In this paper we present a nonsmooth algorithm to minimize the maximum eige
nvalue of matrices belonging to an affine subspace of n x n symmetric matri
ces. 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 pro
ve that quadratic convergence is achieved asymptotically, some strict compl
ementarity and non-degeneracy assumptions are needed. We also introduce new
variants of bundle methods for semidefinite programming.