On level sets of marginal functions

Oliver Stein

Abstract. The investigation of level sets of marginal functions is motivated by several aspects of standard and generalized semi-infinite programming. The feasible set $M$ of such a problem is easily seen to be a level set of the marginal function corresponding to the lower level problem. In the present paper we study the local structure of $M$ at feasible boundary points in the generic case. A codimension formula shows that there is a wide range of these generic situations, but that the number of active indices is always bounded by the state space dimension. We restrict our attention to two special subcases.

In the first case, where the number of active indices is maximal, $M$ is shown to be locally diffeomorphic to the non-negative orthant. This situation is well-known from finite and also from standard semi-infinite programming. However, in the second case a generic situation arises which is typical for generalized semi-infinite programming. Here, the active index set is a singleton, and $M$ can exhibit a re-entrant corner or even local non-closedness, depending on whether the Mangasarian-Fromovitz constraint qualification holds at the active index. If an objective function is minimized over $M$, then in the setting of the second case a local minimizer cannot occur.

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