Minimization of a One Class of Maximum Functions with Applications to Some Eigenvalue Problems

In this paper we consider the problem of minimization of a convex function that can be expressed as a maximum of affine linear functions. A convergence theorem is established and the minimizing sequence is constructed. The computational time is linear with respect to the step number. The obtained results are applied to eigenvalue problems such as eigenvalue minimization, symmetric stabilization, and the Riccati matrix inequalities. A number of examples are provided.