On Column sufficiency and Extended Horizontal Linear Complementarity Problem
Punit Kumar Yadav, K. Palpandi
TL;DR
The article addresses extending the column-sufficient concept to a family of matrices by introducing the column sufficient-W (cS-W) property for an ordered set ${\bf C}=(C_0,\dots,C_k)$ and studying the Extended Horizontal Linear Complementarity Problem (EHLCP). It proves that when ${\bf C}$ has the cS-W property, the solution set $\mathrm{SOL}({\bf C},{\bf d},q)$ is convex for all $q$ and all strictly positive ${\bf d}$, and it establishes a uniqueness result for $q>0$ provided $C_0$ is an $M$-matrix. The paper further analyzes the relationships between the column-$W$, column-$W_0$, and column-ND-$W$ properties, showing that the column-$W$ property implies ND-$W$ and exploring the limits of these implications, with additional results for a cone variant in the $Z$-matrix setting. Overall, these contributions advance the theoretical understanding of the EHLCP solution structure and provide conditions under which the problem is well-posed and its solution set tractable for applications in optimization and equilibrium analysis.
Abstract
In this article, we introduce the concept of the column-sufficient W-property for a set of matrices and prove the convexity of the solution set for the Extended Horizontal Linear Complementarity Problem. Additionally, we present an uniqueness result and establish several results related to the column-sufficient W-property.