Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials
Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications,...
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| Vydané v: | Optimization letters Ročník 15; číslo 4; s. 1175 - 1194 |
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01.06.2021
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| Abstract | Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications, Springer, Berlin, 2018), we establish strong Karush–Kuhn–Tucker type necessary optimality conditions for the weak efficiency of a nonsmooth nonconvex multiobjective programming problem with set, inequality and equality constraints. We also provide several new definitions for the Mordukhovich-pseudoconvexity and Mordukhovich-quasiconvexity with extended-real-valued functions, and then provide sufficient optimality conditions for weak efficiency to such problem in terms of Mordukhovich subdifferentials. |
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| AbstractList | Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications, Springer, Berlin, 2018), we establish strong Karush–Kuhn–Tucker type necessary optimality conditions for the weak efficiency of a nonsmooth nonconvex multiobjective programming problem with set, inequality and equality constraints. We also provide several new definitions for the Mordukhovich-pseudoconvexity and Mordukhovich-quasiconvexity with extended-real-valued functions, and then provide sufficient optimality conditions for weak efficiency to such problem in terms of Mordukhovich subdifferentials. |
| Author | Van Su, Tran Hien, Nguyen Duc |
| Author_xml | – sequence: 1 givenname: Tran orcidid: 0000-0001-7488-242X surname: Van Su fullname: Van Su, Tran email: vansudhqntt@gmail.com organization: Department of Mathematics, Quang Nam University – sequence: 2 givenname: Nguyen Duc surname: Hien fullname: Hien, Nguyen Duc organization: Faculty of Natural Sciences, Duy Tan University, Institute of Research and Development, Duy Tan University |
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| Keywords | Weak efficiency Nonsmooth multiobjective programming problem with constraints Strong Karush–Kuhn–Tucker optimality conditions Mordukhovich subdifferentials |
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| References_xml | – reference: BaoTQMordukhovichBSNecessary conditions in multiobjective optimization with equilibrium constraintsJ. Optim. Theory Appl.20071352179203234653010.1007/s10957-007-9209-x – reference: SuTVHienNDStudniarski’s derivatives and efficiency conditions for constrained vector equilibrium problems with applicationsOptimization201910.1080/02331934.2019.170298507313461 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation II: Applications2006BerlinSpringer10.1007/3-540-31246-3 – reference: MordukhovichBSVariational Analysis and Applications2018BerlinSpringer10.1007/978-3-319-92775-6 – reference: KhanATammerCZalinescuCSet-Valued Optimization2015BerlinSpringer10.1007/978-3-642-54265-7 – reference: ZhaoKQStrong Kuhn-Tucker optimality in nonsmooth multiobjective optimization problemsPac. J. Optim.20151148349433845521327.90296 – reference: LuuDVNecessary and sufficient conditions for efficiency via convexificatorsJ. Optim. Theory Appl.2014160510526318098110.1007/s10957-013-0377-6 – reference: LuuDVNecessary efficiency conditions for vector equilibrium problems with general inequality constraints via convexificatorsBull. Braz. Math. Soc. New Ser.201950685704399318910.1007/s00574-018-00124-x – reference: LuuDVNecessary conditions for efficiency in terms of the Michel–Penot subdifferentialsOptimization20126110991117296612010.1080/02331934.2010.539688 – reference: ConstantinEFirst-order necessary conditions in locally Lipschitz multiobjective optimizationOptimization20186714471460387795910.1080/02331934.2018.1474880 – reference: JiménezBNovoVA finite dimensional extension of Lyusternik theorem with applications to multiobjective optimizationJ. Math. Anal. Appl.2002270340356191570310.1016/S0022-247X(02)00064-1 – reference: AubinJPFrankowskaHSet-Valued Analysis1990BostonBirkhauser0713.49021 – reference: SuTVHangDDOptimality conditions for the efficient solutions of vector equilibrium problems with constraints in terms of directional derivatives and applicationsBull. Iran. Math. Soc.201945616191650402674410.1007/s41980-019-00219-1 – reference: BaoTQMordukhovichBSNecessary conditions for super minimizers in constrained multiobjective optimizationJ. Global. Optim.2009434533552249607610.1007/s10898-008-9336-4 – reference: JiménezBNovoVFirst order optimality conditions in vector optimization involving stable functionsOptimization2008573449471241207710.1080/02331930601120516 – reference: ChuongTDKimDSNonsmooth semi-infinite multiobjective optimization problemsJ. Optim. Theory Appl.2014160748762318099410.1007/s10957-013-0314-8 – reference: GobernaMALopézMALinear Semi-Infinite Optimization1998ChichesterWiley0909.90257 – reference: BurachikRSRizviMMOn weak and strong KuhnTucker conditions for smooth multiobjective optimizationJ. Optim. Theory Appl.2012155477491299405610.1007/s10957-012-0078-6 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation I: Basic Theory2006BerlinSpringer10.1007/3-540-31246-3 – reference: LiuJJZhaoKQYangXMOptimality and regularity conditions using Mordukhovich’s subdifferentialJ. Nonlinear Convex Anal.20171343531339.90302 – reference: LuuDVSuTVContingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraintsRAIRO - Oper. Res. bf201852543559388054310.1051/ro/2017042 – reference: LuuDVSecond-order necessary efficiency conditions for nonsmooth vector equilibrium problemsJ. Glob. Optim.201870437453376126610.1007/s10898-017-0556-3 – reference: MangasarianOLNonlinear Programming1969New YorkMcGraw-Hill0194.20201 – reference: BaoTQMordukhovichBSRelative Pareto minimizers for multiobjective problems: existence and optimality conditionsMath. Program.20101222301247254633410.1007/s10107-008-0249-2 – reference: Hiriart-UrrutyJBLemaréchalCConvex Analysis and Minimization Algorithms I1993BerlinSpringer10.1007/978-3-662-02796-7 – reference: RockafellarRTConvex Analysis Princeton Mathematical Series1970PrincetonPrinceton University Press – volume-title: Variational Analysis and Generalized Differentiation I: Basic Theory year: 2006 ident: 1620_CR20 doi: 10.1007/3-540-31246-3 – volume-title: Convex Analysis Princeton Mathematical Series year: 1970 ident: 1620_CR23 – volume: 155 start-page: 477 year: 2012 ident: 1620_CR5 publication-title: J. Optim. 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| Title | Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials |
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