy: see Neumaier (1990, Section 1.4)), there are several ways to avoid the cluster problem, at least when the global minimizer is nondegenerate, i.e., satisfies the second-order sufficient conditions for a local minimizer.

15.1. Using Hessian information Explicit global Hessian information can be used, as in GlobSol (Kearfott 19966) and Numerica (Van Hentenryck et al. 19976), by interval Newton methods (see Section 11) applied to the Karush-John conditions discussed in Section 5. These may verify the existence of a unique solution of the KarushJohn conditions (Theorem 5.1 and equation (5.4)) in some box around the best point found, and hence allow us to shrink that box to a single point. Alternatively, we may use global Hessian information to verify the secondorder sufficient conditions for a global minimizer given in Neumaier (1996). They apply to smooth nonlinear programs of the form

f(x) s.t.

x e x,

F(x) = 0.

Thus it is necessary to introduce slack variables to rewrite general inequality constraints as equality constraints. The sufficient condition is as follows.

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T h e o r e m 15.1. Let x be a Kuhn-Tucker point for the nonlinear program (15.1), with associated multiplier z, and let y := f'(x)T

- F'(x)Tz,

(15.2) (15.3)

If, for some continuously differentiable function ip : R m —> R with p(0) = 0,


(15.4)

the generalized augmented Lagrangian L(x) := f{x) -
(15.5)

is convex in [u,v], then a; is a global solution of (15.1). Moreover, if L(x) is strictly convex in [u,v], this solution is unique. A choice for ip that works in some neighbourhood of a strong global minimizer (i.e., one in which sufficient second-order conditions for local optimality hold) is given in Neumaier (1996), together with further implementation hints. The convexity can be checked by means of interval arithmetic; see Section 11. If these conditions hold in some box, we can shrink this box to a single point. We can use any of these techniques to construct boxes y that are guaranteed to contain no global minimizer unless already detected, resulting in exclusion constraints. An exclusion constraint is a constraint of the form

It can be used to reduce an arbitrary box x by intersecting it with y and taking the interval hull, which may result in a smaller box. If there were no reduction but the intersection were strictly contained in x, then we might also want to resort to multisection: cf. (12.4). Interesting exclusion boxes are those that are constructed around local minimizers, since this helps fighting the cluster problem. It is possible (though probably not most efficient) to base global optimization algorithms on exclusion methods alone; see the work of Georg et al. (Allgower, Erdmann and Georg 2002, Georg 2001, Georg 2002), who also give associated complexity results. 15.2. Backboxing Whenever we have a tentative approximate global minimizer x, we try to find simultaneously a large box x and a tiny box z such that any global minimizer x G x satisfies x £ z. This allows us to use x as an exclusion

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region while z is stored in an output list as a box containing a putative minimizer: this is termed backboxing. (After terminating the branching process, these boxes need to be checked again for possible elimination.) Since we expect that x has a function value optimal within O(e), but knowing that this only enforces that x has an accuracy of O(y/e) (possibly less in the case of singular Hessians), we start with a box x = [x — y/eu, x + y/eu] for some vector u reflecting the scaling of the variables, and apply the available reduction techniques until no significant improvement results. Call the resulting box z. If second-order techniques are used to do the box reduction, then z is usually a tiny box or empty. If z is empty, x was not a good approximation but we know that x contains no solution. If z is nonempty, it is likely that z contains a solution. Indeed this is always the case if only interval Newton-like reduction techniques are used and z C int x. (This requires some qualifying conditions that ensure that we can verify sufficient existence conditions such as those in Neumaier (1990, Chapter 5.3-4).) Thus we may store z in a list of output boxes together with a flag whether existence (and possibly uniqueness) was verified. If z is still a box of significant size, we must have been close to a degeneracy; splitting would probably not improve this and lead to an exponential number of boxes; thus it is preferable to put this box also in the list of output boxes to indicate that a low resolution candidate for a solution has been found. (This way of handling degeneracies is due to Kearfott (1996a).) No matter what case we have been in, we always know that x cannot contain a solution not yet in the output list. Therefore, we may add the exclusion constraint x £ x to the problem description. However, one can often make x even bigger. So we try recursively x° = x, z° = z, x

but z° = mid(x) if z = 0,

x'' = 2x'~ 0 z'" z'"11,

zl = reduce(x');

using the available ways of reducing xl, stopping when zl C x ' " 1 or zl = xl. (For the inner subtraction 0 , see (11.8).) Then we have the generally stronger new exclusion constraint x <£ x'. (This way of generating exclusion constraints, using interval Newton methods, is due to Van Iwaarden (1986), who calls the technique backboxing, and is part of GlobSol (Kearfott 19966).) Recent methods by Schichl and Neumaier (2004) for constructing large exclusion boxes can be combined with this iterative approach. 15.3. Finite

termination

Closely related to the cluster problem is the question of finite termination. i.e., whether branch and bound algorithms find (assuming exact arithmetic)

