On Generalized Gradients in Optimization Problems with Set-Valued Constraints

We investigate the value function f(x) = inf φ (x, y) where the infimum is over all y ∈ a(x) for some given set-valued map a. Under specified conditions we provide an inner approximation for its generalized gradient. 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.