A Geometric Problem in Simplicial Cones with Applications to Linear Complementarity Problems

Abstract

We consider the following geometric question: suppose we are given a simplicial cone K in R^n. Can we find a point @) in the interior of K satisfying the property that the orthogonal projection of @) onto the linear hull of every face of K is in the relative interior of that fence? This question plays an important role in determining whether a certain class of linear complementarity problems (LCP 's) can be solved efficiently by a pivotal algorithm. The answer to this question is always in the affirmative if n=2, but not so for n=3. We establish some conditions for the answer to this question to be yes, and relate them to other well known properties of square matrices. e.g., world: simplicial cones, orthogonal projections, faces, linear complementarity problem, LCP, pivotal algorithms, P-matrices, symmetric positive definite matrices, 2-matrices, M-matrices.