Recent advances in nonsmooth optimization

RECENT ADVANCES IN0NSM001H OPTIMIZATION

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RECENT ADVANCES INONSMOOTH OPTIMIZATION Editors

Ding-Zhu Du Computer Science Department University of Minnesota Minneapoiis, USA

LiqunQi & Robert S. Womersley School of Mathematics University of New South Wales Sydney, Australia

World Scientifie Silgapore

- New Jersey

London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

RECENT ADVANCES IN NONSMOOTH OPTIMIZATION Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA.

ISBN 981-02-2265-3

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

V

Preface The field of nonsmooth optimization is significant, not only because of the exis­ tence of nondifferentiable functions arising directly in applications, but also because several important methods for solving difficult smooth problems lead directly to the need to solve nonsmooth problems, which are either smaller in dimension or sim­ pler in structure. For example, decomposition methods for solving very large scale smooth problems produce lower-dimensional nonsmooth problems; penalty methods for solving constrained smooth problems result in unconstrained nonsmooth problems; nonsmooth equation methods for solving smooth variational inequalities and smooth nonlinear complementarity problems give rise to systems of nonsmooth equations. After the early work in nonsmooth optimization, many books and proceedings have appeared in this field. The area of nonsmooth optimization is lively, with rapid developments in its applications, theoretical foundations, and available computational methods. There are three features of nonsmooth optimization. The first feature is the variety of classes of nonsmooth optimization problems. They include not only optimization problems whose objective and constraint functions are nondifferentiable, but also optimization problems whose objective and constraint functions are first-order but not second-order differentiable. Superlinear convergence rates for the latter problems cannot be directly established using the smooth opti­ mization techniques. Furthermore, nonsmooth variational inequality problems, nons­ mooth equations, set-valued problems, bilevel programming, etc, are all in the general scope of nonsmooth optimization. Such a variety produces a, wide range of analytical tools and algorithms. The second feature is the rapid expansion of nonsmooth analysis. Starting from convex analysis, it now also includes generalized second-order derivatives, set-valued analysis, generalized convexity, and many other topics. On one hand, nonsmooth analysis is now a subject itself. On the other hand, it continuously provides useful tools for nonsmooth optimization. Terry Rockafellar plays a leading role in this field. This book includes two papers by him and his collaborators. We would like to dedicate this book to him on the occasion of his 60th birthday in October 1995. The third feature is the development of new computational methods in nonsmooth optimization. From convergence analysis to numerical implementation, we observe a renewed surge in this frontier. Early approaches included bundle methods and the adaption of smooth methods to convex composite problems. Now there is a much wider range of methods exploiting the structure of the nonsmooth problem. More than half of this book is devoted to this frontier. The book contains twenty five papers written by forty six authors from twenty countries in five continents. This illustrates the strong interest in nonsmooth opti­ mization. We hope this collection will provide the reader with a glimpse of the recent advances of nonsmooth optimization and their features.

vi We would like to thank the authors of the papers, the anonymous referees, our associate and assistants, Dr. Xiaojun Chen, Mr. Houyuan Jiang and Mr. Zengxin Wei, and the publisher for helping us to produce this excellent collection of papers. Ding-Zhu Du, Liqun Qi and Robert S. Womersley University of Minnesota and University of New South Wales May 1995

VII

Contents Hybrid Methods for Finding the Nearest Euclidean Distance Matrix S. Al-Homidan and R. Fletcher Subdifferential Characterization of Convexity R. Cornea, A. Jofre and L. Thibault

1 18

A Simple Triangulation of Rn with Fewer Simplices for Solving Nonsmooth Convex Programming C.-Y. Dang

24

On Generalized Differentiability of Optimal Solutions and its Application to an Algorithm for Solving Bilevel Optimization Problems S. Dempe

36

Projected Gradient Methods for Nonlinear Complementarity Problems via Normal Maps M. C. Ferris and D. Ralph

57

An NCP-Function and its Use for the Solution of Complementarity Problems A. Fischer

88

An Elementary Rate of Convergence Proof for the Deep Cut Ellipsoid Algorithm J. B. G. Frenk and J. Gromicho

106

Solving Nonsmooth Equations by Means of Quasi-Newton Methods with Globalization M. A. G. Ruggiero, J. M. Martinez and S. A. Santos

121

Superlinear Convergence of Approximate Newton Methods for LC1 Optimization Problems without Strict Complementarity J. Han and D.-F. Sun

141

On Second-Order Directional Derivatives in Nonsmooth Optimization L. R. Huang and K. F. Ng On the Solution of Optimum Design Problems with Variational Inequalities M. Kocvara and J. V. Outrata Monotonicity and Quasimonotonicity in Nonsmooth Analysis S. Komlosi

159

172 193

viii

Sensitivity of Solutions in Nonlinear Programming Problems with Nonunique Multipliers A. B. Levy and R. T. Rockafellar

215

Generalized Convexity and Higher Order Duality of the Non-Linear Programming Problem with Non-Negative Variables B. MondandJ.-Y. Zhang

224

Prederivatives and Second Order Conditions for Infinite Optimization Problems W. Oettli and Pham H. Sach

244

Necessary and Sufficient Conditions for Solution Stability of Parametric Nonsmooth Equations J.-S. Pang

261

Miscellaneous Incidences of Convergence Theories in Optimization and Nonlinear Analysis, Part II: Applications in Nonsmooth Analysis J.-P. Penot

289

Second-Order Nonsmooth Analysis in Nonlinear Programming R. Poliquin and T. Rockafellar Characterizations of Optimality for Homogeneous Programming Problems with Applications A. M. Rubinov and B. M. Glover On Regularized Duality in Convex Optimization A. Ruszczynriski

322

351 381

An Interior Point Method for Solving a Class of Linear-Quadratic Stochastic Programming Problems J. Sun, K.-E. Wee and J.-S. Zhu

392

A Globally Convergent Newton Method for Solving Variational Inequality Problems with Inequality Constraints K. Taji and M. Fukushima

405

Upper Bounds on a Parabolic Second Order Directional Derivative of the Marginal Function D. Ward

418

A SLP Method with a Quadratic Correction Step for Nonsmooth Optimization J.-Z. Zhang, C.-X. Xu and Y.-A. Fan

438

A Successive Approximation Quasi-Newton Process for Nonlinear Complementarity Problem S.-Z. Zhou, D.-H. Li and J.-P. Zeng

459

Euclidean

Distance

Matrices

1

itecent Advances in Nonsmooth Optimization, pp. 1-17 Eds. D.-Z. Du, L. Qi and R.S. Womersley ©1995 World Scientific Publishing Co Pte Ltd

Hybrid Methods for Finding the Nearest Euclidean Distance Matrix Suliman Al-Homidan Department of Mathematics, Saudi Arabia

Roger Fletcher Department of Mathematics DDl 4HN, Scotland, UK

King Saud University,

and Computer

Science,

Riyadh

11451, PO Box

University

of Dundee,

4511,

Dundee

Abstract

A concise characterization is presented for a Euclidean distance matrix in terms of null-space matrices, and methods for the solution of the Euclidean distance matrix problem are considered. One approach (Glunt et al. [8]) is to formulate the problem as a constrained least distance problem in which the constraint is the intersection of two convex sets. The Dykstra-Han projection algorithm can then be used to solve the problem. This method is globally convergent but the rate of convergence is slow. However the method does have the capability of determining the correct rank of the solution matrix, and this can be done in relatively few iterations. If the correct rank of the solution matrix is known, it is shown how to formulate the problem as a smooth unconstrained minimization problem, for which rapid convergence can be obtained by for example the BFGS method. This paper studies hybrid methods that attempt to combine the best features of both types of method. An important feature concerns the interfacing of the component methods. Thus it has to be decided which method to use first, and when to switch between methods. Also it may not be straightforward, as we shall see here, to use the output of one method to start the other method. Difficulties such as these are addressed in the paper. Comparative numerical results are reported.

S. Al-Homidan and R. Fletcher

2

1

Introduction

Symmetric matrices that have non-negative offdiagonal elements and zero diagonal elements arise as data in many experimental sciences. This occurs when the values are measurements of squared distances between points (e.g. atoms, stars, cities) in a Euclidean space. Such a matrix is referred to as a Euclidean distance matrix. Because of data errors such a matrix may not be exactly Euclidean and it is desirable to find the best Euclidean matrix which approximates the non-Euclidean matrix. The aim of this paper is to study methods for solving this problem. An important application arises in the conformation of molecular structures from nuclear magnetic resonance data (see Havel et al. [10] and Crippen [4], [5]). Here a Euclidean distance matrix is used to represent the squares of distances between the atoms of a molecular structure. An attempt to determine such a structure by nuclear magnetic resonance experiments gives rise to a distance matrix F which, because of data errors, may not be Euclidean. There are many other applications in subjects as diverse as archeology, cartography, genetics, geography and multivariate analysis. Pertinent references are given by Al-Homidan [1]. Characterization theorems for the Euclidean distance matrix have been given in many forms over the years. In Section 2 we show that a very concise form of this result can be proved in terms of null-space matrices, that brings out the underlying structure and is readily applicable to the algorithms that follow. Many advances have taken place in constrained optimization over the last forty years or so. There are now effective methods for situations in which the objective and constraint functions are smooth functions. Under reasonable assumptions, these methods can be shown to converge globally (that is from any starting point) to a point which satisfies optimality conditions for the problems. Also the rate of convergence can often be shown to be superlinear. Some progress has also been made for prob­ lems in which non-smooth functions occur. If these functions are a composition of a convex polyhedral function and a smooth function, then again globally and superlinear convergent methods have been suggested. A rather more difficult non-smooth optimization problem occurs when some matrix, defined in terms of the problem vari­ ables, has to be positive semi-definite. One way to handle this problem is to impose a functional constraint in which the least eigenvalue of the matrix is non-negative. However, if there are multiple eigenvalues at the solution, which is usually the case, such a constraint is non-smooth, and this non-smoothness cannot be modelled by a convex polyhedral composite function. An important factor is the determination of the multiplicity of the zero eigenvalues, or alternatively the rank of the matrix at the solution. If this rank is known it is usually possible to solve the problem by conven­ tional techniques. For the Euclidean distance matrix problem, a certain transformed matrix (Dj in Theorem 2.2 below) has to be positive semi-definite. It follows that the Euclidean distance matrix constraint shares the same non-smooth characteristics as the positive semi-definite matrix constraint. This observation is clear from the

Euclidean Distance Matrices

3

characterisation result for the normal cone given by Glunt et al. [8]. One approach [8] is to formulate the Euclidean distance matrix problem as a constrained least distance problem in which the constraint is the intersection of two convex sets. The Dykstra-Han alternating projection algorithm can then be used to solve the problem. This idea is outlined in Section 3. This method is globally convergent but the rate of convergence is linear or slower. It is this latter feature that has probably contributed to the relatively little interest that has been shown in such methods. However the method does have the capability of determining the correct rank of the solution matrix, and this can be done in relatively few iterations. If the correct rank of the solution matrix is known, it is shown in Section 4 how to formulate the problem as a smooth unconstrained minimization problem, for which rapid convergence can be obtained by for example the BFGS method. We discuss how best to parametrize the problem, and give expressions for the objective function and its first derivatives. A trial and error approach to estimating the correct rank is possible, but is not very appealing. Thus we are led to study hybrid methods in Section 5 of the paper. The hybrid method has two different modes of operation. One is a projection method which provides global convergence and enables the correct rank to be determined. The other is a quasi-Newton method which enables rapid convergence to be obtained. An important feature concerns the interfacing of these modes of operation. Thus it has to be decided which method to use first, and when to switch between methods. Also it may not be straightforward, as we shall see here, to use the output of one method to start the other method. Difficulties such as these are addressed in the paper. Numerical experiments are reported in Section 6. Recently, and since the research in this paper was carried out, there has been much interest in interior point methods applied to problems with semi-definite matrix constraints (e.g. Alizadeh et al. [2]). It would certainly be of interest to compare this approach with the hybrid methods described in our paper. Throughout this paper the lower case boldface letters such as x, y, v are used to denote vectors. Matrices are denoted by capital letters such as A, B, C. We use the notation Diag(y4) to denote diag(a„), i = 1,...,n . Superscript (fc) generally denotes quantities related to the fcth iterate, for example f' ', jf'" etc,. Quantities relating to the solution are superscripted with an asterisk, e.g. r~, £>*, etc.

