Set-valued Optimization: An Introduction with Applications

781 Pages·2015·4.344 MB·English

by  Akhtar A. Khan, Christiane Tammer, Constantin Zalinescu

Vector Optimization
Akhtar A. Khan
Christiane Tammer
Constantin Zălinescu
Set-valued
Optimization
An Introduction with Applications
Vector Optimization
SeriesEditor:
JohannesJahn
UniversityofErlangen-Nürnberg
DepartmentofMathematics
Cauerstr.11
81058Erlangen
Germany
[email protected]
Vector Optimization
The series in Vector Optimization contains publications in various fields of opti-
mization with vector-valued objective functions, such as multiobjective optimiza-
tion, 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
arealsoincluded.Theseriescoversmathematicaltheory,methodsandapplications
in economics and engineering. These publications being written in English are
primarily monographs and multiple author works containing current advances in
thesefields.
Moreinformationaboutthisseriesat
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Akhtar A. Khan • Christiane Tammer •
Constantin Za˘linescu
Set-valued Optimization
An Introduction with Applications
123
AkhtarA.Khan ChristianeTammer
RochesterInstituteofTechnologySchool Halle
ofMathematicalSciences Germany
Rochester
NewYork
USA
ConstantinZa˘linescu
University“Al.I.Cuza”Iasi
FacultyofMathematics
Iasi
Romania
ISSN1867-8971 ISSN1867-898X(electronic)
ISBN978-3-642-54264-0 ISBN978-3-642-54265-7(eBook)
DOI10.1007/978-3-642-54265-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014951215
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Withour deepestappreciation,wededicate
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Prof.A.Göpfert

Acknowledgments
WetakethisopportunitytoexpressourmostsinceregratitudetoTruongBao,Mihail
Barbosu, Patricia Clark, Marius Durea, Joydeep Dutta, Rosalind Elster, Franco
Giannessi, Mark Gockenbach, Joachim Gwinner, Alfred Göpfert, Frank Heyde,
Jonas Ide, Johannes Jahn, Kathrin Klamroth, Elisabeth Köbis, Andreas Löhne,
Sophia Maggelakis, Juan Enrique Martìnez-Legaz, Antonino Maugeri, Douglas
Meadows,DumitruMotreanu,BorisMordukhovich,ZuhairNashed, Fabio Raciti,
MiguelSama,AnitaSchöbel,AbulHasanSiddiqi,RaduStrugariu,DougWard,and
PetraWeidnerforthejointworkandfortheirsupport.
WewouldalsoliketothankRosalindElster,AlfredGöpfert,MiguelSama,and
Doug Ward for carefully reading the manuscript and many valuable suggestions.
Moreover,wearegratefultoAndreasLöhneandBrianWinklerfordrawingsome
ofthefiguresforthisbook.
I(AAK)takeanimmensepleasureinexpressingmygratitudeanddeeploveto
my mother Shahida Begum who, since my birth, alone bore all the hardships of
life and made uncountable sacrifices for me. It is also a pleasure to acknowledge
thesupportofmylateuncleandhisfamily.IthankmylovingwifeBascaandmy
adorabledaughtersSophieandAmelieforalltheirloveandsupport.Inaddition,I
wouldliketothankMaryHoeftforherboundlessmotherlylove.
We areextremelythankfultoMr.ChristianRauscherofSpringer-Verlagforhis
limitlesspatienceandunfailingsupportthroughoutthisprocess.
Finally, all the authors are grateful to their coauthors for all their support,
understanding, and hard work over this long and fruitful collaboration. We also
acknowledge that Sects. 2.3, 2.4 (partially), 3.1 and 3.2; Chapters 6, 7, 9, and 10
were written by CZ and the remaining parts were preparedby AAK and CT. The
workofA.A.KhanwaspartiallysupportedbyagrantfromtheSimonsFoundation
vii
viii Acknowledgments
(#210443toAkhtarKhan).TheresearchofC.Za˘linescuwassupportedpartiallyby
thegrantPN-II-ID-PCE-2011-3-0084,CNCS-UEFISCDI,Romania.
Rochester,NY,USA AkhtarA.Khan
Halle,Germany ChristianeTammer
Iasi,Romania ConstantinZa˘linescu
List of Symbols
R Thesetofrealnumbers
R RWDR[f(cid:2)1g[fC1g
R Setofnon-negativerealnumbers
C
.a;b/;(cid:2)a;bŒ;.a;b(cid:2);(cid:2)a;b(cid:2);Œa;b/;Œa;bŒ;Œa;b(cid:2) IntervalsinR
P Setofpositiverealnumbers
N,N(cid:2) Setofnon-negativeintegers,N(cid:2) D
Nnf0g
X;Y;Z;::: Real linear spaces or topological
linearspaces
jj(cid:3)jj,jj(cid:3)jj NorminX,norminX(cid:2)
(cid:2)
Y(cid:3) Y(cid:3) WD Y [ f(cid:2)1g [ fC1g
(extendedspace)
C;K Cones(inY,Z)
CC,C(cid:4) WD(cid:2)CC Positive and negative dual cones
ofC
C# Quasi-interiorofthedualconeCC
BV ,B Open and closed unit balls in the
X X
n.v.s.X
B.k0;"/ Closed ball with the center k0 and
theradius"
FWX (cid:2)Y Set-valuedmapping
fWX !Y(cid:3) Single-valuedmap,function
' ' .y/ WD infft 2 R j y 2 tk0(cid:2)
A;k0 A;k0
Ag(y 2Y)
N .x/ Thesetofneighborhoodsofx 2X
X
inthet.v.s. .X;(cid:3)/
N The set ofbalancedneighborhoods
X
of02X inthet.v.s..X;(cid:3)/
ix