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Projected Splitting Methods for Vertical Linear Complementarity Problems

Author

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  • Francesco Mezzadri

    (University of Modena and Reggio Emilia)

  • Emanuele Galligani

    (University of Modena and Reggio Emilia)

Abstract

In this paper, we generalize the projected Jacobi and the projected Gauss–Seidel methods to vertical linear complementarity problems (VLCPs) characterized by matrices with positive diagonal entries. First, we formulate the methods and show that the subproblems that must be solved at each iteration have an explicit solution, which is easy to compute. Then, we prove the convergence of the proposed procedures when the matrices of the problem satisfy some assumptions of strict or irreducible diagonal dominance. In this context, for simplicity, we first analyze the convergence in the special case of VLCPs of dimension $$2n\times n$$ 2 n × n , and we then generalize the results to VLCPs of an arbitrary dimension $$\ell n\times n$$ ℓ n × n . Finally, we provide several numerical experiments (involving both full and sparse matrices) that show the effectiveness of the proposed approaches. In this context, our methods are compared with existing solution methods for VLCPs. A parallel implementation of the projected Jacobi method in CUDA is also presented and analyzed.

Suggested Citation

  • Francesco Mezzadri & Emanuele Galligani, 2022. "Projected Splitting Methods for Vertical Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 598-620, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01922-y
    DOI: 10.1007/s10957-021-01922-y
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    References listed on IDEAS

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    1. Nagae, Takeshi & Akamatsu, Takashi, 2008. "A generalized complementarity approach to solving real option problems," Journal of Economic Dynamics and Control, Elsevier, vol. 32(6), pages 1754-1779, June.
    2. Francesco Mezzadri & Emanuele Galligani, 2019. "Splitting Methods for a Class of Horizontal Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 500-517, February.
    3. Edalatpour, Vahid & Hezari, Davod & Khojasteh Salkuyeh, Davod, 2017. "A generalization of the Gauss–Seidel iteration method for solving absolute value equations," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 156-167.
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    Cited by:

    1. Ting Zhang & Yong Wang & Zheng-Hai Huang, 2024. "Projected fixed-point method for vertical tensor complementarity problems," Computational Optimization and Applications, Springer, vol. 89(1), pages 219-245, September.
    2. Ali, Rashid & Akgul, Ali, 2024. "A new matrix splitting generalized iteration method for linear complementarity problems," Applied Mathematics and Computation, Elsevier, vol. 464(C).

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