Table Of ContentOPTIMIZATION WITH
MULTIVALUED MAPPINGS
Theory, Applications, and Algorithms
Edited by
STEPHAN DEMPE
TU Bergakademie Freiberg, Germany
VYACHESLAV KALASHNIKOV
ITESM, Monterrey, Mexico
1 3
Library of Congress Control Number: 2006925111
ISBN-10: 0-387-34220-6 e-ISBN: 0-387-34221-4
ISBN-13: 978-0-387-34220-7
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Contents
Preface ........................................................ ix
Part I Bilevel Programming
Optimality conditions for bilevel programming problems
Stephan Dempe, Vyatcheslav V. Kalashnikov and Nataliya Kalashnykova 3
Path-based formulations of a bilevel toll setting problem
Mohamed Didi-Biha, Patrice Marcotte and Gilles Savard.............. 29
Bilevel programming with convex lower level problems
Joydeep Dutta and Stephan Dempe ................................. 51
Optimality criteria for bilevel programming problems using
the radial subdifferential
D. Fangha¨nel.................................................... 73
On approximate mixed Nash equilibria and average marginal
functions for two-stage three-players games
Lina Mallozzi and Jacqueline Morgan............................... 97
Part II Mathematical Programs with Equilibrium Constraints
A direct proof for M-stationarity under MPEC-GCQ for
mathematical programs with equilibrium constraints
Michael L. Flegel and Christian Kanzow ............................111
On the use of bilevel programming for solving a structural
optimization problem with discrete variables
Joaquim J. Ju´dice, Ana M. Faustino, Isabel M. Ribeiro and A. Serra
Neves ..........................................................123
viii Contents
On the control of an evolutionary equilibrium in
micromagnetics
Michal Koˇcvara, Martin Kruˇz´ık, Jiˇr´ı V. Outrata .....................143
Complementarity constraints as nonlinear equations: Theory
and numerical experience
Sven Leyffer.....................................................169
A semi-infinite approach to design centering
Oliver Stein .....................................................209
Part III Set-Valued Optimization
Contraction mapping fixed point algorithms for solving
multivalued mixed variational inequalities
Pham Ngoc Anh and Le Dung Muu.................................231
Optimality conditions for a d.c. set-valued problem via the
extremal principle
N. Gadhi .......................................................251
First and second order optimality conditions in set
optimization
V. Kalashnikov, B. Jadamba, A.A. Khan............................265
Preface
Optimization problems involving multivalued mappings in constraints or
as the objective function are investigated intensely in the area of non-
differentiable non-convex optimization. Such problems are well-known under
the names bilevel programming problems [2, 4, 5], mathematical problems
with equilibrium (complementarity) constraints (MPEC) [6, 9], equilibrium
problems with equilibrium constraints (EPEC) [8, 10], set-valued optimiza-
tion problems [3] and so on. Edited volumes on the field are [1, 7]. Since the
publicationofthesevolumestherehasbeenatremendousdevelopmentinthe
field which includes the formulation of optimality conditions using different
kinds of generalized derivatives for set-valued mappings (as e.g. the coderiva-
tive of Mordukhovich), the opening of new applications (as the calibration
of water supply systems), or the elaboration of new solution algorithms (as
for instance smoothing methods). We are grateful to the contributors of this
volumethatthey agreed topublish their newest resultsin thisvolume. These
results reflect most of the recent developments in the field.
The contributions are classified into three parts. Focus in the first part is
on bilevel programming.
Different promising possibilities for the construction of optimality condi-
tions are the topic of the paper “Optimality conditions for bilevel program-
ming problems.” Moreover, the relations between the different problems in-
vestigated in this volume are carefully considered.
Thecomputationofbesttollstobepayedbytheusersofatransportation
network is one important application of bilevel programming. Especially in
relation to recent economic developments in traffic systems this problem has
attractedlargeinterest.Inthepaper“Path-basedformulationsofabileveltoll
settingproblem,”M.Didi-Biha,P.MarcotteandG.Savarddescribedifferent
formulations of this problem and develop efficient solution algorithms.
Applying Mordukhovich’s coderivative to the normal cone mapping for
the feasible set of the lower level problem, J. Dutta and S. Dempe derive
necessary optimality conditions in “Bilevel programming with convex lower
level problems.”
x Preface
Most papers in bilevel programming concentrate on continuous problems
in both levels of hierarchy. D. Fangha¨nel investigates problems with discrete
lowerlevelproblemsinherpaper“Optimalitycriteriaforbilevelprogramming
problems using the radial subdifferential.” Important for her developments
are the notions of the optimistic resp. the pessimistic optimal solution. Both
notions reduce the bilevel problem to the minimization of a discontinuous
function. To develop the optimality conditions she uses the analytic tool of a
radial-directional derivative and the radial subdifferential.
