F faccinei j s pang finite dimensional variational inequalities and complementarity problems volume i

Francisco Facchinei Jong-Shi PangDipartimento di Informatica e Department of Mathematical soler@dis.uniroma1.it Series Editors: Department of Management Science Department of Industrial

Finite-Dimensional Variational Inequalities and Complementarity

Problems, Volume I

Francisco Facchinei

Jong-Shi Pang

Springer

Peter W Glynn Stephen M Robinson

This page intentionally left blank

Variational Inequalities and Complementarity

Problems

Volume I

With 18 Figures

Francisco Facchinei Jong-Shi Pang

Dipartimento di Informatica e Department of Mathematical

soler@dis.uniroma1.it

Series Editors:

Department of Management Science Department of Industrial Engineering

Terman Engineering Center 1513 University Avenue

glynn@leland.stanford.edu

Mathematics Subject Classification (2000): 90-01, 90C33, 65K05, 47J20

Library of Congress Cataloging-in-Publication Data

Facchinei, Francisco

Finite-dimensional variational inequalities and complementarity problems / Francisco Facchinei, Jong-Shi Pang.

p cm.—(Springer series in operations research)

Includes bibliographical references and indexes.

ISBN 0-387-95580-1 (v 1 : alk paper) — ISBN 0-387-95581-X (v 2 : alk paper)

1 Variational inequalities (Mathematics) 2 Linear complementarity problem.

I Facchinei, Francisco II Title III Series.

QA316 P36 2003

515 ′.64—dc21 2002042739 ISBN 0-387-95580-1 Printed on acid-free paper.

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The finite-dimensional nonlinear complementarity problem (NCP) is a tem of finitely many nonlinear inequalities in finitely many nonnegativevariables along with a special equation that expresses the complementaryrelationship between the variables and corresponding inequalities Thiscomplementarity condition is the key feature distinguishing the NCP from

sys-a genersys-al inequsys-ality system, lies sys-at the hesys-art of sys-all constrsys-ained optimizsys-a-tion problems in finite dimensions, provides a powerful framework for themodeling of equilibria of many kinds, and exhibits a natural link betweensmooth and nonsmooth mathematics The finite-dimensional variationalinequality (VI), which is a generalization of the NCP, provides a broadunifying setting for the study of optimization and equilibrium problemsand serves as the main computational framework for the practical solution

optimiza-of a host optimiza-of continuum problems in the mathematical sciences

The systematic study of the finite-dimensional NCP and VI began inthe mid-1960s; in a span of four decades, the subject has developed into avery fruitful discipline in the field of mathematical programming The de-velopments include a rich mathematical theory, a host of effective solutionalgorithms, a multitude of interesting connections to numerous disciplines,and a wide range of important applications in engineering and economics

As a result of their broad associations, the literature of the VI/CP hasbenefited from contributions made by mathematicians (pure, applied, andcomputational), computer scientists, engineers of many kinds (civil, chem-ical, electrical, mechanical, and systems), and economists of diverse exper-tise (agricultural, computational, energy, financial, and spatial) There aremany surveys and special volumes, [67, 240, 243, 244, 275, 332, 668, 687],

varia-v

A Bird’s-Eye View of the Subject

The subject of variational inequalities has its origin in the calculus of ations associated with the minimization of infinite-dimensional function-als The systematic study of the subject began in the early 1960s withthe seminal work of the Italian mathematician Guido Stampacchia andhis collaborators, who used the variational inequality as an analytic toolfor studying free boundary problems defined by nonlinear partial differen-tial operators arising from unilateral problems in elasticity and plasticitytheory and in mechanics Some of the earliest papers on variational in-equalities are [333, 512, 561, 804, 805] In particular, the first theorem ofexistence and uniqueness of the solution of VIs was proved in [804] Thebooks by Baiocchi and Capelo [35] and Kinderlehrer and Stampacchia [410]provide a thorough introduction to the application of variational inequali-ties in infinite-dimensional function spaces; see also [39] The lecture notes[362] treat complementarity problems in abstract spaces The book byGlowinski, Lions, and Tr´emoli`ere [291] is among the earliest references togive a detailed numerical treatment of such VIs There is a huge literature

vari-on the subject of infinite-dimensivari-onal variativari-onal inequalities and relatedproblems Since a VI in an abstract space is in many respects quite dis-tinct from the finite-dimensional VI and since the former problem is notthe main concern of this book, in this section we focus our introduction onthe latter problem only

The development of the finite-dimensional variational inequality andnonlinear complementarity problem also began in the early 1960s but fol-lowed a different path Indeed, the NCP was first identified in the 1964Ph.D thesis of Richard W Cottle [135], who studied under the supervi-sion of the eminent George B Dantzig, “father of linear programming.”Thus, unlike its infinite-dimensional counterpart, which was conceived inthe area of partial differential systems, the finite-dimensional VI/CP was

born in the domain of mathematical programming This origin has had aheavy influence on the subsequent evolution of the field; a brief account

of the history prior to 1990 can be found in the introduction of the vey paper [332]; see also Section 1.2 in [331] In what follows, we give amore detailed account of the evolutionary process of the field, covering fourdecades of major events and notable highlights

sur-In the 1960s, largely as a result of the celebrated almost tary pivoting algorithm of Lemke and Howson for solving a bimatrix gameformulated as a linear complementarity problem (LCP) [491] and the subse-quent extension by Lemke to a general LCP [490], much focus was devoted

complemen-to the study of the latter problem Cottle, Pang, and Scomplemen-tone presented acomprehensive treatment of the LCP in the 1992 monograph [142] Amongother things, this monograph contains an extensive bibliography of the LCP

up to 1990 and also detailed notes, comments, and historical accounts aboutthis fundamental problem Today, research on the LCP remains active andnew applications continue to be uncovered Since much of the pre-1990details about the LCP are already documented in the cited monograph,

we rely on the latter for most of the background results for the LCP andwill touch on the more contemporary developments of this problem whereappropriate

In 1967, Scarf [759] developed the first constructive iterative method forapproximating a fixed point of a continuous mapping Scarf’s seminal workled to the development of the entire family of fixed-point methods and of thepiecewise homotopy approach to the computation of economic equilibria

The field of equilibrium programming was thus born In essence, the term

“equilibrium programming” broadly refers to the modeling, analysis, andcomputation of equilibria of various kinds via the methodology of mathe-matical programming Since the infant days of linear programming, it wasclear that complementarity problems have much to do with equilibrium pro-grams For instance, the primal-dual relation of a linear program providesclear evidence of the interplay between complementarity and equilibrium.Indeed, all the equilibrium problems that were amenable to solution by thefixed-point methods, including the renowned Walrasian problem in generalequilibrium theory and variations of this problem [760, 842, 866], were infact VIs/CPs

