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Equivalences among the chi measure, Hoffman constant, and Renegar's distance to ill-posedness

Javier Pena(jfp***at***andrew.cmu.edu)
Juan Vera(j.c.veralizcano***at***uvt.nl)
Luis Zuluaga(luis.zuluaga***at***lehigh.edu)

Abstract: We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a procedure to estimate $\chi(A)$ thereby opening an avenue to identify classes of linear programs solvable in polynomial time in the real model of computation.

Keywords: condition measures, chi measure, Hoffman constant, distance to ill-posedness, signed matrices

Category 1: Combinatorial Optimization (Polyhedra )

Category 2: Linear, Cone and Semidefinite Programming (Linear Programming )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Technical Report, Carnegie Mellon University, May 2019.

Download: [PDF]

Entry Submitted: 05/15/2019
Entry Accepted: 05/15/2019
Entry Last Modified: 05/15/2019

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