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a global optimizer with a finite amount of branching only. This is not easy to achieve, and in practice, most algorithms are content with working towards e-optimality, i.e., finding a (nearly) feasible point within e of the true but unknown optimal function value. Theoretical finite termination guarantees are available only for problems where the optimum is attained at extreme points (Al-Khayyal and Sherali 2000, Shectman and Sahinidis 1998). However, in practice, algorithms based on the explicit use of second-order interval information (either via interval Newton operators or via second-order sufficient conditions) have finite termination behaviour on problems with a nondegenerate global minimizer, and it is likely that this can be proved theoretically. In the case of degeneracies, behaviour of branch and bound methods can become arbitrarily poor. However, the situation may improve in cases where the degeneracy can be removed by identifying and eliminating redundant constraints causing the degeneracy. To do this rigorously requires care; see Huyer and Neumaier (2004) for first results in this direction. 16. Linear and convex relaxations One of the highly developed sides of global optimization is the use of linear and convex relaxations to find a lower bound for the value of the objective function, which makes it possible to discard boxes where this lower bound is larger than the function value / b e s t of the best feasible point found so far. The details are well covered in several books (Floudas 1995, 1999, Tawarmalani and Sahinidis 2002 b), so we are brief here and only describe the basic issues and recent extensions that are not widely known. 16.1. Reformulation-linearization McCormick (1976) introduced the notion of a factorable function (composed of finitely many unary or binary operations), and constructed non-smooth relations for such functions. Kearfott (1991), and perhaps others before him, noticed that by introducing intermediate variables, every factorable optimization problem can be rewritten in a form in which all constraints are unary, z = (p{x), or binary, z = x o y. Independently, Ryoo and Sahinidis (1996) proposed using in place of these constraints implied linear constraints (so-called linear relaxations) to generate a set of linear inequalities defining a polyhedral outer approximation. Since the objective can be represented by a single variable and another constraint, this allows one to find a linear programming relaxation for arbitrary factorable optimization problems. Linear relaxations for unary operations are easily found by a simple graphical analysis of the various elementary functions. In particular, for a convex function, the secant between the end-points of the graph is an overestimate,

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and any tangent is an underestimate; frequently, taking the two tangents at the end-points is already quite useful. For concave functions, the reverse situation holds, and in the general case one may also need to consider bitangents, and tangent secants. Since powers can be written in terms of exp, log and the product, the only binary operations that need to be analysed are products and quotients. Assuming that bounds x 6 x, y e y for the factors are available, McCormick proposed for the product z = xy the relaxations yx + xy — xy < z < yx + xy — xy, yx + xy — xy < z < yx + xy — xy, which follow immediately from (x — x)(y — y) > 0 and three similar inequalities. Al-Khayyal and Falk (1983) showed later that these inequalities are indeed best-possible in the sense that any other generally valid linear inequality is a consequence of these and the bound constraints. (One says they form the convex and concave envelope.) For the quotient x = z/y, exactly the same formulas are valid with x = z/y, but remarkably, one does not get the envelope in this way. For example, the following inequality, due to Zamora and Grossmann (1999), is not implied. P r o p o s i t i o n 16.1. Let x , y , z be nonnegative intervals. If x € x, y € y, and z = xy € z then

-±^) 2 .

(16.1)

Inequality (16.1) describes a convex set in the nonnegative orthant of M3, although the inequality itself is not convex. However, it is the prototype of a convex conic constraint (see below) and can be exploited by solvers for second-order cone programs. Therefore, adding this constraint gives a relaxation that may be tighter than the McCormick relaxation. The general formulas for the convex envelope of a quotient, derived explicitly in Tawarmalani and Sahinidis (20026), are quite complicated. Crama (1993) showed that the following bounds define the optimal convex relaxation of a product of factors bounded in [0,1]. P r o p o s i t i o n 16.2.

If Xi G [0,1] (i = 1 : n) and z = x\ • • • xn then < 7 < T

< 1 (i — 1 • T)\

^ / i ^ X j ^ J - ^ 4

— 1 . II).

