Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors

2014/11/29 by Song, Yisheng, Qi, Liqun · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1412.0113

Abstract

This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor A such that the tensor complementarity problem (\q, A): finding \x ∈ ℝⁿ such that \x ≥ \0, \q + A\x^m-1 ≥ \0, and \x^\top (\q + A\x^m-1) = 0, has a solution for each vector \q ∈ ℝⁿ. Several subclasses of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors and semi-positive R₀-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its principal diagonal entries are positive, and so the equivalence of Q-tensor, R-tensors, strictly semi-positive tensors is showed if they are nonnegative tensors. We also show that a tensor is a R₀-tensor if and only if the tensor complementarity problem (\0, A) has no non-zero vector solution, and a tensor is a R-tensor if and only if it is a R₀-tensor and the tensor complementarity problem (\e, A) has no non-zero vector solution, where \e=(1,1⋯,1)^\top.

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