The early research in equilibrium programming was to a large extent

a consequence of the landmark discoveries of Lemke and Scarf In ticular, the subject of fixed-point computations via piecewise homotopiesdominated much of the research agenda of equilibrium programming inthe 1970s A major theoretical advantage of the family of fixed-point ho-

par-viii Preface

motopy methods is their global convergence Attracted by this tage and the novelty of the methods, many well-known researchers in-cluding Eaves, Garcia, Gould, Kojima, Megiddo, Saigal, Todd, and Zang-will all made fundamental contributions to the subject The flurry ofresearch activities in this area continued for more than a decade, untilthe occurrence of several significant events that provided clear evidence

advan-of the practical inadequacy advan-of this family advan-of methods for solving tic equilibrium problems These events, to be mentioned momentarily,marked a turning point whereby the fixed-point/homotopy approach tothe computation of equilibria gave way to an alternative set of methodsthat constitute what one may call a contemporary variational inequalityapproach to equilibrium programming For completeness, we mention sev-eral prominent publications that contain important works on the subject

realis-of fixed-point computation via the homotopy approach and its applications[10, 11, 34, 203, 205, 206, 211, 251, 252, 285, 403, 440, 729, 760, 841, 879].For a recent paper on this approach, see [883]

In the same period and in contrast to the aforementioned algorithmicresearch, Karamardian, in a series of papers [398, 399, 400, 401, 402], devel-oped an extensive existence theory for the NCP and its cone generalization

In particular, the basic connection between the CP and the VI,

Proposi-tion 1.1.3, appeared in [400] The 1970s were a period when many

funda-mental articles on the VI/CP first appeared These include the paper by

Eaves [202] where the natural map FnatK was used to prove a basic theorem

of complementarity, important studies by Mor´e [623, 624] and Mor´e andRheinboldt [625], which studied several distinguished classes of nonlinearfunctions and their roles in complementarity problems, and the individualand joint work of Kojima and Megiddo [441, 599, 600, 601], which investi-gated the existence and uniqueness of solutions to the NCP

Although the initial developments of infinite-dimensional variational equalities and finite-dimensional complementarity problems had followeddifferent paths, there were attempts to bring the two fields more closelytogether, with the International School of Mathematics held in Summer

in-1978 in Erice, Italy, being the most prominent one The proceedings ofthis conference were published in [141] The paper [138] is among the ear-liest that describes some physical applications of VIs in infinite dimensionssolvable by LCP methods

One could argue that the final years of the 1970s marked the beginning

of the contemporary chapter on the finite-dimensional VI/CP During thattime, the U.S Department of Energy was employing a market equilibriumsystem known as the Project Independent Evaluation System (PIES) [350,

351] for energy policy studies This system is a large-scale variationalinequality that was solved on a routine basis by a special iterative algorithmknown as the PIES algorithm, yielding remarkably good computationalexperience For a detailed account of the PIES model, see the monograph

by Ahn [5], who showed that the PIES algorithm was a generalization of theclassical Jacobi iterative method for solving system of nonlinear equations[652] For the convergence analysis of the PIES algorithm, see Ahn andHogan [6]; for a recent update of the PIES model, which has become theNational Energy Modeling System (NEMS), see [278]

The original PIES model provided a real-life economic model for whichthe fixed-point methods mentioned earlier were proved to be ineffective.This experience along with several related events inspired a new wave ofresearch into iterative methods for solving VIs/CPs arising from variousapplied equilibrium contexts One of these events is an important algo-rithmic advance, namely, the introduction of Newton’s method for solvinggeneralized equations (see below)

At about the same time as the PIES model appeared, Smith [793] andDafermos [151] formulated the traffic equilibrium problem as a variationalinequality Parallel to the VI formulation, Aashitiani and Magnanti [1]introduced a complementarity formulation for Wardrop’s user equilibriumprinciple [868] and established existence and uniqueness results of trafficequilibria using fixed-point theorems; see also [20, 253] Computationally,the PIES algorithm had served as a model approach for the design of itera-tive methods for solving the traffic equilibrium problem [2, 254, 259] Morebroadly, the variational inequality approach has had a significant impact

on the contemporary point of view of this problem and the closely relatedspatial price equilibrium problem

In two important papers [594, 595], Mathiesen reported computationalresults on the application of a sequential linear complementarity (SLCP)approach to the solution of economic equilibrium problems These resultsfirmly established the potential of this approach and generated substantialinterest among many computational economists, including Manne and his(then Ph.D.) students, most notably, Preckel, Rutherford, and Stone Thevolume edited by Manne [581] contains the papers [697, 814], which givefurther evidence of the computational efficiency of the SLCP approach forsolving economic equilibrium problems; see also [596]

The SLCP method, as it was called in the aforementioned papers,turned out to be Newton’s method developed and studied several yearsearlier by Josephy [389, 390, 391]; see also the later papers by Eaves[209, 210] While the results obtained by the computational economists

x Preface

clearly established the practical effectiveness of Newton’s method throughsheer numerical experience, Josephy’s work provided a sound theoreticalfoundation for the fast convergence of the method In turn, Josephy’sresults were based on the seminal research of Robinson, who in several

landmark papers [728, 730, 732, 734] introduced the generalized equations

as a unifying mathematical framework for optimization problems, mentarity problems, variational inequalities, and related problems As weexplain below, in addition to providing the foundation for the convergencetheory of Newton’s method, Robinson’s work greatly influenced the moderndevelopment of sensitivity analysis of mathematical programs

comple-While Josephy’s contributions marked a breakthrough in algorithmicadvances of the field, they left many questions unanswered From a com-putational perspective, Rutherford [754] recognized early on the lack ofrobustness in Newton’s method applied to some of the most challengingeconomic equilibrium problems Although ad hoc remedies and special-ized treatments had lessened the numerical difficulty in solving these prob-lems, the heuristic aids employed were far from satisfactory in resolving thepractical deficiency of the method, which was caused by the lack of a suit-able stabilizing strategy for global convergence Motivated by the need for

a computationally robust Newton method with guaranteed global gence, Pang [663] developed the B-differentiable Newton method with a linesearch and established that the method is globally convergent and locallysuperlinearly convergent While this is arguably the first work on globalNewton methods for solving nonsmooth equations, Pang’s method suffersfrom a theoretical drawback in that its convergence requires a Fr´echet dif-ferentiability assumption at a limit point of the produced sequence.Newton’s method for solving nondifferentiable equations had been in-vestigated before Pang’s work Kojima and Shindo [454] discussed such amethod for PC1functions Kummer [466] studied this method for generalnondifferentiable functions Both papers dealt with the local convergencebut did not address the globalization of the method Generalizing the class

conver-of semismooth functions conver-of one variable defined by Mifflin [607], Qi andSun [701] introduced the class of vector semismooth functions and estab-lished the local convergence of Newton’s method for this class of functions.The latter result of Qi and Sun is actually a special case of the generaltheory of Kummer Since its introduction, the class of vector semismoothfunctions has played a central role throughout the subsequent algorithmicdevelopments of the field Although focused mainly on the smooth case,the two recent papers [282, 878] present an enlightening summary of thehistorical developments of the convergence theory of Newton’s method

