On minimizing the largest eigenvalue of a symmetric matrix

The problem of minimizing the largest eigenvalue over an affine family of symmetric matrices is considered. This problem has a variety of applications, such as the stability analysis of dynamic systems or the computation of structured singular values. Given ∈⩾0, an optimality condition is given which ensures that the largest eigenvalue is within ∈ error bound of the solution. A novel line search rule is proposed and shown to have good descent property. When the multiplicity of the largest eigenvalue at solution is known, a novel algorithm for the optimization problem under consideration is derived. Numerical experiments show that the algorithm has good convergence behavior