Robust nonsmooth optimality conditions for uncertain multiobjective programs involving stable functions

In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints (( UCMOP ), for brevity). First, we introduce the constraint qualification of the ( GRSCQ ) type and then establish some robust necessa...

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Veröffentlicht in:	Positivity : an international journal devoted to the theory and applications of positivity in analysis Jg. 28; H. 4; S. 60
1. Verfasser:	Van Su, Tran
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.09.2024 Springer Nature B.V
Schlagworte:	Applied mathematics Calculus of Variations and Optimal Control; Optimization Constraints Convexity Econometrics Euclidean space Fourier Analysis Mathematical programming Mathematics Mathematics and Statistics Multiple objective analysis Operator Theory Optimization Potential Theory Robustness Uncertainty analysis Robust duality theorems Robust efficient solutions Robust nonsmooth optimality conditions Uncertain nonsmooth multiobjective programming 90C46 90C25 65K10 90C29 Wolfe and Mond–Weir types dual problem
ISSN:	1385-1292, 1572-9281
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Abstract	In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints (( UCMOP ), for brevity). First, we introduce the constraint qualification of the ( GRSCQ ) type and then establish some robust necessary optimality conditions in terms of the generalized subdifferentials for some types of minima (including robust weakly efficient and robust properly efficient) to such problem involving stable functions. Under suitable assumptions on the pseudo-convexity of objective and constraint functions, robust necessary nonsmooth optimality conditions become robust sufficient optimality conditions. An application of the obtained results for its Wolfe and Mond–Weir types dual problem is presented and some illustrative examples are also provided for our findings.
AbstractList	In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints ((UCMOP), for brevity). First, we introduce the constraint qualification of the (GRSCQ) type and then establish some robust necessary optimality conditions in terms of the generalized subdifferentials for some types of minima (including robust weakly efficient and robust properly efficient) to such problem involving stable functions. Under suitable assumptions on the pseudo-convexity of objective and constraint functions, robust necessary nonsmooth optimality conditions become robust sufficient optimality conditions. An application of the obtained results for its Wolfe and Mond–Weir types dual problem is presented and some illustrative examples are also provided for our findings. In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints (( UCMOP ), for brevity). First, we introduce the constraint qualification of the ( GRSCQ ) type and then establish some robust necessary optimality conditions in terms of the generalized subdifferentials for some types of minima (including robust weakly efficient and robust properly efficient) to such problem involving stable functions. Under suitable assumptions on the pseudo-convexity of objective and constraint functions, robust necessary nonsmooth optimality conditions become robust sufficient optimality conditions. An application of the obtained results for its Wolfe and Mond–Weir types dual problem is presented and some illustrative examples are also provided for our findings.
ArticleNumber	60
Author	Van Su, Tran
Author_xml	– sequence: 1 givenname: Tran surname: Van Su fullname: Van Su, Tran email: tvsu@ued.udn.vn organization: Faculty of Mathematics, The University of Danang - University of Science and Education
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Cites_doi	10.1007/s10957-018-1437-8 10.1016/j.ejor.2013.10.028 10.1007/BF01594928 10.1080/02331939208843804 10.1007/s10898-009-9522-z 10.1007/978-3-642-50280-4 10.1016/j.jmaa.2007.01.110 10.1080/02331934.2022.2031189 10.1016/j.na.2011.04.006 10.1080/01630563.2016.1155158 10.1090/qam/135625 10.1515/9781400873173 10.1016/j.na.2016.01.002 10.1137/100791841 10.1016/j.ejor.2016.12.045 10.1007/s10957-014-0564-0 10.1007/s10479-016-2363-5 10.1080/02331930601120516 10.1016/j.jmaa.2006.04.060 10.1007/s00158-004-0450-8 10.1007/s00186-014-0471-z 10.1080/02331934.2013.769104 10.1016/j.ejor.2017.04.012 10.1080/02331934.2018.1442448 10.1080/02331934.2022.2038154 10.1007/s10107-006-0092-2 10.1080/02331934.2020.1836636 10.4134/CKMS.2013.28.3.597 10.1080/02331934.2022.2046740
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Keywords	Robust duality theorems Robust efficient solutions Robust nonsmooth optimality conditions Uncertain nonsmooth multiobjective programming 90C46 90C25 65K10 90C29 Wolfe and Mond–Weir types dual problem
