Convergence Theory of Probability-one Homotopies for Model Order Reduction

The optimal H-square model reduction problem is an inherently nonconvex problem and thus provides a nontrivial computational challenge. This paper systematically examines the requirements of probability-one homotopy methods to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems, and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the optimal projection equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computation- al implementation of the homotopy algorithms.