On Eigenvalue Optimization

In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. We present a general framework for a smooth (differentiable) approach to such problems. It is based on the concept of transversality borrowed from differential geometry. In that framework we discuss first-and second-order optimality conditions and rates of convergence of the corresponding second-order algorithms. Finally we present some results on the sensitivity analysis of such problems.