On regular minimax optimization

Abstract

We consider functions of maximum type (max functions for short), subject to (in)equality constraints. The space dimension is finite, and the maximum is taken over a compact manifold with boundary. Effective local minimization algorithms based on Newton's method can be derived in the case where a local minimum is nondegenerate (in a two-level sense). In fact, nondegeneracy refers on the one hand to a local (implicit) reduction of the original max function to another one, where the maximum is taken over a finite set. On the other hand, it refers to strict complementarity and nondegeneracy of the underlying quadratic form with respect to the reduced stationary situation. As the main goal, we show that the set ofn-parameter families of functions, for which the stationary points of the corresponding max function are nondegenerate, constitutes an open and dense subset in the space of alln-parameter families (the topology used takes derivatives up to second order into account). An application to approximation problems of Chebyshev type is presented.