A Second-Order Bundle Method to Minimize the Maximum Eigenvalue Function

Francois Oustry

In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n x n symmetric matrices. We show how a simple bundle method, the approximate eigenvalue method can be used to globalize the second-order method developed by M. L. Overton in the eighties and recently revisited in the framework of the U-Lagrangian theory. With no additional assumption, the resulting algorithm generates a minimizing sequence. A geometrical and constructive proof is given. To prove that quadratic convergence is achieved asymptotically, some strict complementarity and non-degeneracy assumptions are needed. We also introduce a new generation of bundle methods for semidefinite programming.

RR-3738, INRIA Rhone-Alpes, ZIRST - 655 avenue de l'Europe, F-38330 Montbonnot Saint-Martin, July 1999.

Contact: Francois.Oustry@inria.fr