Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods

We study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions. We show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point. We examine in which way standard and projected Newton steps may be combined with a nonsmooth first-order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary