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On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems

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Abstract

In this paper, we analyze the relationship between projected and (possibly accelerated) modulus-based matrix splitting methods for linear complementarity problems. In particular, first we show that some well-known projected splitting methods are equivalent, iteration by iteration, to some (accelerated) modulus-based matrix splitting methods with a specific choice of the parameter \({\varOmega }\). We then generalize this result to any \({\varOmega }\) by formulating new classes of projected splitting methods and also provide a formal projection-based formulation for general (accelerated) modulus-based matrix splitting methods. Finally, we introduce and solve several test problems to evaluate also numerically the equivalence between the analyzed methods.

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Notes

  1. Note that, although, for compactness, we report just their first two digits, the residuals are identical up to machine precision. For instance, consider \(\gamma =1\). The complete residuals for MJ and for PJ are 4.057703018665017e−07 and that for PGS is 4.057703015488710e−07, respectively. Similar considerations hold for all other cases.

References

  1. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2009). https://doi.org/10.1137/1.9780898719000

  2. Liao, L.-Z.: A neural network for the linear complementarity problem. Math. Comput. Model. 29, 9–18 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Cryer, C.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control 9, 385–392 (1971)

    Article  MathSciNet  Google Scholar 

  4. Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)

    Article  MathSciNet  Google Scholar 

  5. Németh, S.Z.: Iterative methods for nonlinear complementarity problems on isoton projection cones. J. Math. Anal. Appl. 350, 340–347 (2009)

    Article  MathSciNet  Google Scholar 

  6. Xia, Z., Li, C.: Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems. Appl. Math. Comput. 271, 34–42 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Zheng, H.: Improved convergence theorems of modulus-based matrix splitting iteration method for nonlinear complementarity problems of \(H\)-matrices. Calcolo 54(4), 1481–1490 (2017)

    Article  MathSciNet  Google Scholar 

  8. Zheng, H., Vong, S.: The modulus-based nonsmooth Newton’s method for solving a class of nonlinear complementarity problems of \(P\)-matrices. Calcolo 55, 37 (2018)

    Article  MathSciNet  Google Scholar 

  9. Zhang, Y.: On the convergence of a class on infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4(1), 208–227 (1994)

    Article  MathSciNet  Google Scholar 

  10. Mezzadri, F., Galligani, E.: An inexact Newton methods for solving complementarity problems in hydrodynamic lubrication. Calcolo 55, 1 (2018)

    Article  MathSciNet  Google Scholar 

  11. Mezzadri, F., Galligani, E.: Splitting methods for a class of horizontal linear complementarity problems. J. Optim. Theory Appl. 180(2), 500–517 (2019)

    Article  MathSciNet  Google Scholar 

  12. Mezzadri, F., Galligani, E.: Modulus-based matrix splitting methods for horizontal linear complementarity problems. Numer. Algorithm (in press). https://doi.org/10.1007/s11075-019-00677-y

  13. Mangasarian, O.L.: Solutions of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465–485 (1977)

    Article  MathSciNet  Google Scholar 

  14. Li, Y., Dai, P.: Generalized AOR methods for linear complementarity problem. Appl. Math. Comput. 188, 7–18 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Bai, Z.-Z., Evans, D.: Matrix multisplitting relaxation methods for linear complementarity problems. Int. J. Comput. Math. 63(3–4), 309–326 (1997)

    Article  MathSciNet  Google Scholar 

  16. Leone, R.D., Mangasarian, O.L., Shiau, T.-H.: Multi-sweep asynchronous parallel successive overrelaxation for the nonsymmetric linear complementarity problem. Ann. Oper. Res. 22(1), 43–54 (1990)

    Article  MathSciNet  Google Scholar 

  17. Jiang, M., Dong, J.: On the convergence of two-stage splitting methods for linear complementarity problems. J. Comput. Appl. Math 181, 58–69 (2005)

    Article  MathSciNet  Google Scholar 

  18. van Bokhoven, W.: Piecewise-Linear Modelling and Analysis. Proefschrift, Eindhoven (1981)

    Google Scholar 

  19. Murty, K.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    MATH  Google Scholar 

  20. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)

    Article  MathSciNet  Google Scholar 

  21. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms 62, 59–77 (2013)

    Article  MathSciNet  Google Scholar 

  22. Zheng, N., Yin, J.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Algorithms 64, 245–262 (2013)

    Article  MathSciNet  Google Scholar 

  23. Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of H-matrices. Appl. Math. Lett. 26, 1159–1164 (2013)

    Article  MathSciNet  Google Scholar 

  24. Liu, S., Zheng, H., Li, W.: A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. Calcolo 53(2), 189–199 (2016)

    Article  MathSciNet  Google Scholar 

  25. Hadjidimos, A., Zhang, L.-L.: Comparison of three classes of algorithms for the solution of the linear complementarity problem with an \(H_+\)-matrix. J. Comput. Appl. Math 336, 175–191 (2018)

    Article  MathSciNet  Google Scholar 

  26. Zheng, H., Li, W., Vong, S.: A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems. Numer. Algorithms 74(1), 137–152 (2017)

    Article  MathSciNet  Google Scholar 

  27. Varga, R.: Matrix Iterative Analysis, 2nd edn. Springer, Berlin (2000)

    Book  Google Scholar 

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Acknowledgements

The author desires to thank the anonymous referees for their 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

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  1. Francesco Mezzadri
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Mezzadri, F. On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems. Calcolo 56, 41 (2019). https://doi.org/10.1007/s10092-019-0337-0

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