In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n × 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 new variants of bundle methods for semidefinite programming.
