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Numerical Algorithms

, Volume 67, Issue 3, pp 477–489 | Cite as

A Barzilai–Borwein type method for stochastic linear complementarity problems

  • Yakui HuangEmail author
  • Hongwei Liu
  • Sha Zhou
Original Paper

Abstract

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.

Keywords

Stochastic linear complementarity problem Barzilai–Borwein type method 

Mathematics Subject Classifications (2010)

90C30 90C33 

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References

  1. 1.
    Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comp. Optim. Appl. 5, 97–138 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, X.J., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X.J., Zhang, C.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627–649 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, X.J., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fang, H.T., Chen, X.J., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, X.L., Liu, H.W., Sun, X.J.: Feasible smooth method based on BarzilaiCBorwein method for stochastic linear complementarity problem. Numer. Algor. 57, 207–215 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ling, C., Qi, L., Zhou, G.L., Caccetta, L.: The SC 1 property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456–460 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, H.W., Huang, Y.K., Li, X.L.: Partial projected Newton method for a class of stochastic linear complementarity problems. Numer. Algor. 58, 593–618 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, H.W., Huang, Y.K., Li, X.L.: New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems. Appl. Math. Comput. 217, 9723–9740 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, H.W., Li, X.L., Huang, Y.K.: Solving equations via the trust region and its application to a class of stochastic linear complementarity problems. J. Comput. Math. Appl. 61, 1646–1664 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tang, J., Ma, C.F.: A smoothing Newton method for solving a class of stochastic linear complementarity problems. Nonlinear Anal. RWA 6, 3585–3601 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xie, Y.J., Ma, C.F.: A smoothing Levenberg-Marquardt algorithm for solving a class of stochastic linear complementarity problem. Appl. Math. Comput. 217, 4459–4472 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14, 1043–1056 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhou, G.L., Caccetta, L.: Feasible semismooth Newton method for a class of stochastic linear complementarity problems. J. Optim. Theory Appl. 139, 379–392 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China
  2. 2.College of Mathematics and Computing ScienceGuilin University of Electronic TechnologyGuilinPeople’s Republic of China

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