Characterization of Q-Matrices Using Bordered Matrix Algorithm

Abstract

Given a matrix $M \in \mathscr {R}^{n\times n}$ and a vector $q \in \mathscr {R}^{n}$, the Linear Complementarity Problem LCP(q, M) is to find a solution (w, z) that satisfies $w - Mz = q, w,z \ge 0$ and $w^{T}z = 0$. This article introduces a variant of Lemke’s algorithm called Bordered Matrix Algorithm (BMA) that aims to process the linear complementarity problem, LCP($q,M_{n \times n}$) where $M = \begin{bmatrix} A &{} v\\ u^{T} &{} \alpha \end{bmatrix}$, by applying Lemke’s algorithm to the parametric LCP($\overline{q}+\theta v$,A) where v is the artificial vector and $\overline{q}$ is the vector of the first $(n-1)$ components of q. The algorithmic properties of BMA are used to study the relationship between A and M toward characterization of Q-matrices.