Z-tensors and complementarity problems

TL;DR

This paper investigates the properties of Z-tensors in relation to homogeneous nonlinear complementarity problems, establishing conditions for solvability and uniqueness, and highlighting differences between global and unique solutions.

Contribution

It introduces new conditions for the global and unique solvability of complementarity problems induced by Z-tensors, based on degree-theoretic methods.

Findings

Z-tensors have specific equivalent conditions for global solvability.

Global solvability does not necessarily imply unique solvability.

A sufficient condition for unique solvability is provided.

Abstract

Tensors are multidimensional analogs of matrices. In this paper, based on degree-theoretic ideas, we study homogeneous nonlinear complementarity problems induced by tensors. By specializing this to $Z$-tensors (which are tensors with non-positive off-diagonal entries), we describe various equivalent conditions for a $Z$-tensor to have the global solvability property. We show by an example that the global solvability need not imply unique solvability and provide a sufficient and easily checkable condition for unique solvability.