A Simplicial Algorithm for Computing an Integer Zero Point of a Mapping with the Direction Preserving Property (Journal of Computational Mathematics, 2021)
A mapping $f:Z^n→ R^n$ is said to possess the direction preserving property if $f_i(x)\gt 0$ implies $f_i(y)≥ 0$ for any integer points $x$ and $y$ with $\|x-y\|_{∞}≤ 1$. 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 $c≤ x^0≤ d$ satisfying that $\max_{1≤ i≤ n}(x_i-x^0_i)f_i(x)\ge0$ for any integer point $x$ with $f(x) eq 0$ on the boundary of $H=\{x\in R^n\;|\;c-e≤ x≤ d+e\}$, where $c$ and $d$ are two finite integer points with $c≤ d$ and $e=(1,1,…,1)^{\top}\in R^n$. 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.
Keywords: Integer Zero Point; Direction Preserving; Simplicial Algorithm; Triangulation; Existence.