Table Of ContentRefonnulation: Nonsmooth, Piecewise Smooth,
Semismooth and Smoothing Methods
Applied Optimization
Volume 22
Series Editors:
Panos M. Pardalos
University ofF lorida, U.S.A.
Donald Hearn
University ofF lorida, U.S.A.
The titles published in this series are listed at the end oft his volume.
Refonnulation: Nonsmooth,
Piecewise Smooth,
Semismooth and
Smoothing Methods
Edited by
Masao Fukushima
Kyoto University,
Kyoto, Japan
and
Liqun Qi
The University o/New South Wales,
Sydney, Australia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4805-2 ISBN 978-1-4757-6388-1 (eBook)
DOI 10.1007/978-1-4757-6388-1
Printed on acid-free paper
All Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
Solving Complementarity Problems by Means of a New Smooth Constrained 1
Nonlinear Solver
Roberto Andreani and Jose Mario Martinez
c-Enlargements of Maximal Monotone Operators: Theory and Applications 25
Regina S. Burachik, Claudia A. Sagastizdbal and B. F. Svaiter
A Non-Interior Predictor-Corrector Path-Following Method for LCP 45
James V. Burke and Song Xu
Smoothing Newton Methods for Nonsmooth Dirichlet Problems 65
Xiaojun Chen, Nami Matsunaga and Tetsuro Yamamoto
Frictional Contact Algorithms Based on Semismooth Newton Methods 81
Peter W. Christensen and Jong-Shi Pang
Well-Posed Problems and Error Bounds in Optimization 117
Sien Deng
Modeling and Solution Environments for MPEC: GAMS 8£ MATLAB 127
Steven P. Dirkse and Michael C. Ferris
Merit Functions and Stability for Complementarity Problems 149
Andreas Fischer
Minimax and Triality Theory in Nonsmooth Variational Problems 161
David Yang Gao
Global and Local Superiinear Convergence Analysis of Newton-Type Methods 181
for Semismooth Equations with Smooth Least Squares
Houyuan Jiang and Daniel Ralph
Inexact Trust-Region Methods for Nonlinear Complementarity Problems 211
Christian Kanzow and Martin Zupke
vi REFORMULATION
Regularized Newton Methods for Minimization of Convex Quadratic Splines 235
with Singular Hessians
Wu Li and John Swetits
Regularized Linear Programs with Equilibrium Constraints 259
Olvi L. Mangasarian
Reformulations of a Bicriterion Equilibrium Model 269
Patrice Marcotte
A Smoothing Function and its Applications 293
Ji-Ming Peng
On the Local Super-Linear Convergence of a Matrix Secant Implementation 317
of the Variable Metric Proximal Point Algorithm for Monotone Operators
Maijian Qian and James V. Burke
Reformulation of a Problem of Economic Equilibrium 335
Alexander M. Rubinov and Bevil M. Glover
A Globally Convergent Inexact Newton Method for Systems of Monotone 355
Equations
Michael V. Solodov and Benar F. Svaiter
On the Limiting Behavior of the Trajectory of Regularized Solutions of a Po- 371
Complementarity Problem
Roman Sznajder and M. Seetharama Gowda
Analysis of a Non-Interior Continuation Method Based on Chen-Mangasarian 381
Smoothing Functions for Complementarity Problems
Paul Tseng
A New Merit Function and a Descent Method for Semidefinite Complementar- 405
ity Problems
Nobuo Yamashita and Masao FUkushima
Numerical Experiments for a Class of Squared Smoothing Newton Methods for 421
Box Constrained Variational Inequality Problems
Guanglu Zhou, De/eng Sun and Liqun Qi
Preface
The concept of "reformulation" has long been playing an important role in
mathematical programming. A classical example is the penalization technique
in constrained optimization that transforms the constraints into the objective
function via a penalty function thereby reformulating a constrained problem as
an equivalent or approximately equivalent unconstrained problem. More recent
trends consist of the reformulation of various mathematical programming prob
lems, including variational inequalities and complementarity problems, into
equivalent systems of possibly nonsmooth, piecewise smooth or semismooth
nonlinear equations, or equivalent unconstrained optimization problems that
are usually differentiable, but in general not twice differentiable. Because of
the recent advent of various tools in nonsmooth analysis, the reformulation
approach has become increasingly profound and diversified.
In view of growing interests in this active field, we planned to organize
a cluster of sessions entitled "Reformulation - Nonsmooth, Piecewise Smooth,
Semismooth and Smoothing Methods" in the 16th International Symposium on
Mathematical Programming (ismp97) held at Lausanne EPFL, Switzerland on
August 24-29, 1997. Responding to our invitation, thirty-eight people agreed
to give a talk within the cluster, which enabled us to organize thirteen sessions
in total. We think that it was one of the largest and most exciting clusters
in the symposium. Thanks to the earnest support by the speakers and the
chairpersons, the sessions attracted much attention of the participants and
were filled with great enthusiasm of the audience.
To take advantage of this opportunity and to popularize this interesting
research field further, we decided to edit a book that primarily consists of
papers based on the talks given in the above-mentioned cluster of sessions.
After more than a year from the first call-for-papers, we are now in the position
of publishing this book containing twenty-two refereed papers. In particular,
these papers cover such diverse and important topics as
o Linear and nonlinear complementarity problems
o Variational inequality problems
o Nonsmooth equations and nonsmooth optimization problems
o Economic and network equilibrium problems
o Semidefinite programming problems
o Maximal monotone operator problems
o Mathematical programs with equilibrium constraints.
The reader will be convinced that the concept of "reformulation" provides ex
tremely useful tools in advancing the study of mathematical programming from
vii
viii REFORMULATION
both theoretical and practical aspects. We hope that this book helps the cur
rent state of the art in mathematical programming to go one step further.
