A Simplicial Algorithm for Computing an Integer Zero Point of a Mapping with the Direction Preserving Property

A mapping f : Z → R is said to possess the direction preserving property if fi(x) > 0 implies fi(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 with c ≤ x ≤ d satisfying that max1≤i≤n(xi − x 0 i )fi(x) > 0 for any integer point x with f(x) 6= 0 on the boundary of H = {x ∈ R | c − e ≤ x ≤ d + e}, where c and d are two finite integer points with c ≤ d and e = (1, 1, · · · , 1) ∈ R. 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, 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. Mathematics subject classification: 90C49.