Generic properties of the complementarity problem

Given f : R + n → R n , the complementarity problem is to find a solution to x ≥ 0, f(x) ≥ 0, and 〈 x, f(x) 〉 = 0. Under the condition that f is continuously differentiable, we prove that for a generic set of such an f , the problem has a discrete solution set. Also, under a set of generic nondegeneracy conditions and a condition that implies existence, we prove that the problem has an odd number of solutions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47915/1/10107_2005_Article_BF01584674.pd

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

A finite characterization of K -matrices in dimensions less than four

The class of real n × n matrices M , known as K -matrices, for which the linear complementarity problem w − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution whenever w − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensions n <4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining to K -matrices are also given. A finite characterization of completely K -matrices ( K -matrices all of whose principal submatrices are also K -matrices) is proved for dimensions <4.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47913/1/10107_2005_Article_BF01589438.pd

Combinatorial integer labeling theorems on finite sets with applications

Tucker’s well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, · · · , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1, ±2, · · · , ±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided