The Reduction Ansatz in absence of lower semi-continuity

Oliver Stein

Abstract. The Reduction Ansatz was developed in order to describe the feasible set of a standard semi-infinite optimization problem locally by finitely many inequality constraints. A feasible set of this type is easily seen to be a level set of a marginal function, where the feasible set of the underlying (lower level) optimization problem is compact and constant. For investigations in parametric standard semi-infinite programming as well as in generalized semi-infinite programming, it is necessary to generalize this approach to the case of a non-constant, upper semi-continuous feasible set of the lower level problem. In particular, as lower semi-continuity of the corresponding set-valued mapping is not required, components of the feasible set may vanish under perturbations of the (lower level) parameter. In the present paper, we give such a generalization of the Reduction Ansatz, and we apply it to explain trap-door points in parametric standard semi-infinite programming, as well as non-closedness of the feasible set in generalized semi-infinite programming.