Mfi

9)

\±<J.6J

fe=i

More generally, arbitrary multilinear functions have an optimal convex relaxation (the envelope) defined by finitely many linear inequalities. For this result, and for other methods for getting linear relaxations of non-

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convex programs based on the reformulation-linearization technique: see Rikun (1997), Sherali (1997), Sherali and Tuncbilec (1995, 1997a, 19976), Al-Khayyal, Larsen and van Voorhis (1995), and Audet, Hansen, Jaumard and Savard (2000). Also related is the lift-and-project technique in mixed integer linear programming; see, e.g., Balas, Ceria and Cornuejols (1993). Note that this approach using the factorable or a nearly factorable form generally results in problems with many more variables than in the original problem formulation. Nevertheless, since the resulting linear or convex programs are extremely sparse, the technique can be very useful, especially for larger problems. In particular, this is the main workhorse of the global optimization packages BARON (Tawarmalani and Sahinidis 20026) and LINGO (Gau and Schrage 2004). Because linear programming solvers are currently much more reliable and faster than general convex solvers, the convex envelopes used in BARON are in fact approximated by a number of linear constraints computed adaptively with a variant of the sandwich algorithm of Rote et al. (Burkard, Hamacher and Rote 1992, Rote 1992). Independently of the way a linear relaxation is produced (see below for alternatives that work without additional variables), the information in the linear relaxation can be exploited not only to get lower bounds on the objective or to eliminate a subproblem, but also to reduce the box. Cheap marginals-based range reduction techniques using Lagrangian multipliers are described in Tawarmalani and Sahinidis (20026) and are implemented in BARON. Recent results of Lebbah et al. (2004) and Lebbah, Rueher and Michel (2002) show that the more expensive approach of minimizing and maximizing each variable with respect to a linear relaxation (which BARON 5.0 did only at the root node of the branch tree) may give a significant speed-up on difficult constraint satisfaction problems, and are now part of the default strategy in BARON 6.0. 16.2. Semidefinite relaxations Starting quite recently, a large number of papers appeared that propose the use of semidefinite relaxations, or convex conic relaxations, to solve polynomial constraint satisfaction and global optimization problems: see, e.g., Chesi and Garulli (2001), Jibetean and De Klerk (2003), Fujie and Kojima (1997), Kim and Kojima (2001, 2003), Kim, Kojima and Waki (2003), Kojima, Kim and Waki (2003), Kojima and Tuncel (2000), Lasserre (2001), Meziat (2003), Gatermann and Parrilo (2004), Parrilo (2003), Parrilo and Lall (2003), Parrilo and Sturmfels (2003). These techniques are implemented in two software packages, GloptiPoly (Henrion and Lasserre 2003a, 20036, 2003c) and SOSTOOLS (Prajna et al. 2002). Being developed completely independently from the mainstream in global optimization, these packages do not incorporate any of the other global techniques, and hence

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are currently restricted to problems with few variables (say, below 20). But since they are able to solve many of these problems to global optimality without doing any branching, their combination with the other techniques, in particular with branch and bound, appears to be highly promising. The background of these methods is that constraints of the form is positive semidefinite,

(16.3)

where the Ak are symmetric (or complex Hermitian) matrices, so-called semidefinite constraints, define convex sets, and that constraints of the form ||Ac-&||2 < aTx + a or \\Ax — 6||| < XiXj,

Xi, Xj G M + ,

so-called second-order cone constraints, describe convex conic sections. Problems with a linear or convex quadratic objective and an arbitrary number of such constraints (in addition to linear constraints) can be efficiently solved using interior point methods. Therefore, convex conic and semidefinite relaxations of nonlinear constraints can be efficiently exploited. Books and surveys emphasizing the nonlinear case include Alizadeh (2003), de Klerk (2002), Lobo, Vandenberghe, Boyd and Lebret (1998), Todd (2001), Vandenberghe and Boyd (1996) and Wolkowicz, Saigal and Vandenberghe (2000); for software, see the semidefinite programming homepage (Helmberg 2003) and the package SeDuMi (see SeDuMi (2001) and Sturm (1999)), on which both GloptiPoly and SOSTOOLS are based. The basic idea behind semidefinite relaxations is the observation that, given any set of basis functions ipi(x) and any nonnegative weight function w(x), the matrix M with components Mik =

w(x)(pi(x)ipk(x)

is always symmetric and positive semidefinite. If the 0 and v(x) > 0, their product is again such a constraint. Thus lots of additional polynomial constraints can be generated and used. Results from algebraic geometry can then be invoked to show that infeasibility and e-optimality can always be achieved by using sufficiently high degrees, without the need for any problem

