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A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors

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Abstract

In this paper, we consider solving the tensor complementarity problem (TCP). We first introduce the concept of the implicit Z-tensor, which is a generalization of Z-tensor. Then, based on a new fixed point reformulation of the TCP, we design an iterative algorithm for solving the TCP with an implicit Z-tensor under the assumption that the feasible set of the problem involved is nonempty. We prove that the proposed fixed point iterative method converges monotonically downward to a solution of the TCP. Furthermore, we establish the global linear rate of convergence of the proposed method under some reasonable assumptions. Compared with the existing related studies, the proposed method not only solves a wider range of TCPs, but also has a lower computational cost. The numerical results verify our theoretical findings.

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Notes

  1. To ensure that all the monomial terms make sense, the index i is required to satisfy \(i+2\le n\) for \(i\in \{1,4,7,\ldots \}\) and \(i+1\le n\) for \(i\in \{2,5,8,\ldots \}\)

  2. Specially, \(t\in [\delta _1,\delta _2]\) reduces to \(t\ge \delta _1\) when \({\hat{a}}_{ii_2\cdots i_m}=0\) for any \(i\in \varOmega ^c\) and \(i_2,\ldots , i_m\in \varOmega \). In this case, the feasibility of i-th inequality \((\hat{{\mathscr {A}}}\bar{{\textbf{x}}}^{m-1}+{\textbf{q}})_i\ge 0\) holds automatically with \(\bar{{\textbf{x}}}\) satisfying \(\bar{{\textbf{x}}}_{\varOmega ^c}={\textbf{0}}\).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300350, People’s Republic of China

    Zheng-Hai Huang & Yong Wang

  2. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, People’s Republic of China

    Yu-Fan Li

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  1. Zheng-Hai Huang
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  2. Yu-Fan Li
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  3. Yong Wang
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Correspondence to Yu-Fan Li.

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This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11871051, 11901600 and 12171357) and the Science and Technology Program of Guangzhou (Grant No. 202002030280).

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Huang, ZH., Li, YF. & Wang, Y. A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors. J Glob Optim 86, 495–520 (2023). https://doi.org/10.1007/s10898-022-01263-8

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