The characterizations of optimal solution set in programming problem under inclusion constrains

The characterizations of the solution set in extremal problem under inclusion constrains: (P)minf(x)s.t.x∈C,0∈F(x)is considered in this paper. When f is continuously convex and F is a set-valued map with convex graph, the Lagrange function of (P) is proved to be a constant on the solution set, and this property is then used to derive various simple Lagrange mulitiplier-based characterizations of the solution set of (P).