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The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

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Abstract

For the second-order cone linear complementarity problems, abbreviated as SOCLCPs, we establish two classes of modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently the SOCLCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The convergence of these modulus-based matrix splitting iteration methods has been established and the optimal iteration parameters of these methods are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving the SOCLCPs.

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References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MathSciNet  Google Scholar 

  2. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems volumes I and II. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  3. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second order cone programming. Linear Algebra Appl. 284(1-3), 193–228 (1998)

    Article  MathSciNet  Google Scholar 

  4. Andersen, E.D., Roos, C., Terlaky, T.: On implementing a primal-dual interior-point method for conic quadratic optimization. Math. Program. 95(2), 249–277 (2003)

    Article  MathSciNet  Google Scholar 

  5. Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Math. Program. 88(1), 61–83 (2000)

    Article  MathSciNet  Google Scholar 

  6. Schmieta, S.H., Alizadeh, F.: Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Math. Oper. Res. 26(3), 543–564 (2001)

    Article  MathSciNet  Google Scholar 

  7. Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point algorithms to symmetric cones. Math. Program. 96(3), 409–438 (2003)

    Article  MathSciNet  Google Scholar 

  8. Yoshise, A.: Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cone. SIAM J. Optim. 17(4), 1129–1153 (2005)

    Article  MathSciNet  Google Scholar 

  9. Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104(2), 293–327 (2005)

    Article  MathSciNet  Google Scholar 

  10. Chen, J.-S.: Two classes of merit functions for the second-order cone complementarity problem. Math. Methods Oper. Res. 64(3), 495–519 (2006)

    Article  MathSciNet  Google Scholar 

  11. Chen, J.-S., Pan, S.: A one-parametric class of merit functions for the second-order cone complementarity problem. Comput. Optim. Appl. 45(3), 581–606 (2010)

    Article  MathSciNet  Google Scholar 

  12. Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12(2), 436–460 (2015)

    Article  MathSciNet  Google Scholar 

  13. Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15(2), 593–615 (2003)

    Article  MathSciNet  Google Scholar 

  14. Chen, X.-D., Sun, D., Sun, J.: Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Comput. Optim. Appl. 25(1), 39–56 (2003)

    Article  MathSciNet  Google Scholar 

  15. Pan, S., Chen, J.-S.: A damped Gauss-Newton method for the second-order cone complementarity problem. Appl. Math. Optim. 59(3), 293–318 (2009)

    Article  MathSciNet  Google Scholar 

  16. Fang, L., Han, C.-Y.: A new one-step smoothing newton method for the second-order cone complementarity problem. Math. Methods Appl. Sci. 34(3), 347–359 (2011)

    Article  MathSciNet  Google Scholar 

  17. Fang, L.: A smoothing-type Newton method for second-order cone programming problems based on a new smooth function. J. Appl. Math. Comput. 34(1), 147–161 (2010)

    Article  MathSciNet  Google Scholar 

  18. Narushima, Y., Sagara, N., Ogasawara, H.: A smoothing newton method with Fischer-Burmeister function for second-order cone complementarity problems. J. Optim. Theory Appl. 149(1), 79–101 (2011)

    Article  MathSciNet  Google Scholar 

  19. Zhang, X.-S., Liu, S.-Y., Liu, Z.-H.: A smoothing method for second order cone complementarity problem. J. Comput. Appl. Math. 228(1), 83–91 (2009)

    Article  MathSciNet  Google Scholar 

  20. Chen, L.-J., Ma, C.-F.: A modified smoothing and regularized Newton method for monotone second-order cone complementarity problems. Comput. Math. Appl. 61(5), 1407–1418 (2011)

    Article  MathSciNet  Google Scholar 

  21. Ma, C.-F.: A regularized smoothing Newton method for solving the symmetric cone complementarity problem. Math. Comput. Model. 54(9-10), 2515–2527 (2011)

    Article  MathSciNet  Google Scholar 

  22. Hayashi, S., Yamaguchi, T., Yamashita, N., Fukushima, M.: A matrix-splitting method for symmetric affine second-order cone complementarity problems. J. Comput. Appl. Math. 175(2), 335–353 (2005)

    Article  MathSciNet  Google Scholar 

  23. Zhang, L.-H., Yang, W.-H.: An efficient matrix splitting method for the second-order cone complementarity problem. SIAM J. Optim. 24, 1178–1205 (2014)

