Combinatorial Characterizations of K-matrices

TL;DR

This paper extends combinatorial characterizations of K-matrices to oriented matroids, simplifies existing proofs, and demonstrates rapid convergence of a principal pivot method for related linear complementarity problems.

Contribution

It generalizes Fiedler and Ptak's theorem to oriented matroids and provides a simplified, elementary proof using duality, with applications to algorithmic convergence.

Findings

Elementary proof leveraging oriented matroid duality

Extension of K-matrix characterization to oriented matroids

Rapid convergence of principal pivot method for K-matrix LCPs

Abstract

We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies the original proof substantially by exploiting the duality of oriented matroids. As an application, we show that a simple principal pivot method applied to the linear complementarity problems with K-matrices converges very quickly, by a purely combinatorial argument.