On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions
The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quanti...
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| Published in | Set-valued and variational analysis Vol. 29; no. 2; pp. 425 - 451 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.06.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1877-0533 1877-0541 1877-0541 |
| DOI | 10.1007/s11228-020-00571-z |
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| Abstract | The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave multifunction term, are provided in a Banach space setting. |
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| AbstractList | The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave multifunction term, are provided in a Banach space setting. |
| Author | Uderzo, A. |
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| Keywords | Optimal value function Parametric optimization Secondary: 49J52, 90C31, 90C48 Primary: 49J53 Set-valued inclusion Solution mapping Lipschitz semicontinuity Calmness |
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| References | PangJ-SError bounds in mathematical programmingMath. Programming1997791-3, Ser. B299332146477210.1007/BF02614322 FabianMJHenrionRKrugerAYOutrataJError bounds: necessary and sufficient conditionsSet-Valued Var. Anal.2010182121149264524610.1007/s11228-010-0133-0 KrugerAYAbout stationarity and regularity in variational analysisTaiwanese J. Math.2009136A17371785258373910.11650/twjm/1500405612 MordukhovichBSVariational Analysis and Applications2018ChamSpringer10.1007/978-3-319-92775-6 IoffeADNonsmooth analysis: differential calculus of nondifferentiable mappingsTrans. Amer. Math. Soc.1981266115661378410.1090/S0002-9947-1981-0613784-7 MordukhovichBSVariational Analysis and Generalized Differentiation. I. Basic Theory2006BerlinSpringer10.1007/3-540-31246-3 HoffmanAJOn approximate solutions of systems of linear inequalitiesJ. Research Nat. Bur. Standards1952492632655127510.6028/jres.049.027 DontchevALRockafellarRTImplicit Functions and Solution Mappings. A View from Variational Analysis20142nd edn.New YorkSpringer1337.26003 KrugerAYError bounds and metric subregularityOptimization20156414979329354010.1080/02331934.2014.938074 IoffeADVariational Analysis of Regular Mappings. Theory and Applications2017ChamSpringer10.1007/978-3-319-64277-2 Ben-TalANemirovskiARobust convex optimizationMath. Oper. Res.1998234769805166241010.1287/moor.23.4.769 RobinsonSMSome continuity properties of polyhedral multifunctionsMath. Programming Stud.19811420621460013010.1007/BFb0120929 RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 MordukhovichBSNamNMWangBMetric regularity of mappings and generalized normals to set imagesSet-Valued Var. Anal.2009174359387255961710.1007/s11228-009-0122-3 BonnansJFShapiroAPerturbation Analysis of Optimization Problems2000New YorkSpringer10.1007/978-1-4612-1394-9 Castellani, M.: Error bounds for set-valued maps. In: Generalized Convexity and Optimization for Economic and Financial Decisions, pp 121–135. Pitagora, Bologna (1999) UderzoAOn a set-covering property of multivalued mappingsPure Appl. Funct. Anal.201721129151362147507296475 AzéDCorvellecJ-NCharacterizations of error bounds for lower semicontinuous functions on metric spacesESAIM Control Optim. Calc. Var.2004103409425208433010.1051/cocv:2004013 BorweinJMZhuQJTechniques of Variational Analysis2005New YorkSpringer1076.49001 UderzoAOn a quantitative semicontinuity property of variational systems with applications to perturbed quasidifferentiable optimization. Constructive nonsmooth analysis and related topics, Springer Optim. Appl., vol. 872014New YorkSpringer115—136115—1361280.49026 IoffeADMetric regularity and subdifferential calculusUspekhi Mat. Nauk2000553(333103162177735210.4213/rm292 RobinsonSMGeneralized equations and their solutions. I. Basic theory. Point-to-set maps and mathematical programmingMath. Programming Stud.19791012814110.1007/BFb0120850 RobinsonSMAn application of error bounds for convex programming in a linear spaceSIAM J. Control19751327127338567110.1137/0313015 WuZYeJJOn error bounds for lower semicontinuous functionsMath. Program.2002922, Ser. A301314190126310.1007/s101070100278 PenotJ-PCalmness and stability properties of marginal and performance functionsNumer. Funct. Anal. Optim.2004253–428730820720701081.49015 AliprantisCDBorderKCInfinite Dimensional Analysis. A Hitchhiker’S Guide2006BerlinSpringer1156.46001 CibulkaRFabianMKrugerAYOn semiregularity of mappingsJ. Math. Anal. Appl.20194732811836391285310.1016/j.jmaa.2018.12.071 KlatteDKummerBNonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications2002DordrechtKluwer Academic Publishers1173.49300 UderzoAOn Lipschitz semicontinuity properties of variational systems with application to parametric optimizationJ. Optim. Theory Appl.201416214778322851510.1007/s10957-013-0455-9 UderzoASolution analysis for a class of set-inclusive generalized equations: a convex analysis approachPure Appl. Funct. Anal.202053769790415860307326961 PenotJ-PCalculus Without Derivatives2013New YorkSpringer10.1007/978-1-4614-4538-8 De GiorgiEMarinoATosquesMProblems of evolution in metric spaces and maximal decreasing curvesAtti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)19806831801876368140465.47041[in Italian] KhanAATammerKZălinescuCSet-Valued Optimization. An Introduction with Applications2015HeidelbergSpringer1308.49004 UderzoAAn implicit multifunction theorem for the hemiregularity of mappings with application to constrained optimizationPure Appl. Funct. Anal.201832371391380047607300546 UderzoAOn some generalized equations with metrically C-increasing mappings: solvability and error bounds with applications to optimizationOptimization201968227253390216410.1080/02331934.2018.1553972 KrugerAYNgaiHVThéraMStability of error bounds for convex constraint systems in Banach spacesSIAM J. Optim.201020632803296273595410.1137/100782206 A Uderzo (571_CR32) 2017; 2 AD Ioffe (571_CR14) 2017 A Ben-Tal (571_CR3) 1998; 23 AY Kruger (571_CR19) 2015; 64 BS Mordukhovich (571_CR21) 2009; 17 D Azé (571_CR2) 2004; 10 A Uderzo (571_CR34) 2019; 68 SM Robinson (571_CR28) 1981; 14 RT Rockafellar (571_CR29) 1998 AA Khan (571_CR15) 2015 AD Ioffe (571_CR13) 2000; 55 SM Robinson (571_CR27) 1979; 10 571_CR6 E De Giorgi (571_CR8) 1980; 68 AL Dontchev (571_CR9) 2014 AD Ioffe (571_CR12) 1981; 266 A Uderzo (571_CR30) 2014; 162 J-P Penot (571_CR24) 2004; 25 MJ Fabian (571_CR10) 2010; 18 Z Wu (571_CR36) 2002; 92 JF Bonnans (571_CR4) 2000 AY Kruger (571_CR18) 2010; 20 CD Aliprantis (571_CR1) 2006 A Uderzo (571_CR35) 2020; 5 AJ Hoffman (571_CR11) 1952; 49 JM Borwein (571_CR5) 2005 BS Mordukhovich (571_CR20) 2006 SM Robinson (571_CR26) 1975; 13 D Klatte (571_CR16) 2002 R Cibulka (571_CR7) 2019; 473 A Uderzo (571_CR31) 2014 J-S Pang (571_CR23) 1997; 79 J-P Penot (571_CR25) 2013 AY Kruger (571_CR17) 2009; 13 BS Mordukhovich (571_CR22) 2018 A Uderzo (571_CR33) 2018; 3 |
