On generalized semi-infinite programming - Discussion

This paper surveys some basic properties of the class of generalized semi-infinite programming problems (GSIP) where the infinite index set of inequality constraints depends on the state variables and all emerging functions are assumed to be continuously differentiable. There exists a wide range of applications which can be modelled as a (GSIP). The paper discusses extensions of the Mangasarian-Fromovitz, Kuhn-Tucker and Abadie constraint qualification to (GSIP) and presents related first order optimality conditions of Fritz-John and Karush-Kuhn-Tucker type. By using directional differentiability properties of the optimal value function of the lower level problem, first and second order necessary and sufficient optimality conditions are discussed. Several examples illustrate the results presented. Key Words: Generalized semi-infinite programming, extended Mangasarian-Fromovitz, Kuhn-Tucker and Abadie constraint qualification, Fritz-John condition, first and second order optimality conditions, optimal value function, directional differentiability, second order epiregularity, second order growth condition