As an alternative to Pang’s line search globalization strategy, Ralph[710] presented a path search algorithm that was implemented by Dirkseand Ferris in their highly successful PATH solver [187], which was awardedthe 1997 Beale-Orchard-Hays Prize for excellence in computational mathe-matical programming; the accompanying paper [186] contains an extensivecollection of MiCP test problems In an important paper that dealt with anoptimization problem [247], Fischer proposed the use of what is now calledthe Fischer-Burmeister function to reformulate the Karush-Kuhn-Tuckerconditions arising from an inequality constrained optimization problem as

a system of nonsmooth equations Collectively, these works paved the wayfor an outburst of activities that started with De Luca, Facchinei, andKanzow [162] The latter paper discussed the application of a globally con-vergent semismooth Newton method to the Fischer-Burmeister reformu-lation of the nonlinear complementarity problem; the algorithm describedtherein provided a model approach for many algorithms that followed Thesemismooth Newton approach led to algorithms that are conceptually andpractically simpler than the B-differentiable Newton method and the pathNewton method, and have, at the same time, better convergence properties.The attractive theoretical properties of the semismooth methods andtheir good performance in practice spurred much research to investigatefurther this class of methods and inspired much of the subsequent studies

In the second half of the 1990s, a large number of papers was devoted to theimprovement, extension, and numerical testing of semismooth algorithms,bringing these algorithms to a high level of sophistication Among otherthings, these developments made it clear that the B-differentiable Newtonmethod is intimately related to the semismooth Newton method applied

to the min reformulation of the complementarity problem, thus confirmingthe breadth of the new approach

The above overview gives a general perspective on the evolution of theVI/CP and documents several major events that have propelled this sub-ject to its modern status as a fruitful and exciting discipline within math-ematical programming There are many other interesting developments,such as sensitivity and stability analysis, piecewise smooth functions, er-ror bounds, interior point methods, smoothing methods, methods of theprojection family, and regularization, as well as the connections with newapplications and other mathematical disciplines, all of which add to therichness and vitality of the field and form the main topics in our work.The notes and comments of these developments are contained at the end

of each chapter

xii Preface

A Synopsis of the Book

Divided into two volumes, the book contains twelve main chapters, lowed by an extensive bibliography, a summary of main results and keyalgorithms, and a subject index The first volume consists of the first sixchapters, which present the basic theory of VIs and CPs The second vol-ume consists of the remaining six chapters, which present algorithms ofvarious kinds for solving VIs and CPs Besides the main text, each chaptercontains (a) an extensive set of exercises, many of which are drawn frompublished papers that supplement the materials in the text, and (b) a set

fol-of notes and comments that document historical accounts, give the sourcesfor the results in the main text, and provide discussions and references onrelated topics and extensions The bibliography contains more than 1,300publications in the literature up to June 2002 This bibliography servestwo purposes: one purpose is to give the source of the results in the chap-ters, wherever applicable; the other purpose is to give a documentation ofpapers written on the VI/CP and related topics

Due to its comprehensiveness, each chapter of the book is by itself

quite lengthy Among the first six sections in Chapter 1, Sections 1.1, 1.2,

1.3, and 1.5 make up the basic introduction to the VI/CP The source

problems in Section 1.4 are of very diverse nature; they fall into several

general categories: mathematical programming, economics, engineering,and finance Depending on an individual’s background, a reader can safelyskip those subsections that are outside his/her interests; for instance, aneconomist can omit the subsection on frictional contact problems, a contactmechanician can omit the subsection on Nash-Cournot production models

Section 1.6 mainly gives the definition of several extended problems; except for (1.6.1), which is re-introduced and employed in Chapter 11, this section

can be omitted at first reading

Chapters 2 and 3 contain the basic theory of existence and multiplicity

of solutions Several sections contain review materials of well-known topics;these are included for the benefit of those readers who are unfamiliar with

the background for the theory Section 2.1 contains the review of degree

theory, which is a basic mathematical tool that we employ throughout thebook; due to its powerful implications, we recommend this to a reader who

is interested in the theoretical part of the subject Sections 2.2, 2.3 (except Subsection 2.3.2), 2.4, and 2.5 (except Subsection 2.5.3) contain funda- mental results While Sections 2.6 and 2.8 can be skipped at first reading, Section 2.7 contains very specialized results for the discrete frictional con-

tact problem and is included herein only to illustrate the application of

the theory developed in the chapter to an important class of mechanicalproblems.

Section 3.1 in Chapter 3 introduces the class of B-differentiable

func-tions that plays a fundamental role throughout the book With the

ex-ception of the nonstandard SBCQ, Section 3.2 is a review of various known CQs in NLP Except for the last two subsections in Section 3.3 and Subsection 3.5.1, which may be omitted at first reading, the remainder of

well-this chapter contains important properties of solutions to the VI/CP

Chapter 4 serves two purposes: One, it is a technical precursor to the

next chapter; and two, it introduces the important classes of PA functions

(Section 4.2) and PC1 functions (Section 4.6) Readers who are not

in-terested in the sensitivity and stability theory of the VI/CP can skip most

of this and the next chapter Nevertheless, in order to appreciate the class

of semismooth functions, which lies at the heart of the contemporary rithms for solving VIs/CPs, and the regularity conditions, which are key tothe convergence of these algorithms, the reader is advised to become famil-iar with certain developments in this chapter, such as the basic notion of

algo-coherent orientation of PA maps (Definition 4.2.3) and its matrix-theoretic characterizations for the special maps MnorK and MnatK (Proposition 4.2.7)

as well as the fundamental role of this notion in the globally unique

solv-ability of AVIs (Theorem 4.3.2) The inverse function Theorem 4.6.5 for

PC1 functions is of fundamental importance in nonsmooth analysis

Sub-sections 4.1.1 and 4.3.1 are interesting in their own right; but they are

not needed in the remainder of the book

Chapter 5 focuses on the single topic of sensitivity and stability of

the VI/CP While stability is the cornerstone to the fast convergence ofNewton’s method, readers who are not interested in this specialized topic

or in the advanced convergence theory of the mentioned method can skipthis entire chapter Notwithstanding this suggestion, Section 5.3 is of

classical importance and contains the most basic results concerning thelocal analysis of an isolated solution

