Tolerances, robustness and parametrization of matrix properties related to optimization problems. (4th March 2019)

• Record Type: Journal Article
• Authors: Hladík, Milan
• Is Part Of: Optimization. Volume 68:Number 2/3(2019)
• Page Start: 667
• Page End: 690
• Publication Date: 2019-03-04
• Subjects: Positive definiteness -- P-matrix -- M-matrix -- totally positive matrix -- regularity radius; 90C31 -- 15Bxx -- 65F35; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1545837
• Languages: English
• ISSNs: 0233-1934

Abstract

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 propertyP . 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; M-matrix, H-matrix and P-matrix property; total positivity; inverse M-matrix property and inverse nonnegativity.