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References	IdeJKobisEConcepts of efficiency for uncertain multiobjective problems based on set order relationsMath. Meth. Oper. Res.2014809912710.1007/s00186-014-0471-z RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/9781400873173 KlamrothKKobisESchobelATammerCA unified approach for different concepts of robustness and stochastic programming via nonlinear scalarizing functionalsOptimization2013625649671306096810.1080/02331934.2013.769104 Rodríguez - Marín, L., Sama, M.: Variational characterization of the contingent epiderivative: J. Math. Anal. Appl. 335, 1374–1382 (2007) BokrantzRFredrikssonANecessary and sufficient conditions for Pareto efficiency in robust multiobjective optimizationEur. J. Oper. Res.2017262682692365476810.1016/j.ejor.2017.04.012 GunawanSAzarmSMultiobjective robust optimization using a sensitivity region conceptStruct. Multidiscrip. Optimization.200529506010.1007/s00158-004-0450-8 WangJLiS-JChenC-RRobust nonsmooth optimality conditions for multiobjective optimization problems with infinitely many uncertain constraintsOptimization202372820392067462046810.1080/02331934.2022.2046740 AubinJ-PFrankowskaHSet-Valued Analysis1990BostonBirkhauser ThuyNTTSuTVRobust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain dataOptimization202372717451775460368710.1080/02331934.2022.2038154 ClasonCKhanAASamaATammerCContingent derivatives and regularization for noncoercive inverse problemsOptimization201968713371364398520110.1080/02331934.2018.1442448 LucDTContingent derivatives of set-valued maps and applications to vector optimizationMath. Program.19915099111109885010.1007/BF01594928 LeeGMLeeJHOn nonsmooth optimality theorems for robust multiobjective optimization problemsJ. Nonlinear Convex Anal.201516203920523422635 JiménezBNovoVFirst order optimality conditions in vector optimization involving stable functionsOptimization2008573449471241207710.1080/02331930601120516 WolfePA duality theorem for nonlinear programmingQ. J. Appl. Math.19611923924410.1090/qam/135625 WeiH-ZChenC-RLiS-JNecessary optimality conditions for nonsmooth robust optimization problemsOptimization202271718171837444824410.1080/02331934.2020.1836636 DuyTQRobust efficiency and well-posedness in uncertain vector optimization problemsOptimization202374937955455923610.1080/02331934.2022.2031189 Rodríguez - Marín, L., Sama, M.: About Contingent epiderivatives: J. Math. Anal. Appl. 327, 745–762 (2007) GopfertARiahiHTammerCZalinescuCVariational Methods in Partially Ordered Spaces2003New YorkSpringer MondMWeirTGenerallized Concavity and Duality. Generallized Concavity in Optimization and Economics1981New YorkAcademic Press ChenJKobisEYaoJ-COptimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraintsJ. Optim. Theory Appl.2019181411436393847510.1007/s10957-018-1437-8 SuTVOptimality conditions for vector equilibrium problems in terms of contingent epiderivativesNumer. Funct. Anal. Optim.201637640665351452210.1080/01630563.2016.1155158 KlamrothKKobisESchobelATammerCA unified approach to uncertain optimizationEur. J. Oper. Res.2017260403420361500010.1016/j.ejor.2016.12.045 FliegeJWernerRRobust multiobjective optimization & applications in portfolio optimizationEur. J. Oper. Res.2014234422433314473110.1016/j.ejor.2013.10.028 JeyakumarVLiGStrong duality in robust convex programming: complete characterizationsSIAM J. Optim.20102033843407276350910.1137/100791841 GiorgiGGuerraggioAOn the notion of tangent cone in mathematical programmingOptimization1992251123123531410.1080/02331939208843804 SunejaSKKhuranaSBhatiaMOptimality and duality in vector optimization involving generalized type I functions over conesJ. Global Optim.2011492335274051210.1007/s10898-009-9522-z Ben-TalANemirovskiAA selected topic in robust convex optimizationMath. Programm. Ser. B.2008112125158232700410.1007/s10107-006-0092-2 Luc, D.T.: Theory of vector optimization. Lecture Notes in Economics and Mathematical Systems, Vol. 39. Springer, Berlin (1989). https://doi.org/10.1007/978-3-642-50280-4 Ben-TalAGhaouiLENemirovskiARobust Optimization. Princeton Series in Applied Mathematics2009PrincetonPrinceton University Press JeyakumarVLeeGMLiGCharacterizing robust solutions sets convex programs under data uncertaintyJ. Optim. Theory Appl.201564407435329797010.1007/s10957-014-0564-0 KimMHDuality theorem and vector saddle point theorem for robust multiobjective optimization problemsCommun. Korean Math. Soc.201328597602308560810.4134/CKMS.2013.28.3.597 LeeJHLeeGMOn optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problemsAnn. Oper. Res.2018269419438384848810.1007/s10479-016-2363-5 ClarkeFHOptimization and Nonsmooth Analysis1983New YorkViley-Interscience ChuongTDOptimality and duality for robust multiobjective optimization problemsNonlinear Anal.2016134127143346262110.1016/j.na.2016.01.002 JeyakumarVLiGLeeGMRobust duality for generalized convex programming problems under data uncertaintyNonlinear Anal.20127513621373286134110.1016/j.na.2011.04.006 RT Rockafellar (1077_CR29) 