Chapter 6 contains another significant yet specialized topic that can be

omitted at first reading From a computational point of view, an importantgoal of this chapter is to establish a sound basis for understanding the con-nection between the exact solutions to a given problem and the computedsolutions of iterative methods under prescribed termination criteria used inpractical implementation As evidenced throughout the chapter and also

in Section 12.6, the theory of error bounds has far-reaching consequences

that extend beyond this goal For instance, since the publication of the

pa-per [222], which is the subject of discussion in Section 6.7, there has been

xiv Preface

an increasing use of error bounds in designing algorithms that can identifyactive constraints accurately, resulting in enhanced theoretical properties

of algorithms and holding promise for superior computational efficiency

Of independent interest, Chapters 7 and 8 contain the preparatory terials for the two subsequent chapters While Sections 7.1 and 7.4 are

ma-both concerned with the fundamentals of nonsmooth functions, the formerpertains to general properties of nonsmooth functions, whereas the latterfocuses on the semismooth functions As far as specific algorithms go, Algo-

rithms 7.3.1 and 7.5.1 in Sections 7.3 and 7.5, respectively, are the most

basic and strongly recommended for anyone interested in the subsequentdevelopments The convergence of the former algorithm depends on the

(strong) stability theory in Chapter 5, whereas that of the latter is rather

simple, provided that one has a good understanding of semismoothness

In contrast to the previous two algorithms, Algorithm 7.2.17 is closest to

a straightforward application of the classical Newton method for smoothsystems of equations to the NCP

The path search Newton method 8.1.9 is the earliest algorithm to be

coded in the highly successful PATH computer software [187] Readers whoare already familiar with the line search and/or trust region methods in

standard nonlinear programming may wish to peruse Subsection 8.3.3 and skip the rest of Chapter 8 in order to proceed directly to the next chapter.

When specialized to C1optimization problems, as is the focus in Chapters 9 and 10, much of the material in Sections 8.3 and 8.4 is classical; these two

sections basically offer a systematic treatment of known techniques andresults and present them in a way that accommodates nonsmooth objectivefunctions

The last four chapters are the core of the algorithmic part of this book

While Chapter 9 focuses on the NCP, Chapter 10 is devoted to the VI.

The first section of the former chapter presents a detailed exposition ofalgorithms based on the FB merit function and their convergence theory

The most basic algorithm, 9.1.10, is described in Subsection 9.1.1 and is accompanied by a comprehensive analysis Algorithm 9.2.3, which com-

bines the min function and the FB merit function in a line search method,

is representative of a mixture of algorithms in one overall scheme

Ex-ample 9.3.3 contains several C-functions that can be used in place of the

FB C-function The box-constrained VI in Subsection 9.4.3 unifies the generalized problems in Section 9.4.

The development in Section 10.1 is very similar to that in Section 9.1.1; the only difference is that the analysis of the first section in Chapter 10

is tailored to the KKT system of a finitely representable VI The other

major development in this chapter is the D-gap function in Section 10.3,

which is preceded by the preparatory discussion of the regularized gap

function in Subsection 10.2.1 The implicit Lagrangian function presented

in Subsection 10.3.1 is an important merit function for the NCP.

Chapter 11 presents interior and smoothing methods for solving CPs

of different kinds, including KKT systems Developed in the abstract ting of constrained equations, the basic interior point method for the im-

set-plicit MiCP, Algorithm 11.5.1, is presented in Section 11.5 An extensive

theoretical study of the latter problem is the subject of the previous

Sec-tion 11.4; in which the important mixed P0 property is introduced (see

Definition 11.4.1) A Newton smoothing method is outlined in tion 11.8.1; this method is applicable to smoothed reformulations of CPs using the smoothing functions discussed in Subsection 11.8.2, particularly those in Example 11.8.11.

Subsec-The twelveth and last chapter discusses various specialized methodsthat are applicable principally to (pseudo) monotone VIs and NCPs of the

P0 type The first four sections of the chapter contain the basic ods and their convergence theories The theory of maximal monotone op-

meth-erators in Subsection 12.3.1 plays a central role in the proximal point method that is the subject of Subsection 12.3.2 Bregman-based meth- ods in Subsection 12.7.2 are well researched in the literature, whereas the interior/barrier methods in Subsection 12.7.4 are recent entrants to the

field

Acknowledgments

Writing a book on this subject has been the goal of the second authorsince he and Harker published their survey paper [332] in 1990 This goalwas not accomplished and ended with Harker giving a lecture series at theUniversit´e Catholique de Louvain in 1992 that was followed by the lecturenotes [331] The second author gratefully acknowledges Harker for thefruitful collaboration and for his keen interest during the formative stage

of this book project

The first author was introduced to optimization by Gianni Di Pillo andLuigi Grippo, who did much to shape his understanding of the disciplineand to inspire his interest in research The second author has been verylucky to have several pioneers of the field as his mentors during his earlycareer They are Richard Cottle, Olvi Mangasarian, and Stephen Robin-son To all these individuals we owe our deepest gratitude Both authorshave benefitted from the fruitful collaboration with their doctoral students

xvi Preface

and many colleagues on various parts of the book We thank them cerely Michael Ferris and Stefan Scholtes have provided useful comments

sin-on a preliminary versisin-on of the book that help to shape its final form

We wish to thank our Series Editor, Achi Dosanjh, the Production tor, Louise Farkas, and the staff members at Springer-New York, for theirskillful editorial assistance

Edi-Facchinei’s research has been supported by grants from the Italian search Ministry, the National Research Council, the European Commission,and the NATO Science Committee The U.S National Science Foundationhas provided continuous research grants through several institutions to sup-port Pang’s work for the last twenty-five years The joint research withMonteiro was supported by the Office of Naval Research as well Pang’sstudents have also benefited from the financial support of these two fundingagencies, to whom we are very grateful