    Article  MathSciNet  Google Scholar 

  24. Malik, M., Mohan, S.R.: On Q and R 0 properties of a quadratic representation in linear complementarity problems over the second-order cone. Linear Algebra Appl. 397(1), 85–97 (2005)

    Article  MathSciNet  Google Scholar 

  25. Pang, J.-S., Sun, D., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28(1), 39–63 (2003)

    Article  MathSciNet  Google Scholar 

  26. Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31(1), 109–123 (2006)

    Article  MathSciNet  Google Scholar 

  27. Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. Optim. 18(2), 461–481 (2007)

    Article  MathSciNet  Google Scholar 

  28. 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 

  29. Faraut, U., Korányi, A.: Analysis on symmetric cones, Oxford Mathematical Monographs. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  30. Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004)

    Article  MathSciNet  Google Scholar 

  31. Yang, W.-H., Yuan, X. -M.: The GUS-property of second-order cone linear complementarity problems. Math. Program. 141, 295–317 (2013)

    Article  MathSciNet  Google Scholar 

  32. Ke, Y.-F., Ma, C.-F.: On the convergence analysis of two-step modulus-based matrix splitting iteration method for linear complementarity problems. Appl. Math. Comput. 243, 413–418 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Murty, K.: Linear complementarity, linear and nonlinear programming. Heldermann, Berlin (1988)

    MATH  Google Scholar 

  34. Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129–143 (2009)

    Article  MathSciNet  Google Scholar 

  35. Zhang, L.-L.: Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer. Algor. 57(1), 83–99 (2011)

    Article  MathSciNet  Google Scholar 

  36. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Algor. 62(1), 100–112 (2013)

    Article  MathSciNet  Google Scholar 

  37. Zhang, L.-L., Ren, Z.-R.: Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl. Math. Lett. 26(6), 638–642 (2013)

    Article  MathSciNet  Google Scholar 

  38. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algor. 62(1), 59–77 (2013)

    Article  MathSciNet  Google Scholar 

  39. Zheng, N., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem. Numer. Algor. 64(2), 245–262 (2013)

    Article  MathSciNet  Google Scholar 

  40. Zheng, N., Yin, J.-F.: Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an H +-matrix. J. Comput. Appl. Math. 260(2), 281–293 (2014)

    Article  MathSciNet  Google Scholar 

  41. Hadjidimos, A., Lapidakis, M., Tzoumas, M.: On iterative solution for linear complementarity problem with an H +-matrix. SIAM J. Matrix Anal. Appl. 33(1), 97–110 (2012)

    Article  MathSciNet  Google Scholar 

  42. Zeng, M.-L., Zhang, G.-F.: Modulus-based GSTS iteration method for linear complementarity problems. J. Math. Study 48(1), 1–17 (2015)

    Article  MathSciNet  Google Scholar 

  43. Liu, S.-M., 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 

  44. Kellogg, R.B.: Another alternating-direction-implicit method. J. Soc. Ind. Appl. Math. 11, 976–979 (1963)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to express their great thankfulness to the referees for the comments and constructive suggestions, which are valuable in improving the quality of the original paper.

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Authors and Affiliations

  1. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yi-Fen Ke & Huai Zhang

  2. College of Mathematics and Informatics and FJKLMAA, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Yi-Fen Ke & Chang-Feng Ma

Authors
  1. Yi-Fen Ke
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  2. Chang-Feng Ma
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  3. Huai Zhang
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Corresponding author

Correspondence to Chang-Feng Ma.

Additional information

This work is supported by China Postdoctoral Science Foundation (No.2017M620878), National Postdoctoral Program for Innovative Talents (No. BX201700234), Fujian Natural Science Foundation (No. 2016J01005), National Basic Research Program of China (No. 2014CB845906), and National Science Foundation of China (No. 41725017, 41590864). It is also partially supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (No. XDB18010202), and the CAS/CAFEA international partnership Program for creative research teams (No. KZZD-EW-TZ-19).

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Ke, YF., Ma, CF. & Zhang, H. The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems. Numer Algor 79, 1283–1303 (2018). https://doi.org/10.1007/s11075-018-0484-4

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  • DOI: https://doi.org/10.1007/s11075-018-0484-4

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