Moreau-Rockafellar type theorem for convex set functions

Let ( X, Γ, μ) be an atomless finite measure space and S ⊂ Γ a convex subfamily. It is proved that the Moreau-Rockafellar theorem, ∂(F 1 + ··· + F n)(Ω) = ∂F 1(Ω) + ··· + ∂F n(Ω) , holds for proper convex set functions F 1, …, F n and Ω ϵ S if all set functions F i , except possibly one, are w ∗-low...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 132; no. 2; pp. 558 - 571
Main Authors Lai, Hang-Chin, Lin, Lai-Jiu
Format Journal Article
LanguageEnglish
Published San Diego, CA Elsevier Inc 01.06.1988
Elsevier
Subjects
Applied sciences
Exact sciences and technology
Mathematical programming
Operational research and scientific management
Operational research. Management science
Kuhn Tucker condition
Convex function
Convex programming
Optimal solution
Optimization
Mathematical programming
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Summary:Let ( X, Γ, μ) be an atomless finite measure space and S ⊂ Γ a convex subfamily. It is proved that the Moreau-Rockafellar theorem, ∂(F 1 + ··· + F n)(Ω) = ∂F 1(Ω) + ··· + ∂F n(Ω) , holds for proper convex set functions F 1, …, F n and Ω ϵ S if all set functions F i , except possibly one, are w ∗-lower semicontinuous on S . As applications, the Kuhn-Tucker type condition for an optimal solution of convex programming problem with set functions and the Fritz John type condition for an optimal solution of vector-valued minimization problem for set functions are obtained.
ISSN:0022-247X
1096-0813
DOI:10.1016/0022-247X(88)90084-4