Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficien...

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Published in:	Journal of global optimization Vol. 51; no. 4; pp. 677 - 688
Main Authors:	Chuong, T. D., Kruger, A. Y., Yao, J.-C.
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.12.2011 Springer Nature B.V
Subjects:	Banach spaces Computer Science Mapping Mathematical programming Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Studies 90C31 49K40 49J52 Coderivative Efficient solution map Implicit multifunction 90C29 Calmness Parametric vector optimization
ISSN:	0925-5001, 1573-2916
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Abstract	The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.
AbstractList	The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.[PUBLICATION ABSTRACT] The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.
Author	Chuong, T. D. Kruger, A. Y. Yao, J.-C.
Author_xml	– sequence: 1 givenname: T. D. surname: Chuong fullname: Chuong, T. D. organization: Department of Mathematics, Dong Thap University – sequence: 2 givenname: A. Y. surname: Kruger fullname: Kruger, A. Y. organization: Centre for Informatics and Applied Optimization, Graduate School of Information Technology & Mathematical Sciences, University of Ballarat – sequence: 3 givenname: J.-C. surname: Yao fullname: Yao, J.-C. email: yaojc@kmu.edu.tw organization: Center for General Education, Kaohsiung Medical University
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Cites_doi	10.1016/j.na.2009.06.038 10.1016/j.ejor.2009.01.004 10.1007/BFb0120850 10.1007/s10107-005-0623-2 10.1007/978-3-642-02431-3 10.1515/9783110883237.2371 10.1023/A:1008775413250 10.1137/050648079 10.1137/S1052623401386071 10.11650/twjm/1500405272 10.1016/j.na.2005.08.029 10.1007/s10957-007-9214-0 10.1007/s11228-008-0076-x 10.1137/080714129 10.1023/A:1023673105317 10.1137/S1052623401395553 10.1016/0022-247X(74)90025-0 10.1016/j.jmaa.2007.05.001 10.1007/s10898-008-9391-x 10.1007/s10479-007-0186-0
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References_xml	– reference: BednarczukE.M.Upper Hölder continuity of minimal pointsJ. Convex Anal.200292327338 – reference: LedyaevY.S.ZhuQ.J.Implicit multifunctions theoremsSet-Valued Anal.1999720923810.1023/A:1008775413250 – reference: ChinchuluunA.PardalosP.M.A survey of recent developments in multiobjective optimizationAnn. Oper. Res.2007154295010.1007/s10479-007-0186-0 – reference: ChuongT.D.YaoJ.-C.YenN.D.Further results on the lower semicontinuity of efficient point multifunctionsPacific J. Optim.20106405422 – reference: PhelpsR.R.Convex Functions, Monotone Operators and Differentiability, 2nd edn. Lecture Notes in Mathematics, vol. 13641993BerlinSpringer – reference: ChuongT.D.HuyN.Q.YaoJ.-C.Pseudo-Lipschitz property of linear semi-infinite vector optimization problemsEur. J. Oper. Res.201020063964410.1016/j.ejor.2009.01.004 – reference: LeeG.M.TamN.N.YenN.D.Normal coderivative for multifunctions and implicit function theoremsJ. Math. Anal. Appl.2008338112210.1016/j.jmaa.2007.05.001 – reference: XiangS.W.YinW.S.Stability results for efficient solutions of vector optimization problemsJ. Optim. Theory Appl.200713438539810.1007/s10957-007-9214-0 – reference: SawaragiY.NakayamaH.TaninoT.Theory of Multiobjective Optimization. Mathematics in Science and Engineering, vol. 1761985Orlando, FLAcademic Press – reference: ZhengX.Y.NgK.F.Metric subregularity and constraint qualifications for convex generalized equations in Banach spacesSIAM J. Optim.200718243746010.1137/050648079 – reference: IoffeA.D.OutrataJ.On metric and calmness qualification conditions in subdifferential calculusSet-Valued Anal.2008162–319922710.1007/s11228-008-0076-x – reference: LucD.T.Theory of Vector Optimization. Lecture Notes in Economics, Math Syst 3191989BerlinSpringer – reference: ZhengX.Y.NgK.F.Calmness for L-subsmooth multifunctions in Banach spacesSIAM J. Optim.20091941648167310.1137/080714129 – reference: Bednarczuk, E.M.: Some stability results for vector optimization problems in partially ordered topological vector. In: Proceedings of First World Congress Nonlinear Analysis (vol. III, pp. 2371–2382). Tampa, FL (1996) – reference: MordukhovichB.S.Variational Analysis and Generalized Differentiation. I: Basic Theory2006BerlinSpringer – reference: GopfertA.RiahiH.TammerC.ZălinescuC.Variational Methods in Partially Ordered Spaces2003New YorkSpringer – reference: HenrionR.JouraniA.Subdifferential conditions for calmness of convex constraintsSIAM J. Optim.20021352053410.1137/S1052623401386071 – reference: HuyN.Q.YaoJ.-C.Stability of implicit multifunctions in Asplund spacesTaiwan. J. Math.2009134765 – reference: KlatteD.KummerB.Nonsmooth Equations in Optimization2002Calculus, Methods and Applications. Kluwer Academic Publisher, Dordrecht-Boston-LondonRegularity – reference: XiangS.W.ZhouY.H.Continuity properties of solutions of vector optimizationNonlinear Anal.2006642496250610.1016/j.na.2005.08.029 – reference: BednarczukE.M.Hölder-like properties of minimal points in vector optimizationControl Cybernet.2002313423438 – reference: FloudasC.A.PardalosP.M.Encyclopedia of Optimization2009BerlinSpringer – reference: HenrionR.JouraniA.OutrataJ.On the calmness of a class of multifunctionsSIAM J. Optim.20021360361810.1137/S1052623401395553 – reference: HenrionR.OutrataJ.Calmness of constraint systems with applicationsMath. Program.200510443746410.1007/s10107-005-0623-2 – reference: KrugerA.Y.On Fréchet subdifferentialsJ. Math. Sci. (N.Y.)200311633325335810.1023/A:1023673105317 – reference: RockafellarR.T.WetsR.J.-B.Variational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 – reference: RobinsonS.M.Generalized equations and their solutions, Part I: Basic theoryMath. Program. 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Snippet	The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of...
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SubjectTerms	Banach spaces Computer Science Mapping Mathematical programming Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Studies
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Title	Calmness of efficient solution maps in parametric vector optimization
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