Optimality for set functions with values in ordered vector spaces

Let (X, Г, μ) be a finite atomless measure space,L a convex subfamily of Φ, andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:L→Y beC-convex andG:L→Z beD-convex set functions. Consider the following optimization problem (P): minimizeF(ω), subject to ω∈L andG(ω)≤Dθ. The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.

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