A SIMPLICIAL ALGORITHM FOR COMPUTING AN INTEGER ZERO POINT OF A MAPPING WITH THE DIRECTION PRESERVING PROPERTY

A mapping $f:Z^n\rightarrow R^n$ is said to possess the direction preserving property if $f_i(x)\gt 0$ implies $f_i(y)\geq 0$ for any integer points $x$ and $y$ with $\|x-y\|_{\infty}\leq 1$. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point $x^0$ with $c\leq x^0\leq d$ satisfying that $\max_{1\leq i\leq n}(x_i-x^0_i)f_i(x)\ge0$ for any integer point $x$ with $f(x)\neq 0$ on the boundary of $H=\{x\in R^n|c-e\leq x\leq d+e\}$, where $c$ and $d$ are two finite integer points with $c\leq d$ and $e=(1,1,\cdots,1)^{\top}\in R^n$. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. (2004). Under this assumption, starting at $x^0$, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities.