Recent Developments in Vector Optimization

576 Pages·2012·4.7 MB·English

by  Gabriele Eichfelder

Vector Optimization
SeriesEditor:
JohannesJahn
UniversityofErlangen-Nu¨rnberg
DepartmentofMathematics
Martensstr.3
81058Erlangen
Germany
[email protected]
Forfurthervolumes:
http://www.springer.com/series/8175
Vector Optimization
The series in Vector Optimization contains publications in various fields of
optimization with vector-valued objective functions, such as multiobjective
optimization, multi criteria decision making, set optimization, vector-valued
game theory and border areas to financial mathematics, biosystems, semidefinite
programming and multiobjective control theory. Studies of continuous, discrete,
combinatorial and stochastic multiobjective models in interesting fields of
operations research are also included. The series covers mathematical theory,
methodsandapplicationsineconomicsandengineering.Thesepublicationsbeing
writteninEnglishareprimarilymonographsandmultipleauthorworkscontaining
currentadvancesinthesefields.
Qamrul Hasan Ansari
Jen-Chih Yao
Editors
Recent Developments
in Vector Optimization
123
Editors
QamrulHasanAnsari Jen-ChihYao
DepartmentofMathematics CenterforGeneralEducation
AligarhMuslimUniversity KaohsiungMedicalUniversity
Aligarh202002,India Kaohsiung80708,Taiwan,ROC
[email protected] [email protected]
ISSN1867-8971 e-ISSN1867-898X
ISBN978-3-642-21113-3 e-ISBN978-3-642-21114-0
DOI10.1007/978-3-642-21114-0
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011936023
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Thisvolumeisdedicatedto Prof.Franco
Giannessiforhiscontributiontothetheory
ofvector variationalinequalitiesand
optimization.Wewouldalsoliketodedicate
thisvolumeto ourfamilies.
•
Preface
We always come cross several decision-making problems in our daily life. Such
problems are always conflicting in which many different view points should be
satisfied. In politics, business, industrial systems, managementscience, networks,
etc. oneoftenencounterssuchkindofproblems.The mostimportantanddifficult
part in such problems is the conflict between various objectives and goals. In
these problems, one has to find the minimum (or maximum)for several objective
functions. Such problems are called vector optimization problems (VOP), multi-
criteriaoptimizationproblemsormulti-objectiveoptimizationproblems.
This volume deals with several different topics/aspects of vector optimization
theoryrangingfromtheverybeginningtothemostrecentone.Itcontainsfourteen
chapterswrittenbydifferentexpertsinthefieldofvectoroptimization.
Chapter 1 deals with the solution concepts in vector optimization and set
optimization, existence results and applications. The concepts of a pre-order and
a partialorderof a setwhichnaturallyinducethenotionofminimalandmaximal
elements for vector and set optimization problems are presented. Some different
optimalitynotionssuchasminimal,weaklyminimal,stronglyminimalandproperly
minimal elements in a pre-ordered linear space are presented and the relations
amongthesenotionsarediscussed.Twodifferentapproaches–thevectorapproach
and the set approach – for defining optimal solutions of set optimization are
presented.Severaloptimalityconceptsforsetoptimizationaswellasorderrelations
forsetsarediscussed.Therelevanceofvectoroptimizationinpracticeisillustrated
withanapplicationtoabicriterialprobleminfielddesignofamagneticresonance
system.
Chapter 2 is devoted to the existence of Lagrange multipliers, (strong) duality
results, linear scalarizations of various classes of solutions to vector optimization
problemsby using theorems of the alternative. The chapter starts by recalling the
Fan–Glicksberg–Hoffmanalternative theorem (1957) for convex functions. Then,
manyequivalentformulationstoageneralGordan-typealternativetheoremvalidfor
(notnecessarilypointed)convexconeswithpossiblyemptyinterior,areestablished.
Theyareexpressedintermsofquasirelativeinterior.Severalclassesofgeneralized
vii
viii Preface
convexity for sets and for vector-valued mappings, are revisited. Applications to
linearcharacterizationsofweaklyefficient,(Benson)properefficientsolutions,and
tocharacterizetheFritz-Johntypeoptimalityconditionin vectoroptimization,are
discussed.Finally,somerecentdevelopmentsaboutproperefficiencyarepresented.
In Chap.3, the generalized notions of infimum and supremum, called infimal
and supremal sets, are studied. It is shown that a set-valued approach to vector
optimization yields a complete lattice such that the infimum and supremum can
be expressed as certain infimal and supremal sets, respectively. This approach is
applied to establish duality results based on a consequent usage of infimum and
supremum. Exemplary, Lagrange and conjugate duality results, being completely
analogoustotheirscalarcounterparts,areproven.
In Chap.4, the vectoroptimization problemswith a variableorderingstructure
are studied. A cone-valued map which associates to each element of the space a
cone of dominated (or preferred) directions defines thereby the variable ordering
structure. The importance of such vector optimization problems is illustrated
with several applicationsfor instance in medical image registration. The different
optimalitynotionswhichareknownsofarintheliteraturearegivenandproperties
