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A New Decomposition Method for Variational Inequalities with Linear Constraints

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Abstract

We propose a new decomposition method for solving a class of monotone variational inequalities with linear constraints. The proposed method needs only to solve a well-conditioned system of nonlinear equations, which is much easier than a variational inequality, the subproblem in the classic alternating direction methods. To make the method more flexible and practical, we solve the sub-problems approximately. We adopt a self-adaptive rule to adjust the parameter, which can improve the numerical performance of the algorithm. Under mild conditions, the underlying mapping be monotone and the solution set of the problem be nonempty, we prove the global convergence of the proposed algorithm. Finally, we report some preliminary computational results, which demonstrate the promising performance of the new algorithm.

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References

  1. Bertsekas, D.P., Gafni, E.M.: Projection method for variational inequalities with applications to the traffic assignment problem. Math. Program. 17, 139–159 (1982)

    MathSciNet  MATH  Google Scholar 

  2. Dafetmos, S.: Traffic equilibrium and variational inequalities. Transp. Sci. 14, 42–54 (1980)

    Article  Google Scholar 

  3. Nagurney, A., Ramanujam, P.: Transportation network policy modeling with goal targets and generalized penalty functions. Transp. Sci. 30, 3–13 (1996)

    Article  MATH  Google Scholar 

  4. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, Berlin (2003)

    Google Scholar 

  5. Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems. North-Holland, Amsterdam (1983)

    Google Scholar 

  6. Han, D.: A modified alternating direction method for variational inequality problems. Appl. Math. Optim. 45, 63–74 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, D., Lo, H.: A new stepsize rule in He and Zhou’s alternating direction method. Appl. Math. Lett. 15, 181–185 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Han, D., Lo, H.: A new alternating direction method for a class of nonlinear variational inequality problems. J. Optim. Theory Appl. 112, 549–560 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. He, B., Zhou, J.: A modified alternation direction method for convex minimization problems. Appl. Math. Lett. 13, 122–130 (2000)

    MathSciNet  Google Scholar 

  10. Wang, S., Yang, H., He, B.: Solving a class of asymmetric variational inequalities via a new alternating direction method. Comput. Math. Appl. 40, 927–937 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Han, D.: A proximal decomposition algorithm for variational inequality problems. J. Comput. Appl. Math. 161, 231–244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Han, D.: A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems. Comput. Math. Appl. 55, 101–115 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. He, B., Yang, H., Meng, Q., Han, D.: Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities. J. Optim. Theory Appl. 112, 129–143 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems. North-Holland, Amsterdam (1983)

    Google Scholar 

  17. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2, 17–40 (1976)

    Article  MATH  Google Scholar 

  18. Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–101 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Methods Softw. 4, 75–83 (1994)

    Article  MathSciNet  Google Scholar 

  20. Eckstein, J., Fukushima, M.: Some reformulation and applications of the alternating directions method of multipliers. In: Hager, W., et al. (eds.) Large Scale Optimization: State of the Art. Kluwer Academic, Norwell (1994)

    Google Scholar 

  21. Esser, E.: Applications of Lagrangian-Based alternating direction methods and connections to split Bregman. UCLA CAM Report 09-31 (2009)

  22. Fukushima, M.: Application of the alternating directions method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992)

    Article  MathSciNet  Google Scholar 

  23. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    MATH  Google Scholar 

  24. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics, Philadelphia, PA (1989)

  25. He, B., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23, 151–161 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kontogiorgis, S., Meyer, R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)

    MathSciNet  MATH  Google Scholar 

  27. Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Control Optim. 7, 951–965 (1997)

    MathSciNet  MATH  Google Scholar 

  28. He, B.: A new method for a class of linear variational inequalities. Math. Program. 66, 137–144 (1994)

    Article  MATH  Google Scholar 

  29. He, B.: Solving a class of linear projection equations. Numer. Math. 68, 71–80 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. He, B.: A class of new methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Zhu, T., Yu, Z.: A simple proof for some important properties of the projection mapping. Math. Inequal. Appl. 7, 453–456 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Dennis, J., Schnabel, R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

  33. Fletcher, R.: Practical Methods of Optimization. Wiley, New York (1985)

    Google Scholar 

  34. Han, D., Lo, H.: Solving variational inequality problems with linear constraints by a proximal decomposition algorithm. J. Glob. Optim. 28, 97–113 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Han, D.: Inexact operator splitting methods with self-adaptive strategy for variational inequality problems. J. Optim. Theory Appl. 132, 227–243 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Taji, K., Fukushima, M., Ibaraki, T.: A globally convergent Newton method for solving strongly monotone variational inequalities. Math. Program. 58, 369–383 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. He, B., Liao, L.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ge, Z., Han, D.: Self-adaptive implicit methods for monotone variant variational inequalities. J. Inequal. Appl. (2009). doi:10.1155/2009/458134

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences and Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing, 210046, P.R. China

    Min Zhang & Deren Han

  2. School of Computer Science, Nanjing Normal University, Nanjing, 210097, P.R. China

    Gang Qian

  3. Department of Mathematics, Taiyuan Normal University, Taiyuan, Shanxi, 030012, China

    Xihong Yan

Authors
  1. Min Zhang
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  2. Deren Han
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  3. Gang Qian
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  4. Xihong Yan
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Corresponding author

Correspondence to Deren Han.

Additional information

Communicated by Nicolas Hadjisavvas.

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Zhang, M., Han, D., Qian, G. et al. A New Decomposition Method for Variational Inequalities with Linear Constraints. J Optim Theory Appl 152, 675–695 (2012). https://doi.org/10.1007/s10957-011-9931-2

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  • DOI: https://doi.org/10.1007/s10957-011-9931-2

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