2

The Euclidean Distance Matrix Problem

In this section the definition of the Euclidean distance matrix is given, and the rela­ tionship between points and distances is summarized. A characterization theorem for the Euclidean distance matrix is proved in a concise way that brings out the under­ lying structure and is readily applicable to the algorithms that follow. The theorem is essentially due to Schoenberg [12] in the case that p = Xi (see below). Young and Householder [13] independently obtain a similar result.

S. Al-Homidan and R. Fletcher

4

It is necessary to distinguish between distance matrices that are obtained in prac­ tice and those that can be derived exactly from n vectors that are irreducibly embed­ ded in IRr where r < n — 1 (this concept is explained below) Definition 2.1. A matrix D e IR"X" is called a distance matrix iff it is the diagonal elements are zero dn = 0

symmetric,

* = 1,..., n,

and the off-diagonal entries are non-positive d<j < 0

V«\± j .

Definition 2.2. A matrix D £ ]Rn*n is called a Euclidean distance matrix iff there exist n points X i , . . . , x n irreducibly embedded in IRr (r < n — 1) such that 0 V x e M}

(3.3)

S. Al-Homidan and R. Fletcher

Q is a convex cone, and Kd = {A: AeJRnxn,

AT = A,

a,, = 0 Vt = l , . . . , n }

(3.4)

is a subspace. Clearly from Theorem 2.1, D G KM n / r' then slow convergence is observed. One reason is that there are more variables in the problem. Also redundancy in the parameter space may have an effect. Thus it makes sense to start with a small value of r, and increase it by one until the solution is recognised. One way to recognise termination is when Z?' r ' agrees sufficiently well with Z?' r+1 ', where £>'r' denotes the Euclidean distance matrix obtained by minimizing (j> when X in (4.1) has r rows. Details of this test, and relevant numerical experience for solving various test problems by this method, are reported in [l]. An obvious alternative to using the BFGS method is to evaluate the Hessian matrix of second derivatives of (X) and use Newton's method. This would be likely

12

S. Al-Homidan and R. Fletcher

to reduce the number of iterations required. It would also enable the algorithm to make progress when Xw is a stationary point (e.g. Xw = 0) that is not a local minimizer. However there is also the disadvantage of increased complexity, and increased housekeeping at each iteration. Moreover it is possible that the Hessian has some negative eigenvalues so a modified form of Newton's method would be required. A simple example serves to illustrate the possibility of a negative eigenvalue. Take n = 2, r = 1 and let F = [^ "*] and X = [0 n ] . Then <j> = 2(1 - x\f. This has global minimizers at xx = ± 1 , a local maximizer at xt = 0, and the Hessian is negative for all xx such that 3xj < 1.

