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On non-ergodic convergence rate of the operator splitting method for a class of variational inequalities

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Abstract

In this paper, we investigate an operator splitting method for solving variational inequalities with partially unknown mappings. According to the global convergence of the operator splitting method, which has been established by Han et al. (Numer Math 111:207–237, 2008), we get the convergence rate of the operator splitting method in non-ergodic sense.

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References

  1. Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  2. Brezis, H.: Opertor maximax monotoes et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam (1973)

    Google Scholar 

  3. Corman, E., Yuan, X.M.: A generalized proximal point algorithm and its convergence rate. SIAM J. Optim. 24(4), 1614–1638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deng, W., Lai, M.J., Peng, Z.M., Yin, W.T.: Parallel multi-block ADMM with \(o(1/k)\) convergence. (2014). arXiv:1312.3040

  5. Ding, X., Han, D.R.: A modification of the forward-backward splitting method for maximal monotone mappings. Numer. Algebra Control Optim. 3, 295–307 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Facchinei, F., Pang, J.S.: Finite-Dimensional Varitional Inequalities and Complementarity Problems, vol. I and II. Spinger, Brelin (2003)

    MATH  Google Scholar 

  8. Gaitsgory, V., Tarnopolskaya, T.: Threshold value of the penalty parameter in the minimization of L1-penalized conditional value-at-risk. J. Ind. Manag. Optim. 9, 191–204 (2013)

    Article  MATH  Google Scholar 

  9. Güler, O.: On the convergence of the proximal point splitting algorithm for convex minimization. SIAM J. Optim. 29(2), 403–419 (1991)

    Article  MATH  Google Scholar 

  10. Han, D.R., Xu, W., Yang, H.: An operator splitting for variational inequalities with partially unknown mappings. Numer. Math. 111, 207–237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. He, B.S., Liao, L.Z., Wang, S.L.: Self-adaptive operator splitting methods variational inequalities. Numer. Math. 94, 715–737 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. He, B.S., Yuan, X.M.: On the \(O(1/n)\) convergence rate of the Douglas–Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, B.S., Yuan, X.M.: On non-ergodic convergence rate of Douglas–Rachford alternating direction method multipliers. Numer. Math. 130(3), 567–577 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, B.S., Yuan, X.M.: On the convergence rate of Douglas–Rachford operator splitting method. Math. Program. 153(2), 715–722 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lee, B.S.: Existence and convergence results for best proximity points in cone metric spaces. Numer. Algebra Control Optim. 4, 133–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lions, P.L., Mercier, B.: Splitting algorithm for the sum of two nonliear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

    Article  MATH  Google Scholar 

  17. Lu, Y., Ge, Y.E., Zhang, L.W.: An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. J. Ind. Manag. Optim. 12, 317–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martinet, B.: Regularization d’Inéquations Variationelles par Approximations Successives. Revue Fransaise d’Informatique Et de Recherche Opérationelle 4, 154–159 (1970)

    MATH  Google Scholar 

  19. Nemirovski, A.: Pro-method with of convergence \(O(1/t)\) for variational inequality with Lipschitz continuous monotone operators and smooth convex–concave saddle point problems. SIAM J. Optim. 15, 229–251 (2005)

    Article  MATH  Google Scholar 

  20. Peaceman, D.H., Rachford, H.H.: The numerical solution of parabolic elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shen, Y., Zhang, W.X., He, B.S.: Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. J. Ind. Manag. Optim. 10, 743–759 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Varga, R.S.: Matrix Iterative Anlysis. Prentice-Hall, Englewood Cliffs (1966)

    Google Scholar 

Download references

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grants: 11171362, 11571055, 11301567 and 11401058), Specialized Research Fund for the Doctoral Program of Higher Education (Grant Number: 20120191110031). The authors are grateful to the two anonymous referees for their valuable comments and suggestions, which helped to improve the paper.

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  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    X. P. Kou & S. J. Li

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  1. X. P. Kou
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  2. S. J. Li
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Correspondence to S. J. Li.

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Kou, X.P., Li, S.J. On non-ergodic convergence rate of the operator splitting method for a class of variational inequalities. Optim Lett 11, 71–80 (2017). https://doi.org/10.1007/s11590-015-0986-0

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