A subspace method for unimodal symmetric eigenvalue optimization problems involving large scale matrices
Jeroen De Vlieger, Karl Meerbergen · Lirias · 2012
We consider the solution of eigenvalue optimization problems involving large symmetric positive definite matrices. To cope with the large scale, we propose a subspace projection method, based on the concept of bundle methods. The theory is generally applicable to matrices that satisfy some smoothness conditions. For the important case of affine parameters, the objective function is quasi convex. We present two methods: one method builds in an iterative way, a small set of smooth constraints, so that the eigenvalue optimization problem can be solved by a smooth optimization method. The second method is a subspace projection or reduction method where in each iteration an eigenvalue optimization problem with small size matrices needs to be solved. Both methods are provably convergent for unimodal problems. Numerical examples are given to illustrate the theory.