Properties of Solution set of Tensor Complementarity Problem

TL;DR

This paper investigates the properties of the solution set of the tensor complementarity problem, establishing conditions for feasibility, boundedness, and bounds related to eigenvalues, thus advancing understanding of structured nonlinear complementarity problems.

Contribution

It proves that a tensor is an S-tensor iff the tensor complementarity problem is feasible and relates boundedness to solution uniqueness, introducing bounds connected to eigenvalues.

Findings

A tensor is an S-tensor iff the tensor complementarity problem is feasible.

Boundedness of solutions is equivalent to solution uniqueness with zero vector.

Global upper bounds are established for problems with strictly semi-positive tensors.

Abstract

The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved that a tensor is an S-tensor if and only if the tensor complementarity problem is feasible, and each Q-tensor is an S-tensor. Furthermore, the boundedness of solution set of the tensor complementarity problem is equivalent to the uniqueness of solution for such a problem with zero vector. For the tensor complementarity problem with a strictly semi-positive tensor, we proved the global upper bounds for solution of such a problem. In particular, the upper bounds keep in close contact with the smallest Pareto $H-$($Z-$)eigenvalue.