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splitting. Apparently, in many cases, relatively low degrees suffice, which is fortunate since the number of intermediate variables would otherwise become excessively large. Moreover, problem symmetry can be exploited by using basis sets with corresponding symmetry properties (Gatermann and Parrilo 2004). The conic and semidefmite relaxations produced in this way also result in problems with many more variables than in the original problem formulation, but since semidefinite relaxations are often much stronger than linear relaxations, the effort required to solve these large problems may be well spent if a subproblem is solved without the need to split it into many smaller pieces. Since problems with semidefinite constraints involving larger matrices are more expensive to solve than those with convex conic constraints, the latter are in principle preferable, but conclusive results on the best way of using or combining the various possible relaxations are not yet available. For semidefinite relaxations of certain fractional functions see Tawarmalani and Sahinidis (2001, 2002a, 20026). 16.3. Relaxations without extra variables In place of introducing additional variables for nonlinear intermediate expressions, it is also possible to relax the original constraints directly. Apart from McCormick's non-smooth convex relations (McCormick 1976), which are difficult to use, this can be done in two different ways. The first possibility is to write the constraints as a difference of convex functions (DC representation). The package aBB (see Adjiman et al. (1996, 1998a, 19986), Adjiman and Floudas (1996), Androulakis et al. (1995)) uses DC-techniques, by separating in each inequality constraint h(x) < 0 a recognizable linear, convex or concave parts from a 'general' remainder. Linear and convex parts are kept, concave parts are overestimated by secant-type constructions, and general terms are made convex by adding a nonpositive separable quadratic function. This ensures that a convex underestimating inequality results. More specifically, if f(x) is twice continuously differentiable at all s in a neighbourhood of a box x and D is a diagonal matrix with nonnegative entries, then frel(x) := f(x) + - (X - X)TD(X - x) is an underestimator of f(x) on the box, and the amount of underestimation is bounded by

- f(x)\ < Jradx r Dradx,

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(16.4)

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attained at the midpoint. (At the vertices there is no overestimation.) If the Hessian G(x) lies in the interval matrix G for all x € x (such a G can be found by interval evaluation of the Hessian, e.g., using automatic differentiation) and all symmetric matrices in G + D are positive semidefinite then /rei is convex. The latter condition can be checked as in Theorem 11.1; the difficulty is to choose D in such a way that this condition holds and the underestimation bound in (16.4) is kept small but the work in getting D remains reasonable (Adjiman et al. 1998a, 19986). Recent, more advanced convexity-enforcing corrections are discussed in Akrotirianakis and Floudas (2004). More general DC-techniques are treated extensively from a mostly theoretical point of view in the book by Horst and Tuy (1990); see also the overview in Tuy (1995). Apart from what is used in aBB (and described above), these techniques have not materialized in available codes; however, see, e.g., Le Thi Hoai An and Pham Dinh Tao (1998) for some recent numerical results. General techniques for recognizing convexity automatically are discussed in forthcoming work of Fourer (2003) and Maheshwari, Neumaier and Schichl (2003). Other questions related to the semidefiniteness of an interval matrix are discussed in Jaulin and Henrion (2004). The second possibility is to use centred forms. The Frontline Interval Global Solver constructs linear enclosures based on a centred form (in fact a first-order Taylor form), + cT(x-z),

f(x)£f

(16.5)

using forward propagation in an automatic differentiation-like manner, described in Kolev and Nenov (2001). Since the coefficients of the linear term in (16.5) are real numbers, this directly gives two parallel linear functions which underestimate and overestimate f(x):

/ + cT(x - z) < f(x) < / + cT(x - z). The COCONUT environment constructs a centred form using slopes and automatic differentiation-like backward propagation, according to formulas given in Schichl and Neumaier (2003), from which linear enclosures are constructed. Indeed, given a centred form

we have the linear underestimator f(x) >i + cT(x-x), where

+ sT(x-z),

a=^

J

'

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'-.

