On self-concordant convex-concave functions

In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of "sufficiently well-behaved" convex-concave functions - those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an exterior penalty polynomial time method for Semidefinite Programming and to the problem of inscribing the largest volume ellipsoid into a given polytope.