Linear Complementarity and the Degree of Mappings

Abstract

Let M be an n × n real matrix and q an n-vector. The problem: Find n-vectors x and y such that 1a × − My = q 1b × > 0, y > 0 1c Either xi = 0 or yi = 0 for 1 < i < n is called the linear complementarity problem. As is explained in [C-D], several significant mathematical programming problems can be formulated as linear complementarity problems. For that reason, linear complementarity has been the subject of a considerable literature (see [K], [M], [L2], [G] and the papers cited there). For the most part in this literature the problem is treated from the algorithmic point of view, with the specification of procedures for solving the problem under various assumptions on the matrix M as a major goal. An exception to this rule is the paper [E-S] of Eaves and Scarf. In that paper, a general class of algorithms is discussed from a geometric point of view, and the linear complementary problem is given a geometric interpretation, making it amenable, for certain M, to the general methods of the paper.The pupose of the present paper is to pursue the investigation of linear complementary from a geometric point of view*.