On generalized gradients in optimization problems with set-valued constraints

The paper investigates the value function f(x)=infℝrsquo;(x,y) where the infimum is over all y∈a(x) for some given set-valued map a. Under specified conditions an inner approximation for its generalized gradient is provided. In some cases a full description of this generalized gradient is given. The results are of use in numerical solution of various optimization problems. Illustrative examples are given.