Coordinate optimization for bi-convex matrix inequalities

We consider optimization of the largest eigenvalue of a smooth selfadjoint matrix valued function Γ(X, Y) of two vector or matrix variables X and Y. We assume that Γ 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 Γ” coordinate optimization always gives a local optimum for the min_Y max _X problem and that it almost never gives a local solution to the min_Y min_X problem