Relations Between PL Maps, Complementary Cones, and Degree in Linear Complementarity Problems

Abstract

Let M be an n × n real matrix and q an n-vector. The problem

$${\text{find z }} \in {{\text{R}}^{\text{n}}}{\text{ and w}} \in {{\text{R}}^{\text{n}}}{\text{such that}}$$

(I, -M, q) w = q + Mz, w ≥ 0, z ≥ 0, and

$${{\rm{w}}^{\rm{T}}}{\rm{z = 0}}\left( {{\rm{i}}{\rm{.e}}{\rm{., }}{{\rm{w}}_{\rm{i}}}{\rm{ = 0 or }} {{\rm{z}}_{\rm{i}}}{\rm{ = 0 for i = l, }}...{\rm{, n}}} \right)$$

is called the linear complementarity problem (LCP). This problem is a canonical form for a variety of significant problems in mathematical progamming, economics and enginerring (e.g., references [2], [6], [9], [18–20], [24], [32]) and there is a wealth of literature on the problem (see, for example, references [3–5], [7], [10–17], [21–23], [25–28], [30–31], [33–43] and papers cited therein).

The authors are indebted to R. W. Cottle and R. E. Stone for insightful discussions and comments on an earlier version of this paper.

The research of this author was partially supported by NSF Grant #MCS 77–15509.

The research of this author was partially supported by NSF Grant #ECS 79–20177 and by the Centre de Recherche de Mathematiques de la Decision of the University of Paris IX. This author also acknowledges with gratitude several fundamental and technical insights provided by T. Dittmer.