First-Order Optimality Conditions for Convex Semi-Infinite Inequality Constrained Programming with Noncompact Sets

In this paper, we study the optimal value function on a noncompact set and obtain the expressions of the subderivative and subdifferential of the optimal value function. Based on some characterizations of subderivative and subdifferential of the objective function and constraint functions, we develop the first-order necessary and sufficient optimality conditions for convex semi-infinite inequality constrained programming in which the index sets are not necessarily compact.