Stability of systems of linear equations and inequalities: distance to ill-posedness and metric regularity

• Published in: Optimization Vol. 56; no. 1-2; pp. 1 - 24
• Main Authors: Cánovas, M. J., Gómez-Senent, F. J., Parra, J.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.02.2007; Taylor & Francis LLC
• Subjects: AMS Subject Classifications 2000: 65F22; Distance to ill-posedness; Euclidean space; Linear constraint systems; Linear equations; Metric regularity; Parameter optimization; Stability; Studies; Systems stability; Well-posedness
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331930600815926

In this article we consider the parameter space of all the linear constraint systems, in the n-dimensional Euclidean space, whose inequality constraints are indexed by an arbitrary, but fixed, set T (possibly infinite) and the number of equations is m≤ n. This parameter space is endowed with the topology of the uniform convergence of the coefficient vectors by means of an extended distance. Our focus is on the stability of the nominal system in terms of whether or not proximal systems preserve consistency/inconsistency. We pay special attention to the different frameworks coming from either splitting each equation into two inequalities, or treating equations as equations. The notable differences arising in the latter setting with respect to the former are emphasized in the article. Ill-posedness is identified with the boundary of the set of consistent systems and a formula for the distance to ill-posedness is obtained. This formula turns out to be closely related with the radius of metric regularity of a set-valued mapping describing homogeneous systems.