A First-Order Optimality Condition in Nonsmooth Generalized Semi-infinite Programming (GSIP)

The purpose of this paper is to develop some first-order necessary optimality conditions for nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. Several authors have studied first-order optimality conditions of GSIPs with continuously differentiable data (see [6,15– 18]), but there are few papers that consider GSIPs in nonsmooth case [9,10]. Extensive references to optimality conditions and constraint qualifications for smooth GSIPs, and their historical notes, can be found in the book by Stein [15]. All the mentioned papers considered lower-level qualification conditions for GSIP. In the present paper we introduce an upper-level constraint qualification in Mangasarian-Fromovitz sense, and we demonstrate some first-order necessary optimality conditions at minimum point of the considered problem.