1970 A Gopfert (1077_CR16) 2003 G Giorgi (1077_CR17) 1992; 25 H-Z Wei (1077_CR34) 2022; 71 A Ben-Tal (1077_CR3) 2008; 112 V Jeyakumar (1077_CR13) 2015; 64 NTT Thuy (1077_CR32) 2023; 72 V Jeyakumar (1077_CR15) 2012; 75 S Gunawan (1077_CR18) 2005; 29 TV Su (1077_CR30) 2016; 37 1077_CR28 1077_CR27 J Chen (1077_CR7) 2019; 181 1077_CR24 M Mond (1077_CR26) 1981 TD Chuong (1077_CR8) 2016; 134 K Klamroth (1077_CR19) 2017; 260 C Clason (1077_CR6) 2019; 68 MH Kim (1077_CR21) 2013; 28 P Wolfe (1077_CR35) 1961; 19 J Ide (1077_CR11) 2014; 80 J Wang (1077_CR33) 2023; 72 B Jiménez (1077_CR12) 2008; 57 V Jeyakumar (1077_CR14) 2010; 20 TQ Duy (1077_CR9) 2023; 7 J Fliege (1077_CR10) 2014; 234 A Ben-Tal (1077_CR2) 2009 JH Lee (1077_CR22) 2018; 269 SK Suneja (1077_CR31) 2011; 49 J-P Aubin (1077_CR1) 1990 FH Clarke (1077_CR5) 1983 GM Lee (1077_CR23) 2015; 16 K Klamroth (1077_CR20) 2013; 62 R Bokrantz (1077_CR4) 2017; 262 DT Luc (1077_CR25) 1991; 50
References_xml	– reference: KlamrothKKobisESchobelATammerCA unified approach for different concepts of robustness and stochastic programming via nonlinear scalarizing functionalsOptimization2013625649671306096810.1080/02331934.2013.769104 – reference: ThuyNTTSuTVRobust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain dataOptimization202372717451775460368710.1080/02331934.2022.2038154 – reference: ClasonCKhanAASamaATammerCContingent derivatives and regularization for noncoercive inverse problemsOptimization201968713371364398520110.1080/02331934.2018.1442448 – reference: GopfertARiahiHTammerCZalinescuCVariational Methods in Partially Ordered Spaces2003New YorkSpringer – reference: RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/9781400873173 – reference: WeiH-ZChenC-RLiS-JNecessary optimality conditions for nonsmooth robust optimization problemsOptimization202271718171837444824410.1080/02331934.2020.1836636 – reference: KlamrothKKobisESchobelATammerCA unified approach to uncertain optimizationEur. J. Oper. Res.2017260403420361500010.1016/j.ejor.2016.12.045 – reference: ChenJKobisEYaoJ-COptimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraintsJ. Optim. Theory Appl.2019181411436393847510.1007/s10957-018-1437-8 – reference: IdeJKobisEConcepts of efficiency for uncertain multiobjective problems based on set order relationsMath. Meth. Oper. Res.2014809912710.1007/s00186-014-0471-z – reference: MondMWeirTGenerallized Concavity and Duality. Generallized Concavity in Optimization and Economics1981New YorkAcademic Press – reference: JeyakumarVLiGLeeGMRobust duality for generalized convex programming problems under data uncertaintyNonlinear Anal.20127513621373286134110.1016/j.na.2011.04.006 – reference: BokrantzRFredrikssonANecessary and sufficient conditions for Pareto efficiency in robust multiobjective optimizationEur. J. Oper. Res.2017262682692365476810.1016/j.ejor.2017.04.012 – reference: ChuongTDOptimality and duality for robust multiobjective optimization problemsNonlinear Anal.2016134127143346262110.1016/j.na.2016.01.002 – reference: KimMHDuality theorem and vector saddle point theorem for robust multiobjective optimization problemsCommun. Korean Math. Soc.201328597602308560810.4134/CKMS.2013.28.3.597 – reference: WolfePA duality theorem for nonlinear programmingQ. J. Appl. Math.19611923924410.1090/qam/135625 – reference: GiorgiGGuerraggioAOn the notion of tangent cone in mathematical programmingOptimization1992251123123531410.1080/02331939208843804 – reference: JeyakumarVLeeGMLiGCharacterizing robust solutions sets convex programs under data uncertaintyJ. Optim. Theory Appl.201564407435329797010.1007/s10957-014-0564-0 – reference: JeyakumarVLiGStrong duality in robust convex programming: complete characterizationsSIAM J. Optim.20102033843407276350910.1137/100791841 – reference: LucDTContingent derivatives of set-valued maps and applications to vector optimizationMath. Program.19915099111109885010.1007/BF01594928 – reference: GunawanSAzarmSMultiobjective robust optimization using a sensitivity region conceptStruct. Multidiscrip. Optimization.200529506010.1007/s00158-004-0450-8 – reference: FliegeJWernerRRobust multiobjective optimization & applications in portfolio optimizationEur. J. Oper. 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Snippet	In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with...
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SubjectTerms	Applied mathematics Calculus of Variations and Optimal Control; Optimization Constraints Convexity Econometrics Euclidean space Fourier Analysis Mathematical programming Mathematics Mathematics and Statistics Multiple objective analysis Operator Theory Optimization Potential Theory Robustness Uncertainty analysis
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Title	Robust nonsmooth optimality conditions for uncertain multiobjective programs involving stable functions
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