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| PROPERTIES OF TENSOR COMPLEMENTARITY PROBLEM AND SOME CLASSES OF STRUCTURED TENSORS |
| Yisheng Song,Liqun Qi |
| (School of Math. and Information Science, Henan Normal University, Xinxiang 453007, Henan, PR China;Dept. of Applied Math., The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China) |
| DOI： |
| 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 an x∈Rn such that x≥ 0, q + Axm-1≥0, and xT(q + Axm-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 was showed if they are nonnegative tensors. We also show that a tensor is an 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 an R0-tensor and the tensor complementarity problem (e, A) has no non-zero vector solution, where e=(1,1\cdots,1)T. |
| Key words: Q-tensor; R-tensor; R0-tensor; strictly semi-positive; tensor complementarity problem |