We wish to mention that Olvi Mangasarian is one of the pioneers of refor
mulation methods. He also contributes a paper to this book. We would like
to dedicate this book to him on the occasion of his 65th birthday in January
1999.
We are grateful to all authors who contributed to this book. We would like
to mention that most of them also served as anonymous referees for submit
ted papers. Moreover we are indebted to the following colleagues who kindly
helped us by reviewing the papers submitted to this book: Bintong Chen, Fran
cisco Facchinei, Alfredo Iusem, Helmut Kleinmichel, Donghui Li, Tom Luo, Jiri
Outrata, Houduo Qi, Stefan Scholtes, Jie Sun, Kouichi Taji, Francis Tin-Loi,
and Akiko Yoshise. Last but not least, we would like to express our special
thanks to Nobuo Yamashita and Guanglu Zhou for their patient assistance and
to Houyuan Jiang for his helpful suggestion during the compilation process.
May 1998 Masao Fukushima, Kyoto University
Liqun Qi, University of New South Wales
Reformulation: Nonsmooth, Piecewise Smooth,
Semismooth and Smoothing Methods, pp. 1-24
Edited by M. Fukush(ma and L. Qi
@1998 Kluwer Acat:':temic Publishers
Solving Complementarity Problems by Means
of a New Smooth Constrained Nonlinear
Solver
Roberto Andreani* and Jose Mario Martlnezt
Abstract Given F : lRn -+ lRm and n a closed and convex set, the problem
of finding x E lRn such that x E n and F(x) = 0 is considered. For solv
ing this problem an algorithm of Inexact-Newton type is defined. Global and
local convergence proofs are presented. As a practical application, the Hori
zontal Nonlinear Complementarity Problem is introduced. It is shown that the
Inexact-Newton algorithm can be applied to this problem. Numerical experi
ments are performed and commented.
Key Words nonlinear systems, inexact-Newton method, global convergence,
convex constraints, box constraints, complementarity.
1 INTRODUCTION
The problem considered in this paper is to find x Ene lRn such that
F(x) = 0, (1.1)
where n is closed and convex, F : n -+ lRm, and first derivatives of F exist
and are continuous on an open set that contains n, except, perhaps, at the
solutions of (1.1).
When m = n this is the constrained nonlinear system problem, considered
recently in [17]. By means of the introduction of slack variables, any nonlinear
"Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, CP
6065, 13081-970 Campinas SP, Brazil. This author was supported by FAPESP (Grant
90-3724-6, 93/02479-6), FINEP and FAEP-UNICAMP.
tDepartment of Mathematics, IMECC-UNICAMP, University of Campinas, CP 6065, 13081-
970 Campinas SP, Brazil. This author was supported by FAPESP (Grant 90-3724-6), FINEP
and FAEP-UNICAMP. email: martinez@ime.unicamp.br
1
2 REFORMULATION
feasibility problem given by a set of equations and inequalities can be reduced
to the form (1.1). See [21] and references therein.
We introduce an Inexact-Newton-like algorithm (see [6]) for solving (1.1).
At each iteration of this algorithm, a search direction d is computed such that
where 11·11 = II ·112, 1·1 is an arbitrary norm, Ll is large and Ok E [0,1). If the
Inexact-Newton step does not exist, the tolerance Ok is increased. Otherwise,
we try to find a new point in the direction of the computed step. We prove that
global convergence holds in the sense that, under some conditions, a solution of
(1.1) is met and, in general, a stationary point of the problem can be found. We
establish conditions under which local convergence results can also be proved.
A practical implementation of the algorithm is given for the case in which n is
an n-dimensional box.
As an application of the Inexact-Newton algorithm we define the Horizontal
Nonlinear Complementarity Problem (HNCP). This is a generalization of the
Horizontal Linear Complementarity Problem (HLCP) (see [2]) for which several
interesting applications exist. The HNCP also generalizes the well-known Non
linear Complementarity Problem. See [5], [9], [14], [15], [20] and [22]. We prove
that, under some conditions on the HNCP, stationary points of this problem
coincide with their global solutions and we also analyze the conditions under
which local convergence results hold. Other applications of problem (1.1) an an
interior-point algorithm for solving it can be found in [21]. Finally, we comment
some numerical experiments.
The Inexact-Newton method for solving (1.1) is given in Section 2 of the
paper. In Section 3 we prove the convergence results. In Section 4 we discuss
the application to HNCP. In Section 5 we show the numerical experiments and
in Section 6 we state some conclusions.
2 THE INEXACT-NEWTON ALGORITHM
Let n c lRn be closed and convex, n c A, A an open set of lRn. We assume
that F: A -+ lRm is continuous and that the Jacobian matrix F'(x) E lRmxn
exists and is continuous for all x E A such that F(x) :j:. O.
We say that x. is a stationary point of problem (1.1) if it is a minimizer of
IIF(x.) + F'(x.)(x - x.)11 subj=ect to x E n. Clearly, this is equivalent to say
that x. is a minimizer of q(x) IIF(x.) + F'(x.)(x - x.)112 subject to x E n.
If the convex set n satisfies a constraint qualification (as it is the case of the
application of Section 4, where n is the positive orthant) this is equivalent
to say that the first order optimality conditions (Karush-Kuhn-Tucker) of the
problem of minimizing the convex quadrat=ic q(x) on n are satisfied at x •.
But, since V'q(x.) = V' f(x.) (where f(x) IIF(x)112) it turns out that the
stationarity of x. corresponds exactly to the satisfaction of the Karush-Kuhn
Tucker conditions for minimizers of f on n.