COMPLETE SEARCH IN CONSTRAINED GLOBAL OPTIMIZATION

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A similar formula provides a linear over estimator. Geometrically, the formulas amount to enclosing the double cone denned by the centred form by a pair of hyperplanes; since linear functions are separable, the formulas derived from an analysis of the univariate case can be applied componentwise. 17. Semilinear constraints and MILP Let us call a constraint semilinear if, for arguments x in a bounded box x, it is equivalent to a finite list of linear constraints and integer constraints; usually the latter involve additional auxiliary variables. The objective function f(x) is called semilinear if the inequality f(x) < xo, where XQ is an additional variable, is semilinear. A semilinear program is an optimization problem with a semilinear objective function and a bounded feasible domain defined by semilinear constraints only. Since we can rewrite an arbitrary global optimization problem min f(x) s.t. x e C in the form min s.t.

XQ

x
it is clear from the definition that any semilinear program can be rewritten as a mixed integer linear program by the introduction of additional variables. The remarkable fact that every factorable optimization problem can be arbitrarily closely approximated by semilinear programs (see Section 18) implies that we can use MILP software to obtain arbitrarily good approximate solutions of factorable optimization problems. To make this observation computationally useful we need to handle two tasks. (i) Find interesting classes of semilinear constraints and constructive procedures for translating such constraints into linear and integer constraints. (ii) Show how to approximate factorable constraints by semilinear constraints: see Section 18. In this section we look at task (i). This is in principle well known, but usually considered to be part of the modelling process. For good overviews of the modelling related issues see, e.g., Floudas (1995, Section 7.4), Williams (1999) and (in German) Kallrath (2002). Here we simply give the underlying mathematical substance. All linear constraints and integer constraints are trivially semilinear. A binary constraint ze{0,i}

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is semilinear, since it can be written in the equivalent form

ze[o,i], zez. We call a list XK of variables constrained by = 1,

x ke

{0,1} (keK),

(17.1)

keK

where K is some index set, a binary special ordered set (BSOS) (c/. Dantzig et al. (1958), Beale and Tomlin (1970)). Clearly, the constraint xK

is a BSOS

(17.2)

is also semilinear. Because (17.1) can hold only if all but one of the Xk (k € K) vanish, (17.2) is equivalent to requiring that xK = e(fc)

for some k,

(17.3)

where e^k> is the unit vector with a one in position k and zeros elsewhere. (A binary special ordered set of size two is just a pair of complementary binary variables, and one of its variables is redundant.) Note that a BSOS is a special ordered set of type 1, and can be handled efficiently by most MILP codes. Since special ordered sets, defined more generally as sets of variables such that at most one - type 1 - or two (which then must be adjacent) - type 2 - are nonzero, are ubiquitous in MILP formulations, any MILP solver has special facilities to make efficient use of special ordered sets. Many techniques for translating semilinear constraints are consequences of the following basic result. Theorem 17.1. Let Fk : Co -> M.mk(k = 1,... ,d) be scalar- or vectorvalued functions such that Fk(x) > Fk

for all x e C o

(17.4)

with F_k € Rmk. Then there is a point x € Co such that Fi(x) > 0 V

•••

V Fd(x)>0

(17.5)

if and only if there are z G Rd and x € Co such that z Fk(x)

is a BSOS,

(17.6)

> Fk{l - z k ) for a l l k = l,...,d.

(17.7)

(The symbol V denotes the logical operation or. The operation and is implicitly given by the comma, and we follow the convention that and is binding more strongly than or.)

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Proof. If (17.5) holds then Fk(x) > 0 for some k, and z = e^ satisfies (17.6) and (17.7). Conversely, if (17.6) holds then z = e^> for some k, and (17.7) implies Fk(x) > 0; the other constraints in (17.7) are automatically satisfied because of (17.4). • Note that (17.4) can always be satisfied if Co is bounded and the Fk are continuous. A constraint of the form (17.5) is called a disjunctive constraint. The theorem implies that linear disjunctive constraints, where all Fk(x) are affine functions of x, are semilinear if the Fk have known, finite lower bounds on the feasible domain (bound qualification), since then (17.7) consists of linear constraints. In the following, we shall always silently assume the bound qualification. (In practice, this is usually enforced where necessary by ad hoc 'big M' domain restrictions. In rigorous solvers, this is of course forbidden.) More generally, linear disjunctive constraints of the form € b>! V

• • • V Adx <E bd

(17.8)

are semilinear, since we can rewrite each Akx &bk in the form

(f>

0 \bk - AkxJ

.

