On Semimonotone Star Matrices and Linear Complementarity Problem

TL;DR

This paper introduces the class of semimonotone star ($E_0^s$) matrices, explores their properties in complementarity problems, and proposes an interior point algorithm for solving LCPs with these matrices.

Contribution

It defines the $E_0^s$ and $ ilde{E_0^s}$ matrix classes, analyzes their properties, and develops an interior point method for LCPs involving these matrices.

Findings

$ ilde{E_0^s}$-matrices are in $P_0$ and in $E_0^f$

LCP$(q, A)$ is processable by Lemke's algorithm if $A otin ilde{E_0^s}$ but in $P_0$

Proposed an interior point algorithm for solving LCP$(q, A)$ with $A otin ilde{E_0^s}$

Abstract

In this article, we introduce the class of semimonotone star ($E_0^s$) matrices. We establish the importance of the class of $E_0^s$-matrices in the context of complementarity theory. We show that the principal pivot transform of $E_0^s$-matrix is not necessarily $E_0^s$ in general. However, we prove that $\tilde{E_0^s}$-matrices, a subclass of the $E_0^s$-matrices with some additional conditions, is in $E_0^f$ by showing this class is in $P_0.$ We prove that LCP$(q, A)$ can be processable by Lemke's algorithm if $A\in \tilde{E_0^s}\cap P_0.$ We find some conditions for which the solution set of LCP$(q, A)$ is bounded and stable under the $\tilde{E^s_0}$-property. We propose an algorithm based on an interior point method to solve LCP$(q, A)$ given $A \in \tilde{E^{s}_{0}}.$