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Splitting Methods for a Class of Horizontal Linear Complementarity Problems

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Abstract

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss–Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations.

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Notes

  1. It is easy to prove that \(\max \{0, w_i^*/a_{ii}\}\) and \(\max \{0, -w_i^*/b_{ii}\}\) are the solution \(x_i^*, y_i^*\) of the HLCP(\(A,B,{\varvec{c}}\)), \(i = 1, \ldots , n\). Indeed, replacing in the i-th row of \(A{\varvec{x}} - B {\varvec{y}} = {\varvec{c}}\), we have \(a_{ii}x_i^* - b_{ii}y_i^*=\max \{0, w_i^*\} - \max \{0, -w_i^*\}=w_i^*\). By the positivity of \(a_{ii}\) and \(b_{ii}\), we then have the nonnegativity of \(x_i\) and \(y_i\) and that \(x_i\) is positive when \(y_i=0\) (and vice versa).

  2. It is easy to notice that this is necessarily true if the hypotheses at the first point of the theorem hold.

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Acknowledgements

The authors desire to thank the anonymous referee for the valuable comments and remarks.

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  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 Francesco Mezzadri.

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Mezzadri, F., Galligani, E. Splitting Methods for a Class of Horizontal Linear Complementarity Problems. J Optim Theory Appl 180, 500–517 (2019). https://doi.org/10.1007/s10957-018-1395-1

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