Note that we can rewrite (17.8) in the equivalent form Akx e bfe for some k € {1 : d}. (17.9) Since many practically relevant constraints can be cast in the form (17.8), this makes the theorem a very useful tool for recognizing semilinear constraints and translating them into a MILP formulation. (There is also an extended literature on disjunctive programming not based on transformations to MILP; for pointers see Balas (1979), Jeroslow (1977), Sherali and Shetty (1980).) For example, semicontinuous (semi-integer) variables are variables Xk constrained by xk = 0 V xk <E a

(17.10)

and xk = 0 V xk E a,

xk E Z,

(17.11)

respectively, which are semilinear constraints. A numerical special ordered set (NSOS) is a vector A € Md such that d

A>0,

2Afc

= 1

'

fc=i

at most two Afe are nonzero, and nonzero A^ must have adjacent indices.

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Since the latter condition can be formulated as Afc + Afc+i = 1 for some k, it is disjunctive; hence the constraint xK is an NSOS

(17.12)

is semilinear. Note that an NSOS is a special ordered set of type 2, and can be handled efficiently by most MILP codes. An exclusion constraint of the form x <£ int x,

(17.13)

where x is a box, is semilinear since it is a disjunction of the constraints

Xk<xk

V xk >xk.

Propositional constraints are semilinear: if xk denotes a binary variable which has the value 1 if and only if a corresponding logical proposition Pk holds, then Px V • • • V P K

iff xx + • • • + xK

> 1,

Pi A • • • A PK iff x k = 1 f o r k = l : K , Pi <^> P 2 iff xi = X2, Pi =>• P2 iff xi < x2, Pi V • • • V PK =• PK+i Conditional

V • • • V PL iff xk < xK+i

linear constraints

+ --- + xL for k =

l:K.

of t h e form

Ax € a

if Bx < b

(17.14)

are semilinear since (17.14) is equivalent to Ax € a

V {Bx)i >bi

V

• • • V {Bx)d > bd,

where d is the number of rows of B and b. (Conditional linear constraints with = or < in place of < in (17.14) are apparently not semilinear in general since their disjunctive form contains strict inequalities, which - according to our definition - are not regarded as linear constraints. However, conditional linear constraints where the condition involves only integer variables and rational coefficients are semilinear since the condition can be replaced by an equivalent strict inequality condition.) Certain minimum and maximum constraints are also semilinear. A constraint of the form aTx < min(^a; - b); i l d

is equivalent to the linear constraints aTx < (Ax — b)i

for i — 1 : d.

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(17.15)

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The reverse constraint aTx > mm(Ax - b)i

(17.16)

i=\:d

is equivalent to the linear disjunctive constraint

aTx >{Ax-b)i

V • • • V aTx > (Ax - b)d.

Similarly, a constraint of the form ( aTx > max(v4z - b)i i=l:d i l d

(17.17)

is equivalent to the linear constraints a1x > (Ax — b)i for i = 1 : d, and the reverse constraint ( aTx < i=l:d max(^x - b)i il:d

(17.18)

is equivalent to the linear disjunctive constraint

aTx <(Ax-b)i

V • • • V aTx < (Ax - b)d.

The constraints aTx = min(Ax - b)i,

(17.19)

i=l:d

aTx = max(Ax - b)i

(17.20)

i=l:d

are also semilinear, since they are equivalent to (17.15), (17.16) and (17.17), (17.18), respectively. In particular, linear complementarity constraints (Billups and Murty 2000), defined by mm(aTx-a,bTx-(3)

=0

(17.21)

are semilinear. Their MILP reformulation needs a single binary variable only since the associated BSOS has size two. Linear complementarity constraints arise in bilevel programming (see, e.g., Ben-Ayed (1993), Luo, Pang and Ralph (1996), Outrata, Kocvara and Zowe (1998), Shimizu, Ishizuka and Bard (1997), Vicente and Calamai (1994)), in which the inner optimization problem is a linear program. See, e.g., Grossmann and Floudas (1987) for solving bilevel programs as mixed integer problems. Constraints of the form aTx-a<

\bTx-l3\,

(17.22)

aTx-a = \bTx-/3\,

(17.23)

aTx - a > \bTx - /3\

(17.24)

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are semilinear since we can write the absolute value in the form \bTx — (3\ = max(/3 — bTx, bTx — /?); again a single binary variable suffices for the MILP formulation. In particular, a constraint a < \x\ < (3

(17.25)

can be modelled as

(a + p)z-(3<x

< (a + /3)z-a,

z6{0,l}.

Certain other piecewise linear constraints are also semilinear. Of particular interest are those of the form aTx <
(17.26)

where