Moreau-Rockafellar type theorem for convex set functions (Q1110462)

Let (X,\(\Gamma\),\(\mu)\) 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 \({\mathcal S}\) of the \(\sigma\)- field \(\Gamma\). Consider an optimization problem as follows: (P) Minimize \(F(\Omega)=(F_ 1(\Omega),F_ 2(\Omega),...,F_ n(\Omega))\) subject to \(\Omega\in {\mathcal S}\) and \(G_ j(\Omega)\leq 0\) \((j=1,2,...,m)\). 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.