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Projected Splitting Methods for Vertical Linear Complementarity Problems

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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 \(2n\times n\), and we then generalize the results to VLCPs of an arbitrary dimension \(\ell n\times n\). 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.

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Data Availability Statement

The data generated and analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors desire to thank the anonymous referees for their helpful comments and remarks.

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Authors and Affiliations

  1. Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Via P. Vivarelli 10/1, Building 26, 41125, Modena, Italy

    Francesco Mezzadri & Emanuele Galligani

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  1. Francesco Mezzadri
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  2. Emanuele Galligani
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Correspondence to Emanuele Galligani.

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Communicated by Qamrul Hasan Ansari.

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Mezzadri, F., Galligani, E. Projected Splitting Methods for Vertical Linear Complementarity Problems. J Optim Theory Appl 193, 598–620 (2022). https://doi.org/10.1007/s10957-021-01922-y

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