| References_xml | – reference: UderzoAOn Lipschitz semicontinuity properties of variational systems with application to parametric optimizationJ. Optim. Theory Appl.201416214778322851510.1007/s10957-013-0455-9 – reference: UderzoAAn implicit multifunction theorem for the hemiregularity of mappings with application to constrained optimizationPure Appl. Funct. Anal.201832371391380047607300546 – reference: FabianMJHenrionRKrugerAYOutrataJError bounds: necessary and sufficient conditionsSet-Valued Var. Anal.2010182121149264524610.1007/s11228-010-0133-0 – reference: CibulkaRFabianMKrugerAYOn semiregularity of mappingsJ. Math. Anal. Appl.20194732811836391285310.1016/j.jmaa.2018.12.071 – reference: DontchevALRockafellarRTImplicit Functions and Solution Mappings. A View from Variational Analysis20142nd edn.New YorkSpringer1337.26003 – reference: KrugerAYNgaiHVThéraMStability of error bounds for convex constraint systems in Banach spacesSIAM J. Optim.201020632803296273595410.1137/100782206 – reference: RobinsonSMGeneralized equations and their solutions. I. Basic theory. Point-to-set maps and mathematical programmingMath. Programming Stud.19791012814110.1007/BFb0120850 – reference: HoffmanAJOn approximate solutions of systems of linear inequalitiesJ. Research Nat. Bur. Standards1952492632655127510.6028/jres.049.027 – reference: UderzoASolution analysis for a class of set-inclusive generalized equations: a convex analysis approachPure Appl. Funct. Anal.202053769790415860307326961 – reference: IoffeADNonsmooth analysis: differential calculus of nondifferentiable mappingsTrans. Amer. Math. Soc.1981266115661378410.1090/S0002-9947-1981-0613784-7 – reference: RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation. I. Basic Theory2006BerlinSpringer10.1007/3-540-31246-3 – reference: AzéDCorvellecJ-NCharacterizations of error bounds for lower semicontinuous functions on metric spacesESAIM Control Optim. Calc. Var.2004103409425208433010.1051/cocv:2004013 – reference: BonnansJFShapiroAPerturbation Analysis of Optimization Problems2000New YorkSpringer10.1007/978-1-4612-1394-9 – reference: KrugerAYAbout stationarity and regularity in variational analysisTaiwanese J. Math.2009136A17371785258373910.11650/twjm/1500405612 – reference: KrugerAYError bounds and metric subregularityOptimization20156414979329354010.1080/02331934.2014.938074 – reference: IoffeADVariational Analysis of Regular Mappings. Theory and Applications2017ChamSpringer10.1007/978-3-319-64277-2 – reference: PenotJ-PCalmness and stability properties of marginal and performance functionsNumer. Funct. Anal. Optim.2004253–428730820720701081.49015 – reference: WuZYeJJOn error bounds for lower semicontinuous functionsMath. Program.2002922, Ser. A301314190126310.1007/s101070100278 – reference: De GiorgiEMarinoATosquesMProblems of evolution in metric spaces and maximal decreasing curvesAtti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)19806831801876368140465.47041[in Italian] – reference: KlatteDKummerBNonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications2002DordrechtKluwer Academic Publishers1173.49300 – reference: KhanAATammerKZălinescuCSet-Valued Optimization. An Introduction with Applications2015HeidelbergSpringer1308.49004 – reference: MordukhovichBSVariational Analysis and Applications2018ChamSpringer10.1007/978-3-319-92775-6 – reference: UderzoAOn a set-covering property of multivalued mappingsPure Appl. Funct. Anal.201721129151362147507296475 – reference: Castellani, M.: Error bounds for set-valued maps. In: Generalized Convexity and Optimization for Economic and Financial Decisions, pp 121–135. Pitagora, Bologna (1999) – reference: AliprantisCDBorderKCInfinite Dimensional Analysis. A Hitchhiker’S Guide2006BerlinSpringer1156.46001 – reference: RobinsonSMAn application of error bounds for convex programming in a linear spaceSIAM J. Control19751327127338567110.1137/0313015 – reference: BorweinJMZhuQJTechniques of Variational Analysis2005New YorkSpringer1076.49001 – reference: UderzoAOn a quantitative semicontinuity property of variational systems with applications to perturbed quasidifferentiable optimization. Constructive nonsmooth analysis and related topics, Springer Optim. Appl., vol. 872014New YorkSpringer115—136115—1361280.49026 – reference: IoffeADMetric regularity and subdifferential calculusUspekhi Mat. Nauk2000553(333103162177735210.4213/rm292 – reference: RobinsonSMSome continuity properties of polyhedral multifunctionsMath. Programming Stud.19811420621460013010.1007/BFb0120929 – reference: PenotJ-PCalculus Without Derivatives2013New YorkSpringer10.1007/978-1-4614-4538-8 – reference: MordukhovichBSNamNMWangBMetric regularity of mappings and generalized normals to set imagesSet-Valued Var. Anal.2009174359387255961710.1007/s11228-009-0122-3 – reference: UderzoAOn some generalized equations with metrically C-increasing mappings: solvability and error bounds with applications to optimizationOptimization201968227253390216410.1080/02331934.2018.1553972 – reference: Ben-TalANemirovskiARobust convex optimizationMath. Oper. Res.1998234769805166241010.1287/moor.23.4.769 – reference: PangJ-SError bounds in mathematical programmingMath. Programming1997791-3, Ser. B299332146477210.1007/BF02614322 – volume: 13 start-page: 271 year: 1975 ident: 571_CR26 publication-title: SIAM J. Control doi: 10.1137/0313015 – volume: 79 start-page: 299 issue: 1-3, Ser. B year: 1997 ident: 571_CR23 publication-title: Math. Programming doi: 10.1007/BF02614322 – volume-title: Variational Analysis of Regular Mappings. Theory and Applications year: 2017 ident: 571_CR14 doi: 10.1007/978-3-319-64277-2 – volume: 23 start-page: 769 issue: 4 year: 1998 ident: 571_CR3 publication-title: Math. Oper. Res. doi: 10.1287/moor.23.4.769 – volume: 68 start-page: 180 issue: 3 year: 1980 ident: 571_CR8 publication-title: Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) – volume: 162 start-page: 47 issue: 1 year: 2014 ident: 571_CR30 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-013-0455-9 – volume: 64 start-page: 49 issue: 1 year: 2015 ident: 571_CR19 publication-title: Optimization doi: 10.1080/02331934.2014.938074 – volume: 55 start-page: 103 issue: 3(333 year: 2000 ident: 571_CR13 publication-title: Uspekhi Mat. Nauk doi: 10.4213/rm292 – ident: 571_CR6 – volume: 266 start-page: 1 issue: 1 year: 1981 ident: 571_CR12 publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1981-0613784-7 – volume-title: Perturbation Analysis of Optimization Problems year: 2000 ident: 571_CR4 doi: 10.1007/978-1-4612-1394-9 – volume: 10 start-page: 128 year: 1979 ident: 571_CR27 publication-title: Math. Programming Stud. doi: 10.1007/BFb0120850 – volume: 49 start-page: 263 year: 1952 ident: 571_CR11 publication-title: J. Research Nat. Bur. Standards doi: 10.6028/jres.049.027 – volume: 13 start-page: 1737 issue: 6A year: 2009 ident: 571_CR17 publication-title: Taiwanese J. Math. doi: 10.11650/twjm/1500405612 – volume: 17 start-page: 359 issue: 4 year: 2009 ident: 571_CR21 publication-title: Set-Valued Var. Anal. doi: 10.1007/s11228-009-0122-3 – volume: 473 start-page: 811 issue: 2 year: 2019 ident: 571_CR7 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2018.12.071 – volume-title: Set-Valued Optimization. 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| Snippet | The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization... |
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| SubjectTerms | Analysis Banach spaces Inclusions Mathematics Mathematics and Statistics Optimization Parameterization Robustness (mathematics) Stability |
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| Title | On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions |
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