Re-Finally, the text of this monograph was typeset by the authors using

LATEX, a document preparation system based on Knuth’s TEX program

We have used the document style files of the book [142] that were prepared

by Richard Cottle and Richard Stone, and based on the LATEX book style

December 11, 2002

Preface v

1.1 Problem Description 2

1.1.1 Affine problems 7

1.2 Relations Between Problem Classes 8

1.3 Integrability and the KKT System 12

1.3.1 Constrained optimization problems 13

1.3.2 The Karush-Kuhn-Tucker system 18

1.4 Source Problems 20

1.4.1 Saddle problems 21

1.4.2 Nash equilibrium problems 24

1.4.3 Nash-Cournot production/distribution 26

1.4.4 Economic equilibrium problems 36

1.4.5 Traffic equilibrium models 41

1.4.6 Frictional contact problems 46

1.4.7 Elastoplastic structural analysis 51

1.4.8 Nonlinear obstacle problems 55

1.4.9 Pricing American options 58

1.4.10 Optimization with equilibrium constraints 65

1.4.11 CPs in SPSD matrices 67

1.5 Equivalent Formulations 71

xvii

xviii Contents

1.5.1 Equation reformulations of the NCP 71

1.5.2 Equation reformulations of the VI 76

1.5.3 Merit functions 87

1.6 Generalizations 95

1.7 Concluding Remarks 98

1.8 Exercises 98

1.9 Notes and Comments 113

2 Solution Analysis I 125 2.1 Degree Theory and Nonlinear Analysis 126

2.1.1 Degree theory 126

2.1.2 Global and local homeomorphisms 134

2.1.3 Elementary set-valued analysis 138

2.1.4 Fixed-point theorems 141

2.1.5 Contractive mappings 143

2.2 Existence Results 145

2.2.1 Applications to source problems 150

2.3 Monotonicity 154

2.3.1 Plus properties and F-uniqueness 162

2.3.2 The dual gap function 166

2.3.3 Boundedness of solutions 168

2.4 Monotone CPs and AVIs 170

2.4.1 Properties of cones 171

2.4.2 Existence results 175

2.4.3 Polyhedrality of the solution set 180

2.5 The VI (K, q, M ) and Copositivity 185

2.5.1 The CP (K, q, M ) 192

2.5.2 The AVI (K, q, M ) 199

2.5.3 Solvability in terms of feasibility 202

2.6 Further Existence Results for CPs 208

2.7 A Frictional Contact Problem 213

2.8 Extended Problems 220

2.9 Exercises 226

2.10 Notes and Comments 235

3 Solution Analysis II 243 3.1 Bouligand Differentiable Functions 244

3.2 Constraint Qualifications 252

3.3 Local Uniqueness of Solutions 266

3.3.1 The critical cone 267

3.3.2 Conditions for local uniqueness 271

3.3.3 Local uniqueness in terms of KKT triples 279

3.3.4 Local uniqueness theory in NLP 283

3.3.5 A nonsmooth-equation approach 287

3.4 Nondegenerate Solutions 289

3.5 VIs on Cartesian Products 292

3.5.1 Semicopositive matrices 294

3.5.2 P properties 298

3.6 Connectedness of Solutions 309

3.6.1 Weakly univalent functions 310

3.7 Exercises 317

3.8 Notes and Comments 330

4 The Euclidean Projector and Piecewise Functions 339 4.1 Polyhedral Projection 340

4.1.1 The normal manifold 345

4.2 Piecewise Affine Maps 352

4.2.1 Coherent orientation 356

4.3 Unique Solvability of AVIs 371

4.3.1 Inverse of MnorK 374

4.4 B-Differentiability under SBCQ 376

4.5 Piecewise Smoothness under CRCQ 384

4.6 Local Properties of PC1Functions 392

4.7 Projection onto a Parametric Set 401

4.8 Exercises 407

4.9 Notes and Comments 414

5 Sensitivity and Stability 419 5.1 Sensitivity of an Isolated Solution 420

5.2 Solution Stability of B-Differentiable Equations 427

5.2.1 Characterizations in terms of the B-derivative 439

5.2.2 Extensions to locally Lipschitz functions 443

5.3 Solution Stability: The Case of a Fixed Set 445

5.3.1 The case of a finitely representable set 452

5.3.2 The NCP and the KKT system 462

5.3.3 Strong stability under CRCQ 469

5.4 Parametric Problems 472

5.4.1 Directional differentiability 482

5.4.2 The strong coherent orientation condition 489

5.4.3 PC1multipliers and more on SCOC 496

xx Contents

5.5 Solution Set Stability 500

5.5.1 Semistability 503

5.5.2 Solvability of perturbed problems and stability 509

5.5.3 Partitioned VIs with P0 pairs 512

5.6 Exercises 516

5.7 Notes and Comments 525

6 Theory of Error Bounds 531 6.1 General Discussion 531

6.2 Pointwise and Local Error Bounds 539

6.2.1 Semistability and error bounds 539

6.2.2 Local error bounds for KKT triples 544

6.2.3 Linearly constrained monotone composite VIs 548

6.3 Global Error Bounds for VIs/CPs 554

6.3.1 Without Lipschitz continuity 557

6.3.2 Affine problems 564

6.4 Monotone AVIs 575

6.4.1 Convex quadratic programs 586

6.5 Global Bounds via a Variational Principle 589

6.6 Analytic Problems 596

6.7 Identification of Active Constraints 600

6.8 Exact Penalization and Some Applications 605

6.9 Exercises 610

6.10 Notes and Comments 616

Subsections are omitted; for details, see Volume II.

7.1 Nonsmooth Analysis I: Clarke’s Calculus 6267.2 Basic Newton-type Methods 6387.3 A Newton Method for VIs 6637.4 Nonsmooth Analysis II: Semismooth Functions 6747.5 Semismooth Newton Methods 6927.6 Exercises 7087.7 Notes and Comments 715

8.1 Path Search Algorithms 7248.2 Dini Stationarity 7368.3 Line Search Methods 7398.4 Trust Region Methods 7718.5 Exercise 7868.6 Notes and Comments 788

9.1 Nonlinear Complementarity Problems 7949.2 Global Algorithms Based on the min Function 8529.3 More C-Functions 8579.4 Extensions 8659.5 Exercises 8779.6 Notes and Comments 882

10.1 KKT Conditions Based Methods 89210.2 Merit Functions for VIs 91210.3 The D-Gap Merit Function 93010.4 Merit Function Based Algorithms 947

xxi

xxii Contents of Volume II

10.5 Exercises 97810.6 Notes and Comments 981

11.1 Preliminary Discussion 99111.2 An Existence Theory 99611.3 A General Algorithmic Framework 100311.4 Analysis of the Implicit MiCP 101211.5 IP Algorithms for the Implicit MiCP 103611.6 The Ralph-Wright IP Approach 105311.7 Path-Following Noninterior Methods 106011.8 Smoothing Methods 107211.9 Excercises 109211.10 Notes and Comments 1097

12.1 Projection Methods 110712.2 Tikhonov Regularization 112512.3 Proximal Point Methods 113512.4 Splitting Methods 114712.5 Applications of Splitting Algorithms 116412.6 Rate of Convergence Analysis 117612.7 Equation Reduction Methods 118312.8 Exercises 121412.9 Notes and Comments 1222

Index of Definitions, Results, and Algorithms II-39

The numbers refer to the pages where the acronyms first appear.AVI, 7 Affine Variational Inequality