5

Hybrid Methods

The algorithms of Sections 3 and 4 have entirely different features, some good, some bad, which suggests that a combination of both approaches might be successful. Projection methods are globally convergent and hence potentially reliable, but the rate of convergence is first order or slower, which can be very inefficient. QuasiNewton methods are reliable and locally superlinearly convergent, but require that the correct rank r* is known. We therefore consider hybrid methods in which the projection algorithm is used sparingly as a way of establishing the correct rank, whilst the BFGS method is used to provide rapid convergence. In order to ensure that each component method is used to best effect, it is important to be able to transfer information from one method to the other. In particular a mechanism must be established so that the result from one method is used to provide the initial data for the other, and vice versa. This mechanism must have a fixed point property, so that if one method finds a solution, then the other method is initialized with an iterate that also corresponds to the solution. We show in this section how this can be done. We have already indicated at the end of Section 3 how the projection method can be initialized with any diagonal matrix A. However if £> and ('*', where ' Fw f(*>l F = Q F(k) = Q $)T , QQ. (5.1) The spectral decomposition FJk) = £ / « A « £ / W T is calculated and £> is determined by £> *> PM(F^) == QQ [ V £ >< == PM(F^)

m

^ V W

f« j Q Q-

(5 2)

Euclidean Distance Matrices

13

It follows from (5.1) and (5.2) that

(£,(*) .- F"")e == Q

Ae = m(*). A(*'e (Dw -- F)e. F)e.

(5.3)

This expression is exact for the projection method. Because A'*' is diagonal, (5.3) can be used to compute a matrix A from any given matrix £>. In our hybrid algorithm we use this as a way of initializing A for the projection method, from the D(k) matrix obtained from the BFGS method. If the BFGS method is using the correct rank r = r' and has found the global solution of <j>, then Z) is the solution D* of (2.2). Hence (5.3) gives the correct solution A* for the projection method. Even if the rank r f r" in the BFGS method, (5.3) enables some information to be extracted from D^ that is hopefully useful. Conversely we let £>'*> be the matrix obtained in (5.2) by the projection method, and consider how to initialize X for the BFGS method. If £> is a Euclidean distance matrix, then it solves (2.2), and by Theorems 2.1 and 2.2, the correct rank r" is the number of positive eigenvalues in the matrix A"1'. We denote this number by A/"(A). In general, when D^ is not a solution, we use Af(Mk)) to determine the row dimension r of X in (4.1) for the BFGS method. To determine the elements of X we again use the construction suggested by (2.8) and (2.6). Thus we define the elements of A from those of D(fc) by i >>2, ay = §( 2. (5.4) Xi) \(d,} --dudu -- ddu) The first row and column of A are zero and are ignored. We then find the spectral decomposition UAUT of the nontrivial part of A. Finally the nontrivial part of X in (4.1) is initialized to the matrix \xJ2Uj where Ar = diag(A,), t = 1,.. . , r contains the r positive eigenvalues of A, and columns of Ur are the corresponding eigenvectors. We have found that it is sufficient to carry out only one iteration of the projection method between each call of the BFGS method. Thus we can express our hybrid algorithm in detail as i. Initialize k = 0, r and A'(0> ii. Minimize (X) using the BFGS method, giving A ' « and £> iii. Use (5.3) to calculate A whenever f(x) = +00. (ii) 8f(x)

is the usual subdifferential of Convex Analysis whenever f is convex.

(Hi) 8f(x) = 6g(x) for any function g which is equal to f near x. (iv) 0 £ Sf(x)

whenever x is a local minimizer of f and f(x) < +00.

(v) for any continuous convex function g % + /)(*) C Sg{x) + lira sup 8f(y) 1 y->X

whenever f is lower semicontinuous near x, where limsup 8f(y) is the weak-star sequential upper limit and y —» x means j - n and f(y) —^ f(x). Examples. All subdifferentials considered in [5] (Clarke subdifferential, approximate subdifferentials and so on, see for example [7], [8], [10], [11], [15], [16]) are presubdifferentials. If E is an Asplund space (resp. a super-reflexive space) the Frechet subgradient set (resp. the proximal subgradient set, see for example [2]) is a presub­ differential. This last fact follows from Fabian [6] or Mordukhovich [12]. Remark. It is not difficult to see from the above properties, that the usual subd­ ifferential of Convex Analysis at each point x is contained in limsup8f(y) for any extended-real-valued function / .

