Tolerances, robustness and parametrization of matrix properties related to optimization problems
arXiv:1709.07629 · doi:10.1080/02331934.2018.1545837
Abstract
When we speak about parametric programming, sensitivity analysis, or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this paper, we turn to another question. Suppose that is a matrix having a specific property . What are the maximal allowable variations of the data such that the property still remains valid for the matrix? We study two basic forms of perturbations. The first is a perturbation in a given direction, which is closely related to parametric programming. The second type consists of all possible data variations in a neighbourhood specified by a certain matrix norm; this is related to the tolerance approach to sensitivity analysis, or to stability. The matrix properties discussed in this paper are positive definiteness; P-matrix, H-matrix and P-matrix property; total positivity; inverse M-matrix property and inverse nonnegativity.