Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. C...

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Bibliographic Details
Published inOptimization Vol. 52; no. 3; pp. 263 - 276
Main Author Song, Wen
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.06.2003
Subjects
Contingent Epiderivative
Mathematics Subject Classification 2000: 49k27
Optimality Conditions
Vector Optimization
Weak Subdifferentials
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ISSN0233-1934
1029-4945
DOI10.1080/0233193031000120051

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Summary:In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.
ISSN:0233-1934
1029-4945
DOI:10.1080/0233193031000120051