On generic one-parametric semi-infinite optimization

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 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 whereas the other three types are typical for the semi-infinite case. We discuss types 7 and 8 in detail. While at points of type 6, the singularity is due to the fact that in the associated lower level problem a Lagrange multiplier corresponding to an active inequality constraint vanishes, at points of type 7 and 8 the gradients of the active constraints in the lower level problem are linearly dependent. If the total number of active constraints in the lower level problem does not exceed the lower level dimension, the point is of type 7, otherwise it is of type 8. Moreover, we distinguish between points of type 8a and 8b, where a point is of type 8a if the Mangasarian-Fromovitz constraint qualification holds in the lower level problem, and of type 8b otherwise. At points of type 8a, the set of generalized critical points is not smooth, but it does not exhibit a turning point. The linear and quadratic indices remain constant when passing along a point of type 8a. Points of type 7 and type 8b are (relative) boundary points of the set of generalized critical points.

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