A mapping f : Z^n → Rn 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 deve...A mapping f : Z^n → Rn 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^0 with c ≤ x^0≤d satisfying that maxl≤i≤(xi - xi^0)fi(x) 〉 0 for any integer point x with f(x) ≠ 0 on the boundary of H = {x ∈R^n [c-e ≤ x〈d+e},wherecanddaretwo finite integer points with c 〈 d and e = (1, 1,... , 1)^T E 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.展开更多
文摘A mapping f : Z^n → Rn 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^0 with c ≤ x^0≤d satisfying that maxl≤i≤(xi - xi^0)fi(x) 〉 0 for any integer point x with f(x) ≠ 0 on the boundary of H = {x ∈R^n [c-e ≤ x〈d+e},wherecanddaretwo finite integer points with c 〈 d and e = (1, 1,... , 1)^T E 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.