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

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 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.

Stability of the linear complementarity problem properties under interval uncertainty

“…We characterize their strong counterparts when entries are interval valued. Other matrix properties were discussed, e.g., in [7,10,12,14,17].…”

Relaxing the Nonsingularity Assumption for Intervals of Totally Nonnegative Matrices

“…For the totally positive matrices, i.e., matrices having all their minors positive (here the perturbed matrix has in turn to be totally positive), it was established in [10], see also [9, Section 9.5], for a few specified entries and in [6] for arbitrary entries. The similar problem for a uniform perturbation of all the coefficients of a totally positive matrix was considered in [13,Section 7].…”