A Vector Labeling Method for Solving Discrete Zero Point and Complementarity Problems

Abstract

In this paper we establish the existence of a discrete zero point of a function from the n‐dimensional integer lattice $\mathbb{Z}^n$ to the n‐dimensional Euclidean space $\mathbb{R}^n$ under very general conditions with respect to the behavior of the function. The proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps. The algorithm provides an efficient method to find an exact solution. We also discuss how to adapt the algorithm for two related problems, namely, to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem. In both cases the modified algorithm provides a constructive existence proof, too. We further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well‐known Lemke’s method to nonlinear environments. An economic application is also presented.