On Subdifferentials of Optimal Value Functions

A general mathematical programming problem in which the constraints are defined by multifunctions and depend on a parameter u, and the resulting value function $m(u)$ are considered. In the context of Banach spaces admitting equivalent Fréchet differentiable norms estimates for the generalized gradient ∂ m of m are established. A special study is made of problems in which the multifunctions defining the constraints take convex values. For these problems, estimates for ∂ m are given in terms of the generalized gradients of the support functions of these multifunctions.