Column competent tensors and tensor complementarity problem

TL;DR

This paper introduces the concept of column competent tensors to extend the idea of column competent matrices, analyzing their properties and applications in tensor complementarity problems to understand solution uniqueness.

Contribution

It defines column competent tensors, explores their properties, and applies them to tensor complementarity problems, extending the theory from matrices to higher-order tensors.

Findings

Inheritance property of column competent tensors

Invariance property of column competent tensors

Application to tensor complementarity problem with supporting examples

Abstract

In multilinear algebra, some special classes of matrices are extended to higher order structured tensors. The local $w$-uniqueness solution to the linear complementarity problem can be identified by the column competent matrix. Motivated by this $w$-uniqueness property, we introduce column competent tensor in the context of tensor complementarity problem. We consider some important properties. In the theory of linear complementarity problem, column competent matrices are introduced to study local $w$-uniqueness property of LCP solution. We present the inheritance property and invariance property of column competent tensors. We study the tensor complementarity problem using column competent tensors and several results are established. Some examples are illustrated to support the results. Keywords: Tensor complementarity problem, column competent tensor, nondegenerate tensor, $\omega$-solution. AMS subject classifications: 90C33, 90C30, 15A69, 46G25.