First-order optimality conditions for two classes of generalized nonsmooth semi-infinite optimization

We investigate two classes of generalized nonsmooth semi-infinite optimization problems in this paper, that is, the generalized convex semi-infinite optimization problem and the generalized Lipscitz semi-infinite optimization problem. Their first order necessary optimality conditions are obtained using either the differentiability properties of the optimal value functions or the bounds for the directional derivatives of the optimal value function.