Polynomial complementarity problems

TL;DR

This paper investigates polynomial complementarity problems, establishing conditions for solution existence and connecting them to tensor complementarity problems, thereby extending and improving existing results in the field.

Contribution

It introduces new theoretical links between polynomial and tensor complementarity problems, including solution existence conditions and the concept of tensor degree.

Findings

Solution sets are nonempty and compact under certain conditions.

Established connections between polynomial and tensor complementarity problems.

Introduced the concept of degree for R_0-tensors.

Abstract

Given a polynomial map f on the Euclidean n-space and a vector q, the polynomial complementarity problem, PCP(f,q), is the nonlinear complementarity problem of finding a nonnegative vector x such that y=f(x)+q is nonnegative and orthogonal to x. It is called a tensor complementarity problem if the polynomial map is homogeneous. In this paper, we establish results connecting the polynomial complementarity problem PCP(f,q) and the tensor complementarity problem PCP(f*,0), where f* is the leading term in the decomposition of f as a sum of homogeneous polynomial maps. We show, for example, that PCP(f,q) has a nonempty compact solution set for every q when zero is the only solution of PCP(f*,0)and the local (topological) degree of min{x,f*(x)} at the origin is nonzero. As a consequence, we establish Karamardian type results for polynomial complementarity problems. By identifying a tensor A of order m and dimension n with its corresponding homogeneous polynomial F(x):= Ax^{m-1}, we relate our results to tensor complementarity problems. These results show that under appropriate conditions, PCP(F+P,q) has a nonempty compact solution set for all polynomial maps P of degree less than m-1 and for all vectors q, thereby substantially improving the existing tensor complementarity results where only problems of the type PCP(F,q) are considered. We introduce the concept of degree of an R_0-tensor and show that the degree of an R-tensor is one. We illustrate our results by constructing matrix based tensors.