R. Correa, A. Jofre and L. Thibault

20

As it has been observed in [5] and [17] the following result by Zagrodny [20] holds for any presubdifferential. Zagrodny Mean Value Theorem. Let f : E -* R U {+00} be a l.s.c. (lower semicontinuous) function and a, 6 £ domf. Then there exist c £]a, &], a sequence (xt) converging to c with f(xk) —> /(c) and x\ £ Sf(xk) such that i) fl^fl lim < *J, a - x , > > / ( a ) - /(&), ii) lim < x*k, a - 6 > > / ( a ) - f(b). We recall that a presubdifferential of a function / is monotone if for all x,y £ E, x' £ 6/(x) and j/* £ A _1 (l - X)(x*k, zx - x) > 0

which is in contradiction with the monotonicity of Sf and 0 6 limsup M U {+00} be a l.s.c. function. assertions are equivalent:

Then the following

i) 8f is monotone, ii) f is convex. Proof. Let x, y £ domf and 2 = Ax + (1 — X)y, with A G]0, l[. It is easy to see from the mean value theorem that there exists a sequence (yk) in dornSf such that yk —> y and f(yk) —+ f(y). Let zk = Xx + (1 — X)yk. From Proposition 2.2, zk € domf. 1) If zk is not a local minimum of / we can choose z'k such that \z'k — zk\ < (1/k) and f(z'k) < f(zk). Applying the Zagrodny mean value theorem on [z,t,zit] we obtain sequences Zfc,„ A ck £}zk,z'k], z'kn 6 Sf(zk< z*kn,x-

Hence

zk,„ > and f(yk) - f(zk,n)

>< z'kn,yk

- zk

and these inequalities and the lower semicontinuity of / imply that Xf(x) + (1 - X)f(yk)

> liminf[/(2 M )+ < z'k^,zk - zk,n >] >

f(ck).

R. Correa, A. Jofre and L. Thibault

22

2) If Zk is a local minimum of / then 0 6 Sf{zk) C dcf(zk). Hence putting c/t = Zk we obtain f(x) > f(ck) and f(yk) > f{ck) which imply in this case that A/(x) + (1 - X)f(yk) >

f(ck).

As f(yk) —* f{y) and Ck —► -z, it follows from the lower semicontinuity of / and from (2) that Xf(x) + (l-X)f(y)>f(z), which is the desired inequality. □ When the presubdifferential is the Frechet or the approximate subgradient set we obtain the following corollary.

Corollary 2.5 Assume that E is an Asplund space (resp. E is a super reflexive space). Then a l.s.c. function f : E —► M U {+00} is convex if and only if its Frechet subgradient (resp. proximal subgradient) set is monotone.

References [1] D. Aussel, J.-N. Corvellec k M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, to appear. [2] J. M. Borwein, D. Preiss, A smooth variational principle with applications to subdifferentiability and differentiability of convex functions, Transactions of the American Mathematical Society 303 (1987) 517-527. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983). [4] R. Correa, A. Jofre and L. Thibault, Characterization of lower semicontinuous convex functions, Proceedings of the American Mathematical Society 116, 1 (1992) 67-72. (Preprint January 1991) [5] R. Correa, A. Jofre and L. Thibault, Subdifferential monotonicity as characteri­ zation of convex functions, Numerical Functional Analysis and Optimization 15 (1994) 531-535. (Preprint October 1991) [6] M. Fabian, Subdifferentiability and trustworthiness in the light of a new vari­ ational principle of Borwein and Preiss, Acta University Carolinae 30 (1989) 51-56. [7] D. Ioffe, Approximate subdifferentials and applications, 2 and 3, Mathematika 33 (1986), 111-128; 36 (1989) 1-38.

Subdifferential

Characterization of Convexity

23

[8] A. Jofre and L. Thibault, Proximal and Frechet normal formulae for some small normal cones in Hilbert spaces, Nonlinear Analysis Theory, Methods and Appli­ cations 19 (1992) 599-612. [9] D. T. Luc, On the monotonicity of subdifferentials, Ada Mathematica ica 8 (1993) 99-106.

