Finding the Least Element of a Nonnegative Solution Set of a Class of Polynomial Inequalities

Abstract

In this paper, we consider the least element problem of a nonnegative solution set of a system of inequalities defined by a homogeneous polynomial mapping and a vector. In the set under consideration, the homogeneous polynomial mapping is defined by a tensor. When the tensor involved is square, the set under consideration is just the feasible region of the tensor complementarity problem (TCP). We first introduce the concept of the generalized -tensor, which is a natural generalization of the -tensor. Then, under the assumption that the considered set is nonempty and the tensor involved is a generalized -tensor, we propose an iterative method for finding the least element of the considered set. Specifically, by recognizing the position indices of positive components in the least element continuously, we solve a series of lower-dimensional system of tensor equations corresponding to these indices and prove that the least element of the set can be obtained within finite step iterations. When the tensor involved is square, the least element obtained is also a solution of the TCP. Compared with the existing methods for finding the least-element solution of the TCP with a -tensor, our method does not require any additional assumptions and has lower computational cost. Preliminary numerical experiments show that the proposed method is effective.