Projected Jacobi Method for Vertical Tensor Complementarity Problems

Zheng-Hai Huang (), Qiumei Zhao () and Yong Wang ()
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Zheng-Hai Huang: Tianjin University
Qiumei Zhao: Tianjin University
Yong Wang: Tianjin University

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 2, No 9, 26 pages

Abstract: Abstract As a generalization of the vertical linear complementarity problem, the vertical tensor complementarity problem (VTCP) has been recently studied . In this paper, we consider a class of VTCPs in which the involved tensors are vertical block implicitly strictly diagonally dominant by mode-1 with positive diagonal entries. By using degree theory, we show that there is at least one solution to this class of VTCPs. Moreover, we also provide a sufficient condition under which the VTCP under consideration has a unique solution. In particular, based on an equivalent reformulation of this class of VTCPs, we propose a projected Jacobi method for solving this class of VTCPs, and discuss the global linear convergence of the method under an additional condition. The proposed method is a generalization of the projected Jacobi method for the vertical linear complementarity problem proposed in the recent literature. The computations required at each iteration of the method are simple. Preliminary numerical experiments show the effectiveness of the method.

Keywords: Vertical tensor complementarity problem; Vertical linear complementarity problem; Projected splitting method; Degree theory; 65K05; 65H10; 90C33 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02699-0

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