Methods for eigenvalue optimization with application to semidefinite optimization

Abstract

This thesis deals with eigenvalue optimization problems and some methods which are applicable in solving those kind of problems. The goal is to develop a novel algorithm which can solve large instances of eigenvalue optimization problems. In the second chapter is shown how semidefinite optimization problems can be turned into eigenvalue optimization problems through a rotation of basis. Although the relation between semidefinite optimization and eigenvalue opti- mization is not new, this approach where a semidefinite optimization problem is turned into a full eigenvalue optimization problem is. In the third chapter SASPA (Subspace Accelerated Sensitive Pole Algorithm) is used to compute the intersection points of eigencurves. This is a novel appli- cation which allows to compute such intersection points with less computation time than through bisection methods. Moreover, SASPA also works for non- simple eigenvalues. The fourth chapter deals with developing a novel algorithm to solve eigen- value optimization problems based on gradients. This algorithm is able to handle large sparse problems within reasonable computation time.