Calculus of directional coderivatives and normal cones in Asplund spaces

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal co...

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Vydané v:	Positivity : an international journal devoted to the theory and applications of positivity in analysis Ročník 21; číslo 3; s. 1115 - 1142
Hlavní autori:	Long, Pujun, Wang, Bingwu, Yang, Xinmin
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.09.2017 Springer Nature B.V
Predmet:	Calculus Calculus of Variations and Optimal Control; Optimization Cones Econometrics Fourier Analysis Mathematics Mathematics and Statistics Operator Theory Potential Theory Directional Mordukhovich coderivative Variational analysis 49J53 49J52 Directional Mordukhovich normal cone 49J50 Strict differentiability Directional sequential normal compactness Generalized differential calculus Directional differentiability
ISSN:	1385-1292, 1572-9281
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Abstract	We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces.
AbstractList	We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces.
Author	Yang, Xinmin Wang, Bingwu Long, Pujun
Author_xml	– sequence: 1 givenname: Pujun surname: Long fullname: Long, Pujun organization: School of Mathematical Sciences, Inner Mongolia University – sequence: 2 givenname: Bingwu orcidid: 0000-0002-1154-1860 surname: Wang fullname: Wang, Bingwu email: bwang@emich.edu organization: Department of Mathematics, Eastern Michigan University – sequence: 3 givenname: Xinmin surname: Yang fullname: Yang, Xinmin organization: College of Mathematical Sciences, Chongqing Normal University
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Cites_doi	10.1016/j.na.2011.02.050 10.1137/0325059 10.1007/s11117-016-0417-1 10.1007/s11228-012-0220-5 10.1007/978-3-642-02431-3 10.1007/s11590-013-0663-0 10.1080/00036811.2013.879384 10.1016/j.na.2012.11.019 10.1016/0362-546X(85)90030-6 10.1006/jmaa.1996.0342 10.1007/BF01581156 10.1007/s10898-010-9615-8 10.1016/S0362-546X(01)00217-6 10.1137/S0363012994266498 10.1016/j.jmaa.2005.11.051 10.1007/b97543 10.1016/S0022-247X(02)00385-2 10.1137/130914449 10.1137/S0363012900374816 10.1287/moor.1050.0147 10.1090/S0002-9939-96-03049-3 10.1016/0022-247X(87)90202-2 10.1007/s11117-011-0142-8 10.1016/S0022-247X(02)00149-X 10.1090/S0002-9947-96-01543-7
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Keywords	Directional Mordukhovich coderivative Variational analysis 49J53 49J52 Directional Mordukhovich normal cone 49J50 Strict differentiability Directional sequential normal compactness Generalized differential calculus Directional differentiability
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References_xml	– reference: MordukhovichBSVariational Analysis and Generalized Differentiation II: Applications2006BerlinSpringer – reference: MordukhovichBSWangBRestrictive metric regularity and generalized subdifferential calculus in Banach spacesInt. J. Math. Math. Sci.20045026502683 – reference: MordukhovichBSWangBSequential normal compactness in variational analysisNonlinear Anal.200147717728197069210.1016/S0362-546X(01)00217-61042.49522 – reference: RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-30888.49001 – reference: MordukhovichBSShaoYExtremal characterizations of Asplund spacesProc. Am. Math. Soc.19961241197205129178810.1090/S0002-9939-96-03049-30849.46010 – reference: MordukhovichBSWangBCalculus of sequential normal compactness in variational analysisJ. Math. Anal. Appl.20032826384200033010.1016/S0022-247X(02)00385-21033.49031 – reference: MordukhovichBSWangBNecessary suboptimality and optimality conditions via variational principlesSIAM J. Control Optim.2002442623640192051410.1137/S03630129003748161020.49018 – reference: WangBWangDOn the fuzzy intersection ruleNonlinear Anal.20127516231634286136210.1016/j.na.2011.02.0501236.49075 – reference: GfrererHOn directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programsSet Valued Var. Anal.2013212151176304824110.1007/s11228-012-0220-51321.49031 – reference: BorweinJMZhuQJTechniques of Variational Analysis2005BerlinSpringer1076.49001 – reference: ChaneyRWSecond-order necessary conditions in constrained semismooth optimizationSIAM J. Control Optim.19872541072108189399910.1137/03250590635.49013 – reference: GinchevIMordukhovichBSOn directionally dependent subdifferentialsC. R. Bulg. Acad. Sci.20116449750828656741289.49021 – reference: FabianMSubdifferentiability and trustworthiness in the light of a new variational principle of Borwein and PreissActa Universitatis Carolinae. Mathematica et Physica1989302515610464450714.49022 – reference: MordukhovichBSNamNMVariational stability and marginal functions via generalized differentiationMath. Oper. Res.2005304800816218581610.1287/moor.1050.01471284.90083 – reference: WangBThe fuzzy intersection rule in variational analysis with applicationsJ. Math. Anal. Appl.200632313651372226018710.1016/j.jmaa.2005.11.0511121.49019 – reference: ClarkeFHOptimization and Nonsmooth Analysis1983New YorkWiley0582.49001 – reference: Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementary Problems, vol. I and II. Springer, New York (2003) – reference: HuangLRNgKFOn some relations between Chaney’s generalized second order directional derivative and that of Ben-Tal and ZoweSIAM J. Control Optim.199634412201235139583110.1137/S03630129942664980877.49019 – reference: PenotJPDirectionally limiting subdifferentials and second-order optimality conditionsOptim. Lett.20138311911200317059710.1007/s11590-013-0663-01320.90084 – reference: MordukhovichBSWangBExtensions of generalized differential calculus in Asplund spacesJ. Math. Anal. Appl.2002272164186193070910.1016/S0022-247X(02)00149-X1006.49007 – reference: GinchevIMordukhovichBSDirectional subdifferentials and optimality conditionsPositivity2012164707737299408010.1007/s11117-011-0142-81254.49007 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation I: Basic Theory2006BerlinSpringer – reference: WangBWangDGeneralized sequential normal compactness in Asplund spacesAppl. Anal.20159499107329507710.1080/00036811.2013.8793841308.49012 – reference: ChaneyRWOn second derivatives for nonsmooth functionsNonlinear Anal.19859111189120981365310.1016/0362-546X(85)90030-60575.49005 – reference: ChaneyRWSecond-order directional derivatives for nonsmooth functionsJ. Math. Anal. Appl.1987128249551191738410.1016/0022-247X(87)90202-20671.49012 – reference: WangBZhuMZhaoYGeneralized sequential normal compactness in Banach spacesNonlinear Anal.201379221232300503810.1016/j.na.2012.11.0191262.49022 – reference: MordukhovichBSShaoYNonsmooth sequential analysis in Asplund spacesTrans. Am. Math. Soc.1996348412351280133339610.1090/S0002-9947-96-01543-70881.49009 – reference: PenotJPThe directional subdifferential of the difference of two convex functionsJ. Glob. Optim.2010493505519277161710.1007/s10898-010-9615-81223.26030 – reference: ChouCCLiXNgKFShiSOptimality conditions for lower semicontinuous functionsJ. Math. Anal. Appl.19962022686701140625610.1006/jmaa.1996.03420856.90127 – reference: LongPWangBYangXMCalculus of directional subdifferentials and coderivatives in Banach spacesPositivity201606731883 – reference: GfrererHOptimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraintsSIAM J. Optim.2014242898931321722210.1137/1309144491298.49021 – reference: HuangLRNgKFSecond-order necessary and sufficient conditions in nonsmooth optimizationMath. Program.1994661379402129707310.1007/BF015811560824.90123 – volume: 75 start-page: 1623 year: 2012 ident: 455_CR30 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2011.02.050 – volume: 25 start-page: 1072 issue: 4 year: 1987 ident: 455_CR4 publication-title: SIAM J. 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Snippet	We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus...
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SubjectTerms	Calculus Calculus of Variations and Optimal Control; Optimization Cones Econometrics Fourier Analysis Mathematics Mathematics and Statistics Operator Theory Potential Theory
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Title	Calculus of directional coderivatives and normal cones in Asplund spaces
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