Vietnam-

[10] Ph. Michel and J. P. Penot, Calcul sous differentiel pour des fonctions lipschitziennes et non lipschitziennes, C.R. Acad. Sci. Paris, 298 (1984), 269-272. [11] B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Mathematics Doklady 22 (1980) 526-530. [12] B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Transactions of the American Mathematical Society, to appear. [13] J. J. Moreau, Fonctionelles convexes, Lecture Notes, Seminaire "Equations aux derivees partielles", College de France, (1966). [14] R. A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Anal­ ysis Theory, Methods and Applications 14 (1990) 305-317. [15] R. T. Rockafellar, Convex Analysis, Princeton University Press, (1970). [16] R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian Journal of Mathematics 32 (1980) 257-280. [17] L. Thibault, A note on the Zagrodny mean value theorem, to appear. [18] L. Thibault, D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, to appear in Journal of Mathematical Analysis and Applications. [19] J. S. Treiman, Shrinking generalized gradients, Nonlinear Analysis Theory, Meth­ ods and Applications 12 (1988) 1429-1449. [20] D. Zagrodny, Approximate mean value theorem for upper subderivatives, Non­ linear Analysis Theory, Methods and Applications 12 (1988) 1413-1438.

24

C. Dang

Recent Advances in Nonsmooth Optimization, pp. 24-35 Eds. D.-Z. Du, L. Qi and R.S. Womersley ©1995 World Scientific Publishing Co Pte Ltd

A Simple Triangulation of Rn with Fewer

f ^ X S o l v i n g Nonsmooth Convex Chuangyin Dang Engineering Science

Department,

Auckland

University,

Auckland,

New

Zealand

Abstract We propose a triangulation of Rn, named Z^-trianguIation, for simplicial algorithms. It has fewer simplices than the Di -triangulation and more or less the same number of simplices as the £>' r triangulation. We give an application of the DJ-triangulation to one of simplicial algorithms for solving nonsmooth convex programming. The DJ-triangulation seems simpler than the D' r triangulation to be used in simplicial algorithms. We expect simplicial algorithms based on the DJ-triangulation to be more efficient.

1

Introduction

In [20] Scarf proposed t h e first elegant constructive proof of existence of a fixed point of a continuous mapping from t h e unit simplex t o itself. S t i m u l a t e d by this pioneering discovery, a number of algorithms, called simplicial algorithms in literature, has been developed for computing fixed points, such as homotopy m e t h o d s in [5], [7], [16], and variable dimension algorithms in [11], [12], [13]. See also [l] a n d [22]. Scarf's algor i t h m uses primitive sets, however all t h e subsequent developments employ simplicial subdivisions or triangulations of the t h e space. It was was investigated in [8], [18] and a n d [23] [8], [18] [23] t h a t efficiency of simplicial algorithms heavily depends on triangulations underlying that t h e m . To speed up u p simplicial algorithms, several triangulations have been proposed. them. See [2], t c . Triangulations tthat h a t can [2], [6], [6], [10], [14], [15], [19], [19], [24], eetc. can easily b bee employed he D in [2], x-triangulation in simplicial algorithms are are tthe ZVtriangulation [2], K\-triangulation AVtriangulation in [9] A*[13], 7 rr riiangulation r i i a n g u l a t i o n in [22], [24]. It was triangulation in [13], [22], and and £>i-triangulation £)' r triangulation in [24]. was and [24] [24] that t h a t the the A-triangulation ^ - t r i a n g u l a t i o n and and DJ-triangulation are better b e t t e r tha.n shown in [2] and DJ-triangulation are than other triangulations according to measures of efficiency of triangulations. The T h e triangulations proposed in [10], [15] and [19] have some nice properties, b u t t h e y are t o o • •

i

i

i

i

Triangulation & Simplicial

Algorithm

25

complicated to be employed in simplicial algorithms. In this paper we give a triangulation that is a modification of the Di-triangulation and is named £)J-triangulation. According to the number of simplices in a unit cube, the £>*-triangulation is better than the Z)i-triangulation and more or less the same as the D\- triangulation. The Z)J-triangulation seems simpler than the D\-triangulation to be used in simplicial algorithms. The paper is organized as follows. We introduce the Z)J-triangulation in Section 2. We discuss how to transform a nonsmooth convex programming problem into an equivalent problem of existence of a fixed point in Section 3. We give a simplicial algorithm for solving nonsmooth convex programming in Section 4.

