Some properties of the solution of the vertical tensor complementarity problem

TL;DR

This paper investigates the existence, uniqueness, and boundedness of solutions for the vertical tensor complementarity problem, introducing new tensor classes and applying degree theory to establish key properties.

Contribution

It introduces the vertical tensor complementarity problem, defines special tensor sets, and provides conditions for solution existence and uniqueness using degree theory.

Findings

Solution set is bounded under certain conditions

Sufficient conditions for existence of solutions

Conditions for uniqueness of solutions

Abstract

In this paper, we mainly focus on the existence and uniqueness of the vertical tensor complementarity problem. Firstly, combining the generalized-order linear complementarity problem with the tensor complementarity problem, the vertical tensor complementarity problem is introduced. Secondly, we define some sets of special tensors, and illustrate the inclusion relationships. Finally, we show that the solution set of the vertical tensor complementarity problem is bounded under certain conditions, and some sufficient conditions for the existence and uniqueness of the solution of the vertical tensor complementarity problem are obtained from the view of the degree theory and the equal form of the minimum function.