B-function, 869 Box-function

CC, 1017 Coerciveness in the Complementary variables

CE, 989 Constrained Equation

C-function, 72 Complementarity function

CP, 4 Complementarity Problem

CQ, 17 Constraint Qualification

Cr, 13 Continuously differentiable of order r

C1,1, 529 = LC1

CRCQ, 262 Constant Rank Constraint Qualification

ESSC, 896 Extended Strong Stability condition

GUS, 122 Globally Uniquely Solvable

FOA, 443 First-Order Approximation

IP, 989 Interior Point

MLCP, 7 Mixed Linear Complementarity Problem

MPEC, 65 Mathematical Program with Equilibrium ConstraintsMPS, 581 Minimum Principle Sufficiency

NCP, 6 Nonlinear Complementarity Problem

NLP, 13 Nonlinear Program

PCr, 384 Piecewise smooth of order r (mainly r = 1)

xxiii

xxiv Acronyms

PL, 344 Piecewise Linear

QP, 15 Quadratic Program

QVI, 16 Quasi-Variational Inequality

SBCQ, 262 Sequentially Bounded Constraint Qualification

SC1, 686 C1functions with Semismooth gradients

SCOC, 490 Strong Coherent Orientation Condition

SLCP, v Sequential Linear Complementarity Problem

SMFCQ 253 Strict Mangasarian-Fromovitz Constraint QualificationSPSD, 67 Symmetric Positive Semidefinite

SQP, 718 Sequential Quadratic Programming

VI, 2 Variational Inequality

WMPS, 1202 Weak Minimum Principle Sufficiency

++ the cone of positive definite matrices inM n

IRn the real n-dimensional space

IRn

+ the nonnegative orthant of IRn

IRn

++ the positive orthant of IRn

IRn ×m the space of n × m real matrices

Matrices

A ≡ (a ij ); a matrix with entries a ij

det A the determinant of a matrix A

tr A the trace of a matrix A

A T the transpose of a matrix A

A −1 the inverse of a matrix A

M/A the Schur complement of A in M

2(A + A T ); the symmetric part of a matrix A

λmax(A) the largest eigenvalue of a matrix A ∈ M n

λmin(A) the smallest eigenvalue of a matrix A ∈ M n

A ≡
λmax(A T A); the Euclidean norm of A ∈ IR n ×n

A • B the Frobenius product of two matrices A and B in IR n ×n

A F ≡ √ A • A; the Frobenius norm of A ∈ IR n ×n

A ·α the columns of A indexed by α

A α · the rows of A indexed by α

A αβ submatrix of A with rows and columns indexed

xxvi Glossary of Notation

Scalars

sgn t the sign, 1, −1, 0, of a positive, negative,

or zero scalar t

t+ ≡ max(0, t); the nonnegative part of a scalar

t − ≡ max(0, −t); the nonpositive part of a scalar

x+ ≡ max(0, x); the nonnegative part of a vector x

x − ≡ max(0, −x); the nonpositive part of a vector x

x α subvector of x with components indexed by α

; the  p -norm of a vector x ∈ IR n

x the 2-norm of x ∈ IR n, unless otherwise specified

1≤i≤n |x i |; the  ∞ -norm of x ∈ IR n

x A ≡ √ x T Ax; the A-norm of x ∈ IR n for A ∈ M n

++

x ≥ y the (usual) partial ordering: x i ≥ y i , i = 1, n

x
y x ≥ y and x = y

x > y the strict ordering: x i > y i , i = 1, , n

min(x, y) the vector whose i-th component is min(x i , y i)

max(x, y) the vector whose i-th component is max(x i , y i)

x ◦ y ≡ (x i y i)n

i=1 ; the Hadamard product of x and y

x ⊥ y x and y are perpendicular

1k k-vector of all ones (subscript often omitted)

Functions

F : D → R a mapping with domain D and range R

F |Ω the restriction of the mapping F to the set Ω

F ◦ G composition of two functions F and G

F −1 the inverse of a mapping F

F (· ; ·) directional derivative of the mapping F

; the Hessian matrix of θ : IR n → IR

θ D(·; ·) Dini directional derivative of θ : IR n → IR

Jac F = ∂ B F the limiting Jacobian or B-subdifferential of

ϕ ∗ (y) the conjugate of a convex function ϕ(x)

ϕ ∞ (d) the recession function of a convex function ϕ(x)

o(t) any function such that lim

t ↓0

o(t)

t = 0O(t) any function such that lim sup

t ↓0

|O(t)|

t < ∞ deg(Φ, Ω, p) the degree of Φ at p relative to Ω

deg(Φ, Ω) ≡ deg(Φ, Ω, 0)

ind(Φ, x) the index of Φ at x

ΠK (x) the Euclidean projection of x on the set K

ΠK,A (x) skewed projection of x on K under the A-norm

ΠA K ≡ Π K,A ◦ A −1

mid(a, b; x) ≡ Π [a,b] (x); the mid function for given a, b ∈ IR n

infS θ(x) the infimum of the function θ on S

supS θ(x) the supremum of the function θ on S

dist(x, W ) Euclidean distance function from vector x to set W

dist∞ (x, W )  ∞ -distance function from vector x to set W

JΦ the resolvent of the multifunction Φ

D f (x, y) ≡ f(x) − f(y) − ∇f(y) T (x − y); Bregman distance

induced by the strictly convex function f

Sets

∈, ∈ element membership, non-membership in a set

∅, ⊆, ⊂ the empty set, set inclusion, proper set inclusion

∪, ∩, × union, intersection, Cartesian product



S i Cartesian product of sets S i

S1\ S2 the difference of two sets S1 and S2

S1+ S2 the vector sum of two sets S1and S2

xxviii Glossary of Notation

Sets (continued)

|S| the cardinality of a finite set S

aff S, lin S the affine, linear hull of a set S, respectively

bd S = ∂S the topological boundary of a set S

cl S, int S the topological closure, interior of a set S, respectively conv S the convex hull of a set S

pos A the conical hull of the columns of A ∈ IR m ×n

ri S the relative interior of a set S

S ∗ , S ∞ the dual cone of a set S, the recession cone of S

S ⊥ the orthogonal complement of a set S

dom Φ the domain of a (multi)function Φ

gph Φ the graph of a (multi)function Φ

ran Φ the range of a (multi)function Φ

IB(x, δ) the open ball with center at x and radius δ

(a neighborhoodN of x) IB(H; ε, S) ε-neighborhood of the function H restricted to the set

S, comprising all continuous functions G such that

G − H S ≡ sup y ∈S G(y) − H(y) < ε

argmaxS θ(x) the set of constrained maximizers of θ on S

argminS θ(x) the set of constrained minimizers of θ on S

supp(x) the support of a vector x

L(x; S) the linearization cone of the set S at a point x ∈ S

N (x; S) the normal cone of the set S at a point x ∈ S

T (x; S) the tangent cone of the set S at a point x ∈ S

C(x; K, F ) critical cone of the pair (K, F ) at x ∈ SOL(K, F )