and relations among them are discussed. The linear and nonlinear scalarization
functionals which allow a characterization of optimal elements are presented and
a basicdualitytheoryisprovided.Specialattentiontovariableorderingstructures
wheretheimagesofthecone-valuedmapareBishop-Phelpsconesispaid.
In Chap.5, some interesting results on optimality conditions for vector opti-
mizationproblemsarepresented.TheVOPbothinfiniteandinfinitedimensionsis
studied.Forexample,thestudyofstrongKKTconditionsisemphasizedandinthe
infinitedimensionalsettingtheoptimalityconditionswhentheorderingconehasan
emptyinterior,isfocused.Ascalar-valuedgapfunctionanditsregularizationisalso
studied and an error boundis presented for a strongly convexvector optimization
problem. Some results on second order sufficient optimality conditionsfor vector
optimizationproblemsarepresented.
Chapter 6 deals with the optimality conditions, which are expressed by means
of vector variational inequalities where the operator is the Gaˆteaux derivative of
the objective function. The image space analysis for VOP with cone constraints
of the form g(x) ∈ D, where D is a closed and convex cone is presented. In
particular, following a vector separation scheme in the image space, generalized
vector Lagrangian functions associated with VOP are introduced, and scalar and
vector saddle points conditions are derived. The Kuhn–Tucker type first order
optimalityconditionsforGaˆteauxdifferentiableVOPwithconeconstraintsandfor
semidifferentiableVOP (in the sense of Giannessi) with inequalityconstraintsare
discussed.
InChap.7,some elementaryconceptsfromnonlinearanalysis,convexanalysis
and invex analysis are reviewed. The directional derivatives, Gaˆteaux derivative,
Dini (lower and upper) directional derivative, Dini-Hadamard (lower and upper)
directional derivative, Clarke directional derivative and their properties are pre-
sented. By treating these directionalderivativesas a bifunction,differentkindsof
invexitiesandinvariantmonotonicitiesforsuchabifunctionareintroduced.Several
Preface ix
propertiesoftheseinvexitiesandinvariantmonotonicitiesarediscussed.Thevector
variational-likeinequalities for bifunctionsare presented in such a way that if we
treattheDiniupperdirectionalderivativeofafunctionasabifunction,thenweget,
so called, the nonsmooth vector variational-like inequalities involving Dini upper
directionalderivative.Someexistenceresultsforthese kindsofvectorvariational-
likeinequalitiesarepresented.Thevectoroptimizationproblembyusingourvector
variational-like inequalities is studied. Several relationships among the weakly
efficientsolutionandefficientsolutionoftheVOP,andthesolutionsofourvector
variational-likeinequalitiesaregiven.
Chapter 8 is concerned with a convex vector optimization problem which is
to minimize a vector objective function consisting of a finite number of real-
valued convex functions with infinitely many convex constraint functions, and an
abstract constraint set (called convex semi-infinite vector problem). For such a
problem,twokindsofapproximateefficientsolutions,namely,ε-efficientsolutions
and weakly ε-efficient solutions are considered. Optimality conditions for these
kinds of approximate solutions are established in terms of Fenchel conjugates
andε-subdifferentialsof the datainvolved.Optimalityconditionsforefficientand
weakly efficient solutions of such a problem are derived as a special case of the
general results. These results are established by using a new version of Farkas
lemmaforsystemsofinfinitelymanyconvexinequalitiesandundernewregularity
conditionswhichgiverisetonewresultsevenforproblemswithfiniteconstraints.
Severalnumericalexamplesaregiventoillustratethemeaningoftheresults.
Chapter9surveyssomeexistingresultsonsolutionstabilityandconnectedness
ofthesolutionsetsoflinearfractionalvectoroptimizationproblemsandofconvex
quadraticvectoroptimizationproblems.Themainconcernisthesituationwherethe
constraintsetisunbounded.Someopenproblemsarealsomentioned.
In Chap.10, various types of Levitin–Polyak (LP, in short) well-posedness for
scalarandvectoroptimizationproblemswithfunctionalconstraintsareintroduced.
Various criteria and characterizations for these types of LP well-posedness are
derived. Relationships among these types of LP well-posedness are presented.
Applications of some types of LP well-posedness to the convergence analysis of
augmented Lagrangian methods and penalty methods for constrained scalar or
vectoroptimizationproblemsarealsogiven.
In Chap.11, new minimal point theorems in product spaces and the corre-
sponding vector variational principles for set-valued functions are presented. As
special cases, many of the existing variational principles of Ekeland’s type are
derived.Moreover,anewapproachisusedtogetextensionsofEkeland’svariational
principles(EVPs)ofIsac-Tammer’sandHa’stypes,aswellasextensionsofEVPs
for bi-functions. An importanttool for deriving variational principles is a general
nonlinearscalarizationtechnique.Someusefulpropertiesofscalarizingfunctionals
is studied. Several applications, especially necessary conditions for solutions of
VOParepresented.
InChap.12,severalversionsoftheFermatruleandtheLagrangemultiplierrule
for various efficient solutions of set-valued optimization problems are presented.
Theserulesareexpressedin termsofcoderivativesin thesenses ofFre´chet,Ioffe,