Coordinate optimization for bi-convex matrix inequalities

• Published in: Proceedings of the 36th IEEE Conference on Decision and Control Vol. 4; pp. 3609 - 3613 vol.4
• Main Authors: Helton, J.W., Merino, O.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 1997
• Subjects: Eigenvalues and eigenfunctions; Level set; Linear matrix inequalities; Mathematics; Minimax techniques; Software algorithms; Terminology; Testing
• ISBN: 0780341872; 9780780341876
• ISSN: 0191-2216
• DOI: 10.1109/CDC.1997.652414

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.