Solving discrete systems of nonlinear equations

Abstract

In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z n of the n-dimensional Euclidean space IR n . It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Z n and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the 'continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

Projection and convolution operations for integrally convex functions

“…The concept of integral convexity found applications, e.g., in economics and game theory. It is used in formulating discrete fixed point theorems [5,6,18] and designing solution algorithms for discrete systems of nonlinear equations [8,17]. In game theory integral concavity of payoff functions guarantees the existence of a pure strategy equilibrium in finite symmetric games [7].…”

Directed discrete midpoint convexity

“…The class of integrally convex functions establishes a general framework of discrete convex functions, including separable convex, L ♮ -convex, M ♮ -convex, L ♮ 2 -convex, M ♮ 2 -convex functions [21], BS-convex and UJ-convex functions [4], and globally/locally discrete midpoint convex functions [18]. The concept of integral convexity is used in formulating discrete fixed point theorems [8,9,29], designing algorithms for discrete systems of nonlinear equations [14,28], and guaranteeing the existence of a pure strategy equilibrium in finite symmetric games [10].…”

On Basic Operations Related to Network Induction of Discrete Convex Functions

“…Jump M-and M ♮ -convex functions find applications in several fields including matching theory [3,20,21,52] and algebra [5]. Integrally convex functions are used in formulating discrete fixed point theorems [14,15,58], and designing solution algorithms for discrete systems of nonlinear equations [23,57]. In game theory the integral concavity of payoff functions guarantees the existence of a pure strategy equilibrium in finite symmetric games [16].…”