Vector Optimization and Monotone Operators via Convex Duality: Recent Advances

282 Pages·2015·1.495 MB·English

by  Sorin-Mihai Grad

Vector Optimization
Sorin-Mihai Grad
Vector Optimization
and Monotone
Operators via
Convex Duality
Recent Advances
Vector Optimization
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UniversityofErlangen-Nürnberg
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Sorin-Mihai Grad
Vector Optimization
and Monotone Operators
via Convex Duality
Recent Advances
123
Sorin-MihaiGrad
FacultyofMathematics
TUChemnitz
Chemnitz
Germany
ISSN1867-8971 ISSN1867-898X(electronic)
ISBN978-3-319-08899-0 ISBN978-3-319-08900-3(eBook)
DOI10.1007/978-3-319-08900-3
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ToCarmen

Preface
Convex analysis plays an increasingly important role in different areas of math-
ematics and its applications, especially in optimization. One of its main tools is
represented by the conjugate functions that, employed in the duality theory, gave
birthtotheconjugatedualityand,inalargersense,toconvexduality.
Theimportanceofthedualityapproachintheoptimizationtheorycomesmainly
from the facts that it provides a lower bound for the objective values of a given
minimization problem and, on the other hand, it leads to necessary and sufficient
optimality conditions. These can be used for determining the optimal solutions
of the primal problem, for instance by employing them in generating different
algorithmicapproachesforsolvingoptimizationproblems.
The step from scalar to vector optimization problems can be made, from the
pointofviewofduality,inseveralways,dependingonthedesiredoutcomes.Inthis
book,severaldualityapproachesforvectoroptimizationproblemsareinvestigated
and,tosomeextent,compared.Moreover,wedealwithdifferentsolutionconcepts
forvectoroptimizationproblems,too.
One can generalize the framework even further, for instance by considering
set-valued multifunctions. Among the problems where they appear, an increasing
interest is attracted by the ones involving monotone operators, especially since
new methods for approaching them by means of convex analysis were developed.
Following this path, we provide in the book several results concerning different
propertiesofthemonotoneoperators,too.
Thisbookstartedasthehabilitationthesisoftheauthor[107],wherehecollected
themajorityofhiscontributionstothefieldsofvectoroptimizationandmonotone
operators during the last decade, most of them obtained by means of duality
theory for convex optimization problems. Several results, remarks, comments and
explanations were added to the original thesis version in order to make the book
morecompleteandreader-friendly.
The book is structured into seven chapters. The first one is dedicated to an
introduction and to several necessary preliminaries meant to make this work as
vii
viii Preface
self-containedaspossible.Inthesecondchapter,wedealwithscalaroptimization
problems. After assigning conjugate dual problems to them, we characterize their
stable"-dualitygapviainclusionsinvolvingepigraphsandsubdifferentials,respec-
tively.Thesecharacterizationsarefurtherusedtorediscover,incase"D0andwhen
theinvolvedfunctionsareendowedwithconvexityandtopologicalproperties,some
recent closedness type regularity conditions from the literature and the formulae
theyguarantee.
Withinthethirdchapter,differentminimalityconceptsforsetsareconsideredand
investigated. Moreover, the corresponding minimality sets are compared. Several
of these minimality concepts are presented in more general frameworks than
consideredsofarintheliteratureanditisshownthatsomeoftheirpropertiesremain
validinthenewsetting,too,insomecasesunderadditionalweakhypotheses.These
minimalitynotionsleadtosolutionconceptsforvectoroptimizationproblems,the
so-calledefficiencyconcepts.
A vector duality approach via a general scalarization for vector optimization
problems is introduced in the fourth chapter. Different scalarization functions
consideredintheliteraturearethenemployedasspecialcasesofthegeneralscalar-
ization, leading to various vector dual problems for a general vector optimization
problem. The latter contains as special cases the classes of both constrained and
unconstrained vector optimization problems and vector duals and corresponding
dualitystatementsarederivedineachcase.
Extending the classical Wolfe and Mond-Weir duality concepts for both scalar
and vector general nondifferentiable optimization problems is the aim of the fifth
chapter. Considered so far in the literature mainly for optimization problems
involvingdifferentiablefunctions,boththesedualityconceptsleadtodualproblems
thatcontainthecorrespondingoptimalityconditionsasconstraints.
In the sixth chapter, new vector dual problems are assigned to the classical
linear vector optimization problem in finitely dimensional spaces with respect to
bothefficientandweaklyefficientsolutionsandthentheyareextendedtoinfinitely
dimensionalspaces,too.Itisshownthatthesenewvectordualsencounternotrouble
in some special cases where some of their older counterparts from the literature
failedtoachievestrongduality.Comparisonsbetweentheknownvectordualsand
thenewonesaredelivered,too.Moreover,weproposeanewvectordualproblem
toasemidefinitevectoroptimizationproblems,too.
Thelastchapterpresentsresultsinvolvingmonotoneoperatorsobtainedmainly
bymeansofconvexanalysis,viaconjugatefunctionsanddualityformulae.Firstwe
deal with Brézis-Haraux type approximations for the range of a sum of monotone
operators,thenweprovidecharacterizations involvingrepresentativefunctionsfor
thesurjectivityofasumofmonotoneoperators,fromwhichclosednesstyperegu-
larityconditionsforvariousinterestingspecialcasescanbederived.Wealsoshow
how approaching the maximal monotonicity of the bifunctions via representative
functions leads to generalizing known results and to positively answering to some
recentlyposedconjectures.
I would like to express my thanks to Gert Wanka for his continuous support
and for giving me the opportunity first to study and then to do my research at
Preface ix
Chemnitz University of Technology. I am very grateful to my mentor and friend
Radu Ioan Bo¸t for his guidance and many useful discussions during all the years
of working together and for his advices regarding this book. I would also like
to thank Ernö Robert Csetnek for his valuable comments concerning some of
the results presented here and for carefully reading a preliminary version of the
manuscript.Thanksarealsoduetotheformerandcurrentmembersoftheresearch
group Approximation Theory within the Faculty of Mathematics of the Chemnitz
University of Technology for discussions and remarks regarding different parts of
thisbookandforcontributingtothefriendlyandinspiringworkingatmospherewe
allenjoyedandbenefitedfrom.Iamgratefultomycoauthorsandtotheanonymous
reviewersofthepapersintegratedwithinthisbookwhocontributedwithcomments
andremarkstotheirimprovement,aswellastothefellowmathematicians forthe
valuablefeed-backprovidedatdifferentconferenceswhereIpresentedpartsofthis
manuscript. The financial support from DFG (German Research Foundation) via
projects WA 922/1-3 and WA 922/8-1 that made possible some of the research
presented in this book is thankfully acknowledged, too. Last but not least, I am
grateful to my family and especially to my better half Carmen for unconditioned
love,patienceandsupport.
Forupdatesanderratathereaderisreferredtotheauthor’swebsite
http://www.tu-chemnitz.de/~gsor
Chemnitz,Germany Sorin-MihaiGrad
May2014