Variants of the Ekeland Variational Principle for a set-valued map involving the Clarke Normal Cone
In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions.
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| Vydáno v: | Journal of mathematical analysis and applications Ročník 316; číslo 1; s. 346 - 356 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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San Diego, CA
Elsevier Inc
01.04.2006
Elsevier |
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| ISSN: | 0022-247X, 1096-0813 |
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| Abstract | In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions. |
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| AbstractList | In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions. |
| Author | Ha, Truong Xuan Duc |
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| Cites_doi | 10.1006/jmaa.2000.7357 10.1006/jmaa.1994.1144 10.1090/S0002-9947-1995-1307998-0 10.1023/A:1004663208905 10.1137/S0363012902411532 10.1016/0022-247X(74)90025-0 10.1016/S0022-247X(03)00482-7 |
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| Keywords | Ekeland's variational principle Set-valued map Clarke's normal cone Coderivative Vector optimization Sufficient condition Mathematical analysis Set-valued mapping Optimization method Variational principle Palais Smale condition Clarke normal cone |
| Language | English |
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| References | Aussel, Corvellec, Lassonde (bib001) 1995; 347 Ekeland (bib004) 1974; 47 Krasnoselskii (bib008) 1964 Clarke (bib003) 1983 Ha (bib005) 2003; 286 Rockafellar (bib012) 1981 Chen, Huang, Hou (bib002) 2000; 106 Hiriart-Urruty (bib006) 1979; 60 Luc (bib009) 1989 Zaffaroni (bib013) 2003; 42 Mordukhovich (bib011) 1994; 183 Mordukhovich (bib010) 1980; 22 Huang (bib007) 2001; 262 Krasnoselskii (10.1016/j.jmaa.2005.04.044_bib008) 1964 Ekeland (10.1016/j.jmaa.2005.04.044_bib004) 1974; 47 Zaffaroni (10.1016/j.jmaa.2005.04.044_bib013) 2003; 42 Ha (10.1016/j.jmaa.2005.04.044_bib005) 2003; 286 Luc (10.1016/j.jmaa.2005.04.044_bib009) 1989 Chen (10.1016/j.jmaa.2005.04.044_bib002) 2000; 106 Hiriart-Urruty (10.1016/j.jmaa.2005.04.044_bib006) 1979; 60 Aussel (10.1016/j.jmaa.2005.04.044_bib001) 1995; 347 Huang (10.1016/j.jmaa.2005.04.044_bib007) 2001; 262 Mordukhovich (10.1016/j.jmaa.2005.04.044_bib011) 1994; 183 Clarke (10.1016/j.jmaa.2005.04.044_bib003) 1983 Mordukhovich (10.1016/j.jmaa.2005.04.044_bib010) 1980; 22 Rockafellar (10.1016/j.jmaa.2005.04.044_bib012) 1981 |
| References_xml | – year: 1983 ident: bib003 article-title: Optimization and Nonsmooth Analysis – volume: 60 start-page: 57 year: 1979 end-page: 85 ident: bib006 article-title: New concepts in nondifferentiable programming publication-title: Bull. Soc. Math. France – volume: 262 start-page: 12 year: 2001 end-page: 23 ident: bib007 article-title: New stability results for Ekeland's publication-title: J. Math. Anal. Appl. – volume: 106 start-page: 151 year: 2000 end-page: 164 ident: bib002 article-title: A general approximate variational principle for set-valued maps publication-title: J. Optim. Theory Appl. – volume: 286 start-page: 509 year: 2003 end-page: 523 ident: bib005 article-title: The Ekeland variational principle for set-valued maps involving coderivatives publication-title: J. Math. Anal. Appl. – year: 1964 ident: bib008 article-title: Positive Solutions of Operator Equations – year: 1989 ident: bib009 article-title: Theory of Vector Optimization – volume: 22 start-page: 526 year: 1980 end-page: 530 ident: bib010 article-title: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems publication-title: Soviet Math. Dokl. – volume: 42 start-page: 1071 year: 2003 end-page: 1086 ident: bib013 article-title: Degrees of efficiency and degrees of minimality publication-title: SIAM J. Control Optim. – volume: 347 start-page: 4147 year: 1995 end-page: 4161 ident: bib001 article-title: Mean value property and subdifferential criteria for lower semicontinuous function publication-title: Trans. Amer. Math. Soc. – volume: 47 start-page: 324 year: 1974 end-page: 353 ident: bib004 article-title: On the variational principle publication-title: J. Math. Anal. Appl. – year: 1981 ident: bib012 article-title: The theory of subgradients and its applications to problems of optimization: Convex and nonconvex functions – volume: 183 start-page: 250 year: 1994 end-page: 288 ident: bib011 article-title: Generalized differential calculus for nonsmooth and set-valued mappings publication-title: J. Math. Anal. Appl. – year: 1981 ident: 10.1016/j.jmaa.2005.04.044_bib012 – volume: 262 start-page: 12 year: 2001 ident: 10.1016/j.jmaa.2005.04.044_bib007 article-title: New stability results for Ekeland's ɛ-variational principles for vector-valued and set-valued maps publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.2000.7357 – volume: 183 start-page: 250 year: 1994 ident: 10.1016/j.jmaa.2005.04.044_bib011 article-title: Generalized differential calculus for nonsmooth and set-valued mappings publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1994.1144 – volume: 60 start-page: 57 year: 1979 ident: 10.1016/j.jmaa.2005.04.044_bib006 article-title: New concepts in nondifferentiable programming publication-title: Bull. Soc. Math. France – volume: 22 start-page: 526 year: 1980 ident: 10.1016/j.jmaa.2005.04.044_bib010 article-title: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems publication-title: Soviet Math. Dokl. – volume: 347 start-page: 4147 year: 1995 ident: 10.1016/j.jmaa.2005.04.044_bib001 article-title: Mean value property and subdifferential criteria for lower semicontinuous function publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1995-1307998-0 – volume: 106 start-page: 151 year: 2000 ident: 10.1016/j.jmaa.2005.04.044_bib002 article-title: A general approximate variational principle for set-valued maps publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1004663208905 – volume: 42 start-page: 1071 year: 2003 ident: 10.1016/j.jmaa.2005.04.044_bib013 article-title: Degrees of efficiency and degrees of minimality publication-title: SIAM J. Control Optim. doi: 10.1137/S0363012902411532 – volume: 47 start-page: 324 year: 1974 ident: 10.1016/j.jmaa.2005.04.044_bib004 article-title: On the variational principle publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(74)90025-0 – year: 1983 ident: 10.1016/j.jmaa.2005.04.044_bib003 – year: 1964 ident: 10.1016/j.jmaa.2005.04.044_bib008 – year: 1989 ident: 10.1016/j.jmaa.2005.04.044_bib009 – volume: 286 start-page: 509 year: 2003 ident: 10.1016/j.jmaa.2005.04.044_bib005 article-title: The Ekeland variational principle for set-valued maps involving coderivatives publication-title: J. Math. Anal. Appl. doi: 10.1016/S0022-247X(03)00482-7 |
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| SubjectTerms | Calculus of variations and optimal control Clarke's normal cone Coderivative Ekeland's variational principle Exact sciences and technology Global analysis, analysis on manifolds Mathematical analysis Mathematics Sciences and techniques of general use Set-valued map Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Vector optimization |
| Title | Variants of the Ekeland Variational Principle for a set-valued map involving the Clarke Normal Cone |
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