In their paper “On approximate mixed Nash equilibria and average
marginal functions for two-stage three-players games,” L. Mallozzi and J.
Morgan investigate bilevel optimization problems where the two lower level
players react computing a Nash equilibrium depending on the leader’s selec-
tion.OtherthaninD.Fangha¨nel’spaper,theyapplyaregularizationapproach
to the bilevel problem on order to get a continuous auxiliary function.
Thesecondpartcollectspapersinvestigatingmathematicalprogramswith
equilibrium constraints.
Due to violation of (most of) the standard regularity conditions at all
feasible points of mathematical programs with equilibrium constraints using
classical approaches only necessary optimality conditions in form of the Fritz
John conditions can be obtained. This leads to the formulation of weaker
optimalityconditionsasA-,B-,C-andM-stationarity.Thisopensthewayfor
theinvestigationofassumptionsguaranteeingthatoneoftheseconditionscan
beshowntobenecessaryforoptimality.Intheirpaper“AdirectproofforM-
stationarity under MPEC-GCQ for mathematical programs with equilibrium
constraints,” M. L. Flegel and C. Kanzow give an interesting proof of such a
result.
Tosolveandtoinvestigatebilevelprogrammingproblemsormathematical
programs with equilibrium constraints these are usually transformed into a
(nondifferentiable) standard optimization problem. In “On the use of bilevel
programmingforsolvingastructuraloptimizationproblemwithdiscretevari-
ables,” J. J. Ju´dice et al. use the opposite approach to solve an applied large-
dimensional mixed integer programming problem. They transform this prob-
lem into a mathematical program with equilibrium constraints and solve the
latter problem using their complementarity active-set algorithm. This then
results in a promising algorithm for the original problem.
A further applied problem is investigated in the paper “On the control of
an evolutionary equilibrium in micromagnetics” by M. Koˇcvara et al. They
model the problem as an MPEC in infinite dimensions. The problem after
discretization can be solved by applying the implicit programming technique
since the solution of the lower level’s evolutionary inequality in uniquely de-
termined. The generalized differential calculus of B. Mordukhovich is used to
compute the needed subgradients of the composite function.
An important issue of the investigations is the formulations of solution
algorithms. Promising attempts of applying certain algorithms of nonlinear
mathematical programming are quite recent. S. Leyffer investigates in his
Preface xi
paper “Complementarity constraints as nonlinear equations: Theory and nu-
merical experience” one such algorithm. The method is based on an exact
smoothing approach to the complementarity conditions. Comprehensive nu-
merical tests show the power of the method.
Again a challenging applied problem is investigated in the paper “A semi-
infiniteapproachtodesigncentering”byO.Stein.Heformulatedthisproblem
as a general semi-infinite optimization problem. After certain reformulations,
the intrinsic bilevel structure of the problem is detected.
The third part on multivalued set-valued optimization starts with the
paper “Contraction mapping fixed point algorithms for solving multivalued
mixedvariationalinequalities”byP.N.AnhandL.D.Muu.Theyuseafixed-
pointapproachtosolvemultivaluedvariationalinequalitiesandBanach’scon-
traction mapping principle to find the convergence rate of the algorithm.
Mordukhovich’s extremal principle is used in the paper “Optimality con-
ditions for a d.c. set-valued problem via the extremal principle” by N. Gadhi
to derive optimality conditions for a set-valued optimization problemwith an
objective function given as difference of two convex mappings. The necessary
optimality conditions are given in form of set inclusions.
Second order necessary optimality conditions for set-valued optimization
problems are derived in the last paper “First and second order optimality
conditions in set optimization” in this volume by V. Kalashnikov et al. The
maintheoreticaltoolinthispaperisanepiderivativeforset-valuedmappings.
We are very thankful to the referees of the enclosed papers who with
their reports for the papers have essentially contributed to the high scientific
quality of the enclosed papers. We are also thankful to Sebastian Lohse for
carefully reading the text and to numerous colleagues for fruitful and helpful
discussions on the topic during the preparation of this volume. Last but not
leastwethankJohnMartindaleandRobertSaleyfromthepublisherfortheir
continuing support.
Freiberg and Monterrey, Stephan Dempe
February 2006 Vyatcheslav V. Kalashnikov
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xii Preface
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