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

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).