**Abstract.** This tutorial presents an introduction to generalized semi-infinite programming (GSIP) which in recent years became a vivid field of active research in mathematical programming. A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables. There exist a wide range of applications which give rise to GSIP models; some of them are discussed in the present paper. Furthermore, geometric and topological properties of the feasible set and, in particular, the difference to the standard semi-infinite case are analyzed. By using first order approximations of the feasible set corresponding constraint qualifications are developed. Then, necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so-called lower level problem are used. Finally, an overview on numerical methods is given.

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# Generalized semi-infinite programming: a tutorial

Francisco Guerra Vázquez, J.-J. Rückmann, Oliver Stein and Georg Still