Linear complementarity and oriented matroids

TL;DR: Linear complementary problems (LCP) were considered in this article, where the authors fixed their notations and considered the solvability of linear complementarity problems with respect to the special properties of the coefficient matrix M.

TL;DR: The finiteness proof of the generalize new criss-cross type algorithms for linear complementarity problems (LCPs) given with sufficient matrices gives a new, constructive proof to Fukuda and Terlaky's LCP duality theorem.

TL;DR: In this article, it was shown that the dual linear complementarity problem can be solved in polynomial time if the matrix is row sufficient, and that all feasible solutions are complementary.

TL;DR: In this paper, the authors prove the existence of a sequence of |B∖B| admissible pivots from any basis B (not necessarily feasible) to the unique optimal basis B, if the given LP has an optimal solution and is fully nondegenerate.

TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.

Abstract: The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations. Mathematical objects of study that appear to be disjoint and independent, such as point and vector configurations, arrangements of hyperplanes, convex polytopes, directed graphs, and linear programs find a common generalization in the language of oriented matroids. The oriented matroid of a finite set of points P extracts relative position and orientation information from the configuration; for example, it can be given by a list of signs that encodes the orientations of all the bases of P . In the passage from a concrete point configuration to its oriented matroid, metrical information is lost, but many structural properties of P have their counterparts at the—purely combinatorial—level of the oriented matroid. (In computational geometry, the oriented matroid data of an unlabelled point configuration are sometimes called the order type.) From the oriented matroid of a configuration of points, one can compute not only that face lattice of the convex hull, but also the set of all its triangulations and subdivisions (cf. Chapter 16). We first introduce oriented matroids in the context of several models and motivations (Section 6.1). Then we present some equivalent axiomatizations (Section 6.2). Finally, we discuss concepts that play central roles in the theory of oriented matroids (Section 6.3), among them duality, realizability, the study of simplicial cells, and the treatment of convexity.

TL;DR: The role of problems of the form w and z satisfying w = q + Mz, w = or 0, z = or0, zw = 0 play a fundamental role in mathematical programming.

TL;DR: In this paper it is shown that every coordinatization (representation) of a matroid over an ordered field induces an orientation of the matroid, and that every unimodular matroid has an orientation that is induced by a coordinatisation and is unique in a certain straightforward sense.