C π (x; K) ≡ C(Π K (x); K, I − x); critical cone of K at x ∈ IR n

I(x) the index set of active constraints at x

M(x) the set of KKT multipliers at x ∈ SOL(K, F )

M π (x) the set of KKT multipliers at ΠK (x)

M e (x) the (finite) set of extreme KKT multipliers inM(x)

P (A, b) {y ∈ IR n : Ay ≤ b}; a polyhedron

I(A, b) family of index sets identifying the faces of P (A, b)

Bbas(A, b) normal family of basis matrix of P (A, b)

[x, y] the closed line segment joining x and y in IR n

(x, y) the open line segment joining x and y in IR n

x ⊥ the orthogonal complement of the vector x

epi ϕ the epigragph of a convex function ϕ

++× IR m), used in IP theory

H+ ≡ H(IR 2n × IR m), used in IP theory

Problem Classes and Fundamental Objects

AVI (K, q, M ) AVI defined by the polyhedron K, vector q,

and matrix M

CE (G, X) constrained equation defined by the function G

and the set X

CP (F, G) vertical CP defined by two functions F and G

CP (K, F ) CP defined by the cone K and the mapping F

CP (K, q, M ) ≡ CP (K, F ) with F (x) ≡ q + Mx

FEA(K, F ) the feasible region of the CP (K, F )

LCP (q, M ) LCP defined by the vector q and matrix M

NCP (F ) NCP defined by the function F : IR n → IR n

SOL(K, F ) solution set of the VI (K, F )

SOL(K, G, A, b) solution set of the VI (K, G, A, b)

SOL(K, q, M ) solution set of the VI (K, q, M )

SOL(q, M ) solution set of the LCP (q, M )

VI (K, F ) VI defined by the set K and the mapping F

copositive matrices M such that x T M x ≥ 0 for all x ≥ 0

nondegenerate matrices with nonzero principal minors

positive definite matrices M such that x T M x > 0 for all x = 0

positive semidefinite matrices M such that x T M x ≥ 0 for all x

positive semi- positive semidefinite + [x T M x = 0 ⇒ Mx = 0]

xxx Glossary of Notation

Matrix Classes (continued)

R0 matrices M such that SOL(0, M ) = {0}

S0 matrices M such that M x ≥ 0 for some x
0

S matrices M such that M x > 0 for some x ≥ 0

CP Functions

ψCCK(a, b) ≡ ψFB(a, b) − τ max(0, a) max(0, b);

the Chen-Chen-Kanzow C-function

ψFB(a, b) ≡ √ a2+ b2− a − b; the Fischer-Burmeister

the Kanzow-Kleinmichel C-function

ψLT(a, b) ≡ (a, b) q − a − b, q > 1; the Luo-Tseng C-function

ψLTKYF(a, b) ≡ φ1(ab) + φ2(−a, −b), the

Luo-Tseng-Kanzow-Yamashita-Fukushima family of C-functions

ψMan(a, b) ≡ ζ(|a − b|) − ζ(b) − ζ(a); Mangasarian’s family

of C-functions, includes the min function

ψU(a, b) Ulbrich’s C-function; see Exercise 1.8.21

ψYYF(a, b) ≡ η

2((ab)+)2+12ψFB(a, b)2, η > 0;

the Yamada-Yamashita-Fukushima C-function

Fψ (x) ≡ (ψ(x i , F i (x))) n

i=1; the reformulation function of the

NCP (F ) for a given C-function ψ

2a − 12b



v2+ 1

2b max( 0, v − b u )2

− 12a max( 0, v − a u )2; for b > a > 0

φQ(τ, τ  ; r, s) Qi’s B-function; see (9.4.7)

HIP(x, y, z) the IP function for implicit MiCP; see (11.1.4)

HCHKS(u, x, y, z) the IP function for the CHKS smoothing of the

min function; see Exercise 11.9.8

FnatK the natural map associated with the pair (K, F )

FnorK the normal map associated with the pair (K, F )

FnatK,D (x) ≡ x − Π K,D (x − D −1 F (x)); the skewed natural map

associated with the pair (K, F ) using Π K,D

FnatK,τ the natural map associated with the VI (K, τ F )

MnatK ≡ Fnat

K for F (x) ≡ Mx

MnorK ≡ Fnor

K for F (x) ≡ Mx

θgap the gap function of a VI

θdual the dual gap function of a VI

θ c the regularized gap function with parameter c > 0

θlin

c the linearized gap function with parameter c > 0

θ ab ≡ θ a − θ b ; the D-gap function for b > a > 0

xxxii Glossary of Notation

Selected Function Classes and Properties

co-coercive on K functions F for which ∃ η > 0 such that

x∈X

x→∞

F (x) = ∞

P0, P, P∗ (σ) see Definition 3.5.8

(pseudo) monotone see Definition 2.3.1

(pseudo) monotone see Definition 2.3.9

uniformly P see Definition 3.5.8

univalent = continuous plus injective

weakly univalent uniform limit of univalent functions

The chapters of the book are numbered from 1 to 12; the sections are

de-noted by decimal numbers of the type 2.3 (meaning Section 3 of Chapter

2) Many sections are further divided into subsections; most subsections arenumbered, some are not The numbered subsections are by decimal num-

bers following the section numbers; e.g., Subsection 1.3.1 means Chapter

1, Section 3, Subsection 1

All definitions, results, and miscellaneous items are numbered

consecu-tively within each section in the form 1.3.5, 1.3.6, meaning Items 5 and 6

in Section 3 of Chapter 1 All items are also identified by their types, for

example, 1.4.1 Proposition., 1.4.2 Remark When an item is referred

to in the text, it is called out as Algorithm 5.2.1, Theorem 4.1.7, and

so forth Equations are numbered consecutively and identified by chapter,section, and equation Thus (3.1.4) means Equation (4) in Section 1 ofChapter 3

xxxiii

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In this chapter, we formally define the problems that form the main topic

of this book: the variational inequality (VI) and the complementarity lem (CP) We identify many major themes that will be discussed in detailthroughout the book The principal body of the present chapter consists

prob-of three parts In the first part, which covers Sections 1.1 through 1.3, we

introduce some basic classifications and associated terminology for variousspecial cases of these problems We explain the interconnection betweenthe VI and the CP as well as their relation to a standard nonlinear pro-

gram In the second part, Section 1.4, we present an extensive set of source

problems from engineering, economics, and finance that can be modeled asVIs and/or CPs These applied contexts provide solid evidence of the wideapplicability of the VI/CP methodology in modeling diverse equilibrium

phenomena In the third part, which covers Sections 1.5 and 1.6, we

de-scribe various equivalent formulations of VIs and CPs as systems of smoothand nonsmooth equations and also as constrained and unconstrained opti-mization problems These formulations provide the basis for the develop-ment of the theory and algorithms for the VI/CP that are the main topics

in the subsequent chapters

Except for some source problems in Section 1.4, we strive to present

the materials throughout this chapter in an elementary fashion, using onlyconcepts and results that are well known in linear and nonlinear program-ming We refer the reader unfamiliar with these concepts and backgroundresults to the commentary where basic references are suggested

