Optimality for set functions with values in ordered vector spaces (Q1115816)

Let (X,\(\Gamma\),\(\mu)\) be a finite atomless measure space, \({\mathcal S}^ a \)convex subfamily of \(\Gamma\), Y and Z locally convex Hausdorff topological vector spaces which are ordered by the cones C and D, respectively. Let F: \({\mathcal S}\to Y\) be C-convex and G: \({\mathcal S}\to Z\) be D-convex set functions. Consider the following optimization problem (P): minimize F(\(\Omega)\), subject to \(\Omega\in {\mathcal S}\) and \(G(\Omega)\leq_ D\theta\). The paper generalizes the Moreau-Rockafellar theorem with set functions. 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.