Necessary and sufficient conditions for a subclass of $P$-tensor

TL;DR

This paper introduces the class of B-Nekrasov tensors within tensor complementarity problems, establishing their properties, relationships with other tensor classes, and providing necessary and sufficient conditions for their characterization.

Contribution

It defines B-Nekrasov tensors, explores their properties, and establishes their relationship with P-tensors and Nekrasov Z-tensors, including a necessary and sufficient condition for their characterization.

Findings

B-Nekrasov tensors contain Nekrasov Z-tensors with positive diagonals.

A necessary and sufficient condition for B-Nekrasov tensors is provided.

The class of P-tensors includes B-Nekrasov tensors.

Abstract

In this article, we introduce the class $B$-Nekrasov tensor in the context of tensor complementarity problem. We study some tensor theoretic properties. We show that the class of B-Nekrasov tensor contains the class of Nekrasov $Z$-tensor with positive diagonal entries. We present a necessary and sufficient condition for a $B$-Nekrasov tensor. We show that the class of $P$-tensor contains the class of $B$-Nekrasov tensor. Keywords: Tensor complementarity problem, $P$-tensors, Nekrasov tensors, Nekrasov $Z$-tensor, $B$-Nekrasov tensors, Nonsingular $H$-tensor, Diagonally dominant tensor.