Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems

This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of local saddle points (local Nash equilibria) is a subset of locally stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results when compared to existing methods are illustrated through numerical examples.