Abstract
In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.
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Acknowledgements
The authors would like to thank the editors and anonymous referees for their valuable suggestions which helped us to improve the paper. This paper was supported by the Chinese NSF Grants 11371154 and 11601188, by the Training Program for Outstanding Young Teachers in Guangdong Province (Grant No. 20140202), and by Educational Commission of Guangdong Province, China (Grant No. 2014KQNCX210).
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Communicated by Guoyin Li.
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Xie, SL., Li, DH. & Xu, HR. An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem. J Optim Theory Appl 175, 119–136 (2017). https://doi.org/10.1007/s10957-017-1157-5
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DOI: https://doi.org/10.1007/s10957-017-1157-5