Coordinate optimization for bi-convex matrix inequalities

Abstract

We consider optimization of the largest eigenvalue of a smooth selfadjoint matrix valued function /spl Gamma/(X, Y) of two vector or matrix variables X and Y. We assume that /spl Gamma/ is concave or convex in Y and separately in X, but possibly has bad joint behavior. A typical problem one faces in control design are matrix versions of minimizing in Y and maximizing in X. Also minimizing in X and Y is an important problem. When joint behavior in X and Y is bad existing commercial software must be applied to each coordinate separately, and so can be used only to give a coordinate optimization algorithm. We give strong evidence in this article that on "well behaved /spl Gamma/" coordinate optimization always gives a local optimum for the min/sub Y/ max/sub X/ problem and that it almost never gives a local solution to the min/sub Y/ min/sub X/ problem.