Variational Analysis and Generalized Differentiation in Optimization and Control: In Honor of Boris S. Mordukhovich

243 Pages·2010·2.62 MB·English

by  Qamrul Hasan Ansari

Springer Optimization and Its Applications
VOLUME47
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor–CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(PrincetonUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(McMasterUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimization has been expanding in all directions at an astonishing rate
during the last few decades. New algorithmic and theoretical techniques
havebeendeveloped,thediffusioninto otherdisciplineshasproceededata
rapidpace,andourknowledgeofallaspectsofthefieldhasgrownevenmore
profound.At thesametime, oneofthe moststriking trendsin optimization
is the constantly increasing emphasis on the interdisciplinary nature of the
field.Optimizationhasbeenabasictoolinallareasofappliedmathematics,
engineering,medicine,economics,andothersciences.
The series Springer Optimization and Its Applications publishes under-
graduate and graduate textbooks, monographs and state-of-the-art exposi-
tory works that focus on algorithms for solving optimization problems and
alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered
includenonlinearoptimization(convexandnonconvex),networkflowprob-
lems,stochasticoptimization,optimalcontrol,discreteoptimization,multi-
objective programming, description of software packages, approximation
techniquesandheuristicapproaches.
Forfurthervolumes:
http://www.springer.com/series/7393
BorisS.Mordukhovich
VARIATIONAL ANALYSIS AND
GENERALIZED DIFFERENTIATION
IN OPTIMIZATION AND CONTROL
In Honor of Boris S. Mordukhovich
EditedBy
REGINAS.BURACHIK
SchoolofMathematicsandStatistics
UniversityofSouthAustralia,Australia
JEN-CHIHYAO
DepartmentofAppliedMathematics,NationalSunYat-Sen
University,Kaohsiung,Taiwan
ABC
Editors
ReginaS.Burachik Jen-ChihYao
SchoolofMathematicsandStatistics DepartmentofAppliedMathematics
UniversityofSouthAustralia NationalSunYat-SenUniversity
MawsonLakesSA5095 Kaohsiung80424
Australia Taiwan,R.O.C.
[email protected] [email protected]
ISSN1931-6828
ISBN978-1-4419-0436-2 e-ISBN978-1-4419-0437-9
DOI10.1007/978-1-4419-0437-9
SpringerNewYorkDordrechtHeidelbergLondon
MathematicsSubjectClassification(2010):49Jxx,49J53
(cid:2)c SpringerScience+BusinessMedia,LLC2010
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
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toproprietaryrights.
Coverillustration:‘Mordukhovichsubdifferentialsof f ifr=2...’
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Preface
This special volume is dedicated to Boris M. Mordukhovich, on the occasion of
his60thbirthday,andaimstocelebratehisfundamentalcontributionstovariational
analysis,generalizeddifferentiationandtheirapplications.Amainexampleofthese
contributionsis Boris’ recentopusmagnus“VariationalAnalysis and Generalized
Differentiation”(vols.IandII)[2,3].Adetailedexplanationandcarefuldescription
ofBoris’researchandachievementscanbefoundin[1].
Boris’ active work and jovial attitude have constantly inspired researchers of
severalgenerations,withwhomhehasgenerouslysharedhisknowledgeandenthu-
siasm,alongwithhiswell-knownwarmthandhumantouch.
Variationalanalysisisarapidlygrowingfieldwithinpureandappliedmathemat-
ics, with numerous applications to optimization, control theory, economics, engi-
neering,andotherdisciplines.Eachofthe12chaptersofthisvolumeisacarefully
reviewedpaperinthefieldofvariationalanalysisandrelatedtopics.
ManychaptersofthisvolumewerepresentedattheInternationalSymposium
on Variational Analysis and Optimization (ISVAO), held in the Department of
Applied Mathematics, National Sun Yat-sen University, Kaohsiung,Taiwan, from
November28toNovember30,2008.Thesymposiumwasorganizedinhonourof
Boris’60thbirthday.ItbroughttogetherBorisandotherresearcherstodiscussstate-
of-the-artresultsinvariationalanalysisanditsapplications,withemphasisonopti-
mizationandcontrol.We thanktheorganizersandparticipantsofthe symposium,
whomadethesymposiumahighlybeneficialandenjoyableevent.
Wearealsogratefultoalltheauthorsofthisspecialvolume,whohavetakenthe
opportunitytocelebrateBoris’birthdayandhisdecadesofcontributionstothearea.
References
1. R. Henrion, A. Kruger and J. Outrata, Boris Mordukhovich, The Never Tiring Traveller,
Celebrates His Sixtieth Birthday, Set-Valued Analysis, Vol.16, Numbers 2-3, pp.125–127,
2008.
v
vi Preface
2. B. S. Mordukhovich, Variational analysis and generalized differentiation, I: Basic theory,
Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol.330, Springer,
Berlin,2006.
3. B. S. Mordukhovich, Variational analysis and generalized differentiation, II: Applications,
Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol.331, Springer,
Berlin,2006.
GuestEditors:
ReginaS.Burachik
Jen-ChihYao
Contents
Preface............................................................ v
ListofContributors ................................................ ix
SystemsofVectorQuasi-equilibriumProblemsandTheirApplications .. 1
QamrulHasanAnsariandJen-ChihYao
PropertiesofDerivatesandSomeApplications....................... 43
MichaelMcAseyandLibinMou
OnComputingtheMordukhovichSubdifferentialUsingDirectedSets
inTwoDimensions .............................................. 59
RobertBaier,ElzaFarkhi,andVeraRoshchina
FutureChallengesforVariationalAnalysis .......................... 95
JonathanM.Borwein
A Deflected Subgradient Method Using a General Augmented
LagrangianDualitywithImplicationsonPenaltyMethods............. 109
ReginaS.BurachikandC.Yalc¸ınKaya
OnWeakMultifunctionEquilibriumProblems ...................... 133
AdelaCapa˘ta˘,Ga´borKassay,andBogla´rkaMosoni
Optimality Conditionsfor a Simple ConvexBilevel Programming
Problem ....................................................... 149
S.Dempe,N.Dinh,andJ.Dutta
SmoothRepresentationsofOptimalSolutionSetsofPiecewiseLinear
ParametricMultiobjectivePrograms ............................... 163
YaPingFangandXiaoQiYang
vii
viii Contents
GlobalOptimalityConditionsforClassesofNon-convexMulti-objective
QuadraticOptimizationProblems ................................. 177
V.Jeyakumar,G.M.Lee,andG.Li
Application of Variational Analysis and Control Theory to
NonparametricMaximumLikelihoodEstimationofaDensityFunction.. 187
IlyaShvartsman
ParametricVariationalSystemwithaSmooth-Boundary
ConstraintSet .................................................. 205
J.-C.YaoandN.D.Yen
ExactPenaltyin ConstrainedOptimizationandtheMordukhovich
BasicSubdifferential............................................. 223
AlexanderJ.Zaslavski
List of Contributors
QamrulHasanAnsari DepartmentofMathematics,AligarhMuslimUniversity,
Aligarh202002,India,[email protected]
RobertBaier ChairofAppliedMathematics,UniversityofBayreuth,
95440Bayreuth,Germany,[email protected]
JonathanM. Borwein CARMA, Universityof Newcastle, Newcastle, NSW,
Australia,[email protected]
ReginaS.Burachik MathematicsandStatistics,UniversityofSouthAustralia,
MawsonLakesSA5095,Australia,[email protected]
AdelaCapa˘ta˘ DepartmentofMathematics,TechnicalUniversityof
Cluj-Napoca,Str.ConstantinDaicoviciu15,400020Cluj-Napoca,Romania,adela.
[email protected]
S.Dempe DepartmentofMathematicsandComputerScience,TUBergakademie
Freiberg,Freiberg,Germany,[email protected]
N.Dinh DepartmentofMathematics,InternationalUniversity,VietnamNational
UniversityofHoChiMinhCity,HoChiMinhCity,Vietnam,[email protected]
J.Dutta DepartmentofMathematicsandStatistics,IndianInstituteofTechnology,
Kanpur,India,[email protected]
Ya Ping Fang Department of Mathematics, Sichuan University, Chengdu,
Sichuan,China,[email protected]
ElzaFarkhi SchoolofMathematicalSciences,SacklerFacultyofExactSciences,
TelAvivUniversity,69978TelAviv,Israel,[email protected]
V.Jeyakumar DepartmentofAppliedMathematics,UniversityofNewSouth
Wales,Sydney2052,Australia,[email protected]
Ga´borKassay Faculty ofMathematicsand ComputerScience,Babes¸-Bolyai
University,Cluj-Napoca,Romania,[email protected]
ix