Tolerances, robustness and parametrization of matrix properties related to optimization problems

TL;DR

This paper investigates the maximum allowable data variations that preserve specific matrix properties relevant to optimization, focusing on perturbation types and their impact on properties like positive definiteness and total positivity.

Contribution

It introduces a framework for analyzing tolerances and robustness of matrix properties under different perturbation models in optimization contexts.

Findings

Derived bounds for perturbations preserving positive definiteness.

Characterized stability regions for P-matrices and H-matrices.

Analyzed the impact of perturbations on total positivity and inverse M-matrix properties.

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 $A$ is a matrix having a specific property $\mathcal{P}$. 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.