Smoothing approximations for two classes of convex eigenvalue optimization problems

Abstract

In this thesis, we consider two problems: minimizing the sum of the I? largest eigenvalues and the sum of the I? largest absolute values of eigenvalues of a parametric linear operator. In order to apply Nesterova??s smoothing algorithm to solve these two problems, we construct two computable smoothing functions whose gradients are Lipschitz continuous. This construction is based on Shia??s thesis [12] and new techniques introduced in this thesis. Numerical results on the performance of Nesterova??s smooth algorithm are also reported.