2

Triangulation

In [2] the so-called Z)1-triangulation of Rn was proposed. Its definition is as follows. Let W be the set given by either {y G Rn | all components of y are odd} or {y € R" | all components of y are even}. Let y be a vector in W. Let ir = (7r(l),7r(2),..., ir(n)) be a permutation of elements of { 1 , 2 , . . . ,n], and s = (si,s2,... ,sn)T a sign vector with s< 6 { 1 , - 1 } , i = 1,2,... ,n. Let p be an integer with 0 < p < n — 1. Let u' be the ith unit vector of Rn for i = 1,2,... ,n. If p = 0, then y° = y, and y1 = y + s^j)u'"u),

j = 1,2,

...,n,

and if 1 < p, then y° = y + s, and r i^-a.uju"0'. i = l,2,...,p-l, y1 =

1 y + s^u^,

j

=p,p+l,...,n.

We use Dx(y, n,s,p) to denote the convex hull of y3, j = 0 , 1 , . . . , n. Then Dx(y,ir,s,p) is a simplex. The collection of all such simplices, denoted Dj, is a triangulation of Rn See [2] for the details. Let N = { 1 , 2 , . . . , n } and W0 = { 0 , 1 , . . . , n } . Let ft € i?" be an integral vector. Let 70(ft) = {i G W | hi is odd} and 7eC0 = {i € N \ h{ is even}. We use r to denote the number of elements of I0{h). Note that the number of elements of Ie(h) is given by n — r. Let A(h) = {x\ - 1 < x, - hi < 1, i g I0(&), and x, = ^ , i € 7e(/i)}

26

C. Dang

and X,:; -- fe, B(h) ={x\ B{ft)= {i | - 1 < x, ft,(ftx) and D{h2) for any two integral vectors ft1 and ft2 is given by the convex hull of 2 2 2 l l ({0} x A(ft>) Aih1) nnA(h A(h ))))Uu({1} ({1}xxB(k B(k ) ) nn B(ft B(ft2)), )),

which is either empty or a common face of both £?(&*) and £(ft 2 ), and UAeZnZ)(/i) /)(ft) = [ 0 , l ] x / ? " . See [11] for the details. triangulation with

We obtain a triangulation Te of R" by using the £>,-

W={yeRn\

all components of y are even},

and a triangulation T0 of Rn by using the ^^triangulation with W = {y G 6 /RTn | all components of j/y are odd}. Let T. be the set of faces of simplices of Tc, and f0 the set of faces of simplices of TB. The restriction of T. to A(h) is given by {aeT A(h) and dim(cr) = r}, Tc\A(h) = {a Ge\crC f. | a C A(ft) and the restriction of T0 to B(h) is given by fT00|B(ft) \B(h) = {;) = 2 + 2" + £ £

(„ + i v -

They also showed that N(D[)/N(Ki) approaches to (e — 2)2 as n goes to infinity. Therefore, the Z)*-triangulation and Z)J-triangulation have more or less the same number of simplices in C. However, the Z?*-triangulation seems simpler than the D[-triangulation to be used in simplicial algorithms. As follows, we give the pivot rules of the ZJJ-triangulation in Figure 1, which tell us how to generate all adjacent simplices of a simplex. Let a = D\(y, n, s,pi,p2) be a given simplex with vertices y', i = — 1,0,..., n. We want to know what is its adjacent simplex, a' = .DJ(i/',7r', s',p[,p'2), opposite to vertex y' We show how to obtain y', 7r', S', P'17 p'2 from y, IT, s, pi, p2 in Figure 1 where y" = y + 2^ (0) w T(0) , ' y + 2^ ( ,_ 1)U ' r