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Moreau-Rockafellar type theorem for convex set functions. (English) Zbl 0656.90097
Let (X,Γ,μ) be a finite atomless measure space and F 1 ,F 2 ,···,F n , G 1 ,G 2 ,···,G m be convex real-valued set functions defined on a convex subfamily 𝒮 of the σ- field Γ. Consider an optimization problem as follows: (P) Minimize F(Ω)=(F 1 (Ω),F 2 (Ω),···,F n (Ω)) subject to Ω𝒮 and G j (Ω)0 (j=1,2,···,m). The authors prove a theorem of Moreau-Rockafellar type for set functions, and then use the theorem to prove a Kuhn-Tucker type condition for an optimal solution of the minimization problem (P) for real valued set functions. If the set functions are vector-valued, the Fritz John type condition for an optimum of the multiobjective minimization problem (P) is established.
Reviewer: Z.Liu

MSC:
90C48Programming in abstract spaces
54C60Set-valued maps (general topology)
90C25Convex programming