Bundle Methods to Minimize the Maximum Eigenvalue Function

Abstract

In the last ten years the study of interior point methods dominated algorithmic research in semidefinite programming. Only recently interest in nonsmooth optimization methods revived again, the impetus coming from two different directions. On the one hand alternative possibilities were sought to solve structured large scale semidefinite programs which were not amenable to current interior point codes [338], on the other hand new developments in the second order theory of nonsmooth convex optimization suggested the specialization of these theoretic techniques to semidefinite programming [597, 598]. We present these new methods under the common framework of bundle methods and survey the underlying theory as well as some implementational aspects. In order to illustrate the efficiency and potential of the algorithms we also present numerical results.