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Parametric semi-infinite programming: jumps in the set of local minimizers

Hubertus Th. Jongen and Oliver Stein

Abstract. For generic one-parametric families of differentiable semi-infinite optimization problems the corresponding set $\Sigma$ of generalized critical points has been classified into eight types in our previous paper. Five of these types also occur in problems with a finite number of inequality constraints whereas the other three types are typical for the semi-infinite case. In the present paper we show that, under the usual compactness assumption on the family of feasible sets, a jump to a different component of $\Sigma$ - consisting of local minimizers - is possible, whenever a path of local minimizers ends at one of the additional singular points. For turning points of type 6 we give a feasible direction of quadratic descent, whereas at points of type 7 and points of type 8b a linear descent direction is given.