Generalized convexity and concavity of the optimal value function in nonlinear programming

In this paper we consider generalized convexity and concavity properties of the optimal value functionf* for the general parametric optimization problemP(ε) of the form minxf(x, ε) s.t.x∈R(ε). Many results on convexity and concavity characterizations off* were presented by the authors in a previous paper. Such properties off* and the solution set mapS* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. We give sufficient conditions for several types of generalized convexity and concavity off*, in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point-to-set mapR. Specializations of these results to the parametric inequality-equality constrained nonlinear programming problem are provided.