Sufficient conditions for optimization of matrix functions

Inequalities involving matrix polynomials and associated optimization problems have become very important in engineering. Commonplace in design problems are performance functions Γ(X,Y) which are convex in X and convex in Y but which are not jointly convex, and the problem is to minimize the highest eigenvalue of Γ. In a previous paper (1997) we derived first order tests for coordinate optimization (the most common approach to these problems) and to first order optimality tests for a true optimum. This article treats second order optimality conditions for optimization of matrix functions. Second order tests are important because optimization of matrix valued Γ based on linearization or coordinate descent will often produce critical points which are not local solutions to the problem. Sufficient conditions, especially if they include second order information, become very valuable in these cases, as they can be used to tell which among the critical points correspond to true local optimal points, and to provide good update directions. Also we introduce and characterize a strong notion of matrix convexity which appears suited to many well behaved engineering problems