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New Parallel Descent-like Method for Solving a Class of Variational Inequalities

  • Z. K. Jiang
  • X. M. Yuan
Article

Abstract

To solve a class of variational inequalities with separable structures, some classical methods such as the augmented Lagrangian method and the alternating direction methods require solving two subvariational inequalities at each iteration. The most recent work (B.S. He in Comput. Optim. Appl. 42(2):195–212, 2009) improved these classical methods by allowing the subvariational inequalities arising at each iteration to be solved in parallel, at the price of executing an additional descent step. This paper aims at developing this strategy further by refining the descent directions in the descent steps, while preserving the practical characteristics suitable for parallel computing. Convergence of the new parallel descent-like method is proved under the same mild assumptions on the problem data.

Keywords

Variational inequalities Parallel computing Descent-like methods Alternating direction methods Augmented Lagrangian method 

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References

  1. 1.
    Baiocchi, C., Capelo, A.: Variational and QuasiVariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1988) MATHGoogle Scholar
  2. 2.
    Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer Series in Operations Research. Springer, New York (2003) Google Scholar
  3. 3.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems and Variational Models. Kluwer Academic, Dordrecht (2001) Google Scholar
  5. 5.
    Kinderlehrer, D., Stampacchia, G.: A Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980) Google Scholar
  6. 6.
    Nagurney, A.: Network Economics, a Variational Inequality Approach. Kluwer Academics, Dordrecht (1993) MATHGoogle Scholar
  7. 7.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation, Numerical Methods. Englewood Cliffs, Prentice-Hall (1989) MATHGoogle Scholar
  8. 8.
    Eckstein, J., Fukushima, M.: Some reformulation and applications of the alternating direction method of multipliers. Large Scale Optimization: State of the Art, pp. 115–134. Kluwer Academic, Dordrecht (1994). Google Scholar
  9. 9.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) MATHGoogle Scholar
  10. 10.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1989) MATHGoogle Scholar
  11. 11.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006) MATHGoogle Scholar
  12. 12.
    Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Methods Softw. 4, 75–83 (1994) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992) MathSciNetGoogle Scholar
  14. 14.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems, pp. 299–331. North-Holland, Amsterdam (1983) CrossRefGoogle Scholar
  15. 15.
    Han, D.R., Lo, H.K.: New alternating direction method for a class of nonlinear variational inequality problems. J. Optim. Theory Appl. 112(3), 549–560 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Xu, M.H.: Proximal alternating directions method for structured variational inequalities. J. Optim. Theory Appl. 134(1), 107–117 (2007) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    He, B.S., Liao, L.Z., Han, D.R., Yang, H.: A new inexact alternating direction method for monotone variational inequalities. Math. Program. 92(1), 103–118 (2002) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    He, B.S., Yang, H., Wang, S.L.: Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theory Appl. 106(2), 337–356 (2000) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998) MathSciNetGoogle Scholar
  20. 20.
    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) MATHCrossRefGoogle Scholar
  21. 21.
    Pardalos, P.M., Phillips, A., Rosen, J.B.: Topics in Parallel Computing in Mathematical Programming. Science Press, Marrickville (1992) MATHGoogle Scholar
  22. 22.
    Pardalos, P.M., Rajasekaran, S.: Advances in Randomized Parallel Computing. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  23. 23.
    Migdalas, A., Pardalos, P.M., Storoy, S.: Parallel Computing in Optimization. Kluwer Academic, Dordrecht (1997) MATHGoogle Scholar
  24. 24.
    Migdalas, A., Toraldo, G., Kumar, V.: Nonlinear optimization and parallel computing. Parallel Comput. 29(4), 375–391 (2003) CrossRefMathSciNetGoogle Scholar
  25. 25.
    D’Apuzzo, M., Marino, M., Migdalas, A., Pardalos, P.M., Toraldo, G.: Parallel computing in global optimization. In: Handbook of Parallel Computing and Statistics, pp. 225–258. Chapman & Hall, London (2006) Google Scholar
  26. 26.
    He, B.S.: Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Comput. Optim. Appl. 42(2), 195–212 (2009) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Eaves, B.C.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Bertsekas, D.P., Gafni, E.M.: Projection method for variational inequalities with applications to the traffic assignment problem. Math. Program. Study 17 (1982) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsHuaiHai Institute of TechnologyLianyungangChina
  2. 2.Department of MathematicsHong Kong Baptist UniversityHong KongChina

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