Right bounds for eigenvalue ranges of interval matrices - Estimation principles vs global optimization

Abstract

The paper develops a study on the evaluation of right bounds for the eigenvalue ranges of interval matrices. Given an arbitrary interval matrix A, a right bound approximates the right end point of the eigenvalue range - defined as an exact value, denoted by I(A), which, generally speaking, is not directly calculable. We consider two classes of methods providing right bounds: (i) I(A) is approximated by a value Ie(A) (with I(A) less or equal Ie(A)), which is calculable from a mathematical expression, especially constructed as an estimation of I(A) by majorization; (ii) I(A) is approximated by a value Ic(A) which is computable as the solution of a global optimization problem with constraints given by the interval coefficients of A. For our study on right bounds, we use three estimation principles, based on different majorization approaches - corresponding to the class of methods (i), and a genetic-algorithm-based optimizer that masters non-smooth cost functions - corresponding to the class of methods (ii). The tests performed on a relevant collection of interval matrices (most of them selected from literature) yield a thorough comparative analysis revealing drawbacks and advantages equally unexpected at a first glance.