Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions

In this paper, we will study optimality conditions of semi‐infinite programs and generalized semi‐infinite programs by employing lower order exact penalty functions and the condition that the generalized second‐order directional derivative of the constraint function at the candidate point along any feasible direction for the linearized constraint set is non‐positive. We consider three types of penalty functions for semi‐infinite program and investigate the relationship among the exactness of these penalty functions. We employ lower order integral exact penalty functions and the second‐order generalized derivative of the constraint function to establish optimality conditions for semi‐infinite programs. We adopt the exact penalty function technique in terms of a classical augmented Lagrangian function for the lower‐level problems of generalized semi‐infinite programs to transform them into standard semi‐infinite programs and then apply our results for semi‐infinite programs to derive the optimality condition for generalized semi‐infinite programs. We will give various examples to illustrate our results and assumptions.