On minimizing the largest eigenvalue of a symmetric matrix Original Research Article

Optimization problems involving eigenvalues arise in many engineering problems. In this paper, we consider the problem of minimizing the largest eigenvalue over an affine family of symmetric matrices. This problem has a variety of applications, such as the stability analysis of dynamic systems or the computation of structured singular values. Given ε ≥ 0, we give an optimality condition which ensures that the largest eigenvalue is within ε error bound of the solution. Also, a new line search rule is proposed, and it is shown to have good descent properties. When the multiplicity of the largest eigenvalue the solution is known, a new algorithm for the optimization problem under consideration is proposed. Some numerical experiments on the proposed algorithm are presented.