Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems

Abstract

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, B-tensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP; and hence, they are very easy to calculate.

Key words： School of Mathematics; Tianjin University; Tianjin 300350; China

References

1 Bai X L, Huang Z H, Li X. Stability of solutions and continuity of solution maps of tensor complementarity problems. Asia-Pac J Oper Res, 2019, 36(2): 1940002 (19pp)

2 Bai X L, Huang Z H, Wang Y. Global uniqueness and solvability for tensor complementarity problems. J Optim Theory Appl, 2016, 170(1): 72–84

3 Balaji R, Palpandi K. Positive definite and Gram tensor complementarity problems. Optim Lett, 2018, 12(3): 639–648

4 Che M L, Qi L, Wei Y M. Positive-definite tensors to nonlinear complementarity problems. J Optim Theory Appl, 2016, 168(2): 475–487

5 Chen H, Qi L, Song Y. Column sufficient tensors and tensor complementarity problems. Front Math China, 2018, 13(2): 255–276

6 Ding W, Luo Z, Qi L. P-tensors, P₀-tensors, and their applications. Linear Algebra Appl, 2018, 555: 336–354

7 Du S, Zhang L. A mixed integer programming approach to the tensor complementarity problem. J Global Optim, 2019, 73(4): 789–800

8 Gowda MS. Polynomial complementarity problems. Pac J Optim, 2017, 13(2): 227–241

9 Gowda M S, Luo Z, Qi L, Xiu N H. Z-tensors and complementarity problems. arXiv: 1510.07933v2

10 Guo Q, Zheng M M, Huang Z H. Properties of S-tensors. Linear Multilinear Algebra, 2019, 67(4): 685–696

11 Han L X. A continuation method for tensor complementarity problems. J Optim Theory Appl, 2019, 180(3): 949–963

12 Hieu Vu T. On the R₀-tensors and the solution map of tensor complementarity problems. J Optim Theory Appl, 2019, 181(1): 163–183

13 Hu S, Huang Z H, Wang J. Error bounds for the solution set of quadratic complementarity problems. J Optim Theory Appl, 2018, 179(3): 983–1000

14 Huang Z H, Qi L. Formulating an n-person noncooperative game as a tensor complementarity problem. Comput Optim Appl, 2017, 66(3): 557–576

15 Huang Z H, Suo Y Y, Wang J. On Q-tensors. arXiv: 1509.03088v1

16 Ling L, He H, Ling C. On error bounds of polynomial complementarity problems with structured tensors. Optimization, 2018, 67(2): 341–358

17 Liu D, Li W, Vong S W. Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear Algebra, 2018, 66(9): 1726–1749

18 Luo Z, Qi L, Xiu N. The sparsest solutions to Z-tensor complementarity problems. Optim Lett, 2017, 11(3): 471–482

19 Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439(1): 228–238

20 Song Y, Mei W. Structural properties of tensors and complementarity problems. J Optim Theory Appl, 2018, 176(2): 289–305

21 Song Y, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165(3): 854–873

22 Song Y, Qi L. Tensor complementarity problem and semi-positive tensors. J Optim Theory Appl, 2016, 169(3): 1069–1078

23 Song Y, Qi L. Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim Lett, 2017, 11(7): 1407–1426

24 Song Y, Qi L. Properties of tensor complementarity problem and some classes of structured tensors. Ann Appl Math, 2017, 33(3): 308–323

25 Song Y, Yu G. Properties of solution set of tensor complementarity problem. J Optim Theory Appl, 2016, 170(1): 85–96

26 Tawhid M A, Rahmati, S. Complementarity problems over a hypermatrix (tensor) set. Optim Lett, 2018, 12(6): 1443–1454

27 Wang J, Hu S, Huang Z H. Solution sets of quadratic complementarity problems. J Optim Theory Appl, 2018, 176(1): 120–136

28 Wang X, Chen H, Wang Y. Solution structures of tensor complementarity problem. Front Math China, 2018, 13(4): 935–945

29 Wang Y, Huang Z H, Bai X L. Exceptionally regular tensors and tensor complementarity problems. Optim Methods Softw, 2016, 31(4): 815–828

30 Wang Y, Huang Z H, Qi L. Global uniqueness and solvability of tensor variational inequalities. J Optim Theory Appl, 2018, 177(1): 137–152

31 Xie S L, Li D H, Xu H R. An iterative method for finding the least solution to the tensor complementarity problem. J Optim Theory Appl, 2017, 175(1): 119–136

32 Zhang K, Chen H, Zhao P. A potential reduction method for tensor complementarity problems. J Ind Manag Optim, 2019, 15(2): 429–443

33 Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35(2): 437–452

34 Zhao X, Fan J. A semidefinite method for tensor complementarity problems. Optim Methods Softw, 2018,

35 Zheng M M, Zhang Y, Huang Z H. Global error bounds for the tensor complementarity problem with a P-tensor. J Ind Manag Optim, 2019, 15(2): 933–946