1

2 1 Introduction

The simplest example of a variational inequality is the classical problem

of solving a system of nonlinear equations Indeed, as we see shortly, thisproblem can be thought of as a VI without constraints In its general form,

a variational inequality is formally defined below

1.1.1 Definition Given a subset K of the Euclidean n-dimensional

space IRn and a mapping F : K → IR n , the variational inequality, noted VI (K, F ), is to find a vector x ∈ K such that

de-( y − x ) T F (x) ≥ 0, ∀ y ∈ K. (1.1.1)

The set of solutions to this problem is denoted SOL(K, F ) 2

Throughout this book, we are interested only in the situation where the

set K is closed and the function F is continuous The latter continuity of

F is understood to mean the continuity of F on an open set containing K;

a similar consideration applies to the differentiability (if appropriate) of F

In most realizations of the VI (K, F ) discussed in the book, the set K is convex Mathematically, some results do not require the convexity of K, however Thus, we do not make the convexity of K a blanket assumption.

See the VI in part (c) of Exercise 1.8.31 and also the one in Exercise 2.9.28.

Since K is closed and F is continuous, it follows that SOL(K, F ) is

al-ways a closed set (albeit it could be empty) Understanding further erties of the solution set of a VI is an important theme that has boththeoretical and practical significance Many results in this book addressquestions pertaining to this theme

prop-A first geometric interpretation of a VI, and more specifically of the

defining inequality (1.1.1), is that a point x in the set K is a solution of the

VI (K, F ) if and only if F (x) forms a non-obtuse angle with every vector

of the form y − x for all y in K We may formalize this observation using the concept of normal cone Specifically, associated with the set K and any vector x  belonging to K, we may define the normal cone to K at x 

to be the following set:

N (x  ; K) ≡ { d ∈ IR n : d T (y − x ) ≤ 0, ∀ y ∈ K }. (1.1.2)

Vectors in this set are called normal vectors to the set K at x . The

inequality (1.1.1) clearly says that a vector x ∈ K solves the VI (K, F ) if

and only if−F (x) is a normal vector to K at x; or equivalently,

Figure 1.1 illustrates this point of view The normal cone is known to play

an important role in convex analysis and nonlinear programming Thisrole persists in the study of the VI The inclusion (1.1.3) is an instance of

a generalized equation.

K x

N (x; K)

−F (x)

Figure 1.1: Solution and the normal cone

In addition to providing a unified mathematical model for a variety ofapplied equilibrium problems, the VI includes many special cases that areimportant in their own right As alluded to in the opening of this section,simplest among these cases is the problem of solving systems of nonlinear

equations, which corresponds to the case where the set K is equal to the

entire space IRn It is not difficult to show that when K = IR n, a vector

x belongs to SOL(K, F ) if and only if x is a zero of the mapping F (i.e.,

F (x) = 0); in other words, SOL(IR n , F ) = F −1(0) To see this, we note

that for any set K, if x ∈ K and F (x) = 0, then clearly x ∈ SOL(K, F ) Thus F −1(0)∩K is always a subset of SOL(K, F ) To establish the reverse inclusion when K = IR n, we note that

x ∈ SOL(IR n

, F ) ⇒ F (x) T

d ≥ 0 ∀ d ∈ IR n

.

In particular with d taken to be −F (x), we deduce that F (x) = 0 Thus

SOL(IRn , F ) ⊆ F −1(0); hence equality holds.

The above argument applies more generally to a solution of the VI

(K, F ) that belongs to the topological interior of K ⊂ IR n Specifically, if

x is a solution of this VI and x belongs to int K, then F (x) = 0 In fact, since x ∈ int K, there exists a scalar τ > 0 sufficiently small such that the

4 1 Introduction vector y ≡ x − τF (x) belongs to K Substituting this vector into (1.1.1),

we deduce that−F (x) T F (x) ≥ 0, which implies F (x) = 0.

In all interesting realizations of the VI, it is invariably the case that none

of the zeros of F , if any, are solutions of the VI (K, F ) In other words,

the VI is a genuinely nontrivial generalization of the classical problem ofsolving equations Nonetheless, as we see throughout the book, the theoryand methods for solving equations are instrumental for the analysis andsolution of the VI

When K is a cone (i.e., x ∈ K ⇒ τx ∈ K for all scalars τ ≥ 0), the VI admits an equivalent form known as a complementarity problem (For an

explanation of the term “complementarity”, see the discussion below.)

1.1.2 Definition Given a cone K and a mapping F : K → IR n, the

complementarity problem, denoted CP (K, F ), is to find a vector x ∈ IR n

satisfying the following conditions:

The precise connection between the VI (K, F ) and the CP (K, F ) when K

is a cone is described in the following elementary result

1.1.3 Proposition Let K be a cone in IR n A vector x solves the VI (K, F ) if and only if x solves the CP (K, F ).

Proof Suppose that x solves the VI (K, F ) Clearly x belongs to K.

Since a cone must contain the origin, by taking y = 0 in (1.1.1), we obtain

x T F (x) ≤ 0.

Furthermore, since x ∈ K and K is a cone, it follows that 2x ∈ K Thus

the CP (K, F ) Conversely, if x solves the CP (K, F ), then it is trivial to

1.1.4 Remark A word about the notation CP (K, F ): when the acronym

CP is attached to the pair (K, F ), it is understood that K is a cone 2 The CP (K, F ) is defined by three conditions: (i) x ∈ K, (ii) F (x) ∈ K ∗,

and (iii) x T F (x) = 0 We introduce some concepts associated with vectors satisfying the first two conditions Specifically, we say that a vector x ∈ IR n

is feasible to the CP (K, F ) if

We say that a vector x ∈ IR n is strictly feasible to the same problem if

x ∈ K and F (x) ∈ int K ∗ .

This definition implicitly assumes that int K ∗ is nonempty, but the

def-inition does not require int K to be nonempty If int K = ∅ and F is continuous, then CP (K, F ) has a strictly feasible vector if and only if there exists a vector x  such that

x  ∈ int K and F (x ) ∈ int K ∗ .

We say that the CP (K, F ) is (strictly) feasible if it has a (strictly) sible vector The feasible region of the CP (K, F ) is the set of all its feasible vectors and is denoted FEA(K, F ) Clearly SOL(K, F ) ⊆ FEA(K, F ); in

fea-set notation, we can write

FEA(K, F ) = K ∩ F −1 (K ∗ ).