On continuous deformations of semi-infinite optimization problems

Rainer Hettich, Hubertus Th. Jongen and Oliver Stein

Abstract. We consider differentiable semi-infinite optimization problems depending on a real parameter. For generic one-parametric families we roughly classify the corresponding set of generalized critical points into eight types. Five of these types also occur in problems with a finite number of inequality constraints. The other three types are typical for the semi-infinite case. We discuss Type 6 in detail. This singularity is due to the fact that in the associated lower level problem a Lagrange parameter corresponding to an active inequality constraint vanishes. The set of generalized critical points is not smooth at such a point. In addition it will be shown that the (quadratic) index either remains constant or changes by at most one when passing along a point of Type 6. An explicit example, exhibiting a turning point, will be provided. The points of Type 1 - 6 generically occur if the index set of inequality constraints is constant and is described by means of finitely many smooth constraints for which the gradients of active constraints are linearly independent throuhghout. The other types (Type 7 and Type 8) are related to the violation of the latter linear independence.