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

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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