On stability of the generalized saddle points

Abstract

In this paper, we introduce the concept of generalized saddle points for a function defined on a product of topological spaces. The generalized saddle points are defined as minima of a special function, that is the supremum of Nikaido-Isoda function. We prove elementary properties of the generalized saddle points and study their dependence on a parameter. We show that under natural assumptions if a function has the only generalized saddle point, then this point is stable. Moreover, for functions defined over a sets of solutions to nonlinear equations with a parameter in finite-dimensional spaces, we provide sufficient conditions of the generalized saddle point stability in terms of λ-truncations.