The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

TL;DR

This paper investigates the weighted horizontal linear complementarity problem within Euclidean Jordan algebras, establishing existence and uniqueness results, and connecting these to interior point systems.

Contribution

It extends the theory of complementarity problems to Euclidean Jordan algebras, introducing new properties and solvability conditions for the weighted horizontal linear complementarity problem.

Findings

Established existence results under topological degree conditions.

Proved uniqueness in the Euclidean space setting.

Connected the problem framework to interior point systems.

Abstract

A weighted complementarity problem (wCP) is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem (wHLCP) in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of R_0, R, and P properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of R^n. We show how our results naturally lead to interior point systems.