Journal of Optimization Theory and Applications

, Volume 132, Issue 2, pp 227–243 | Cite as

Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems



The Peaceman-Rachford and Douglas-Rachford operator splitting methods are advantageous for solving variational inequality problems, since they attack the original problems via solving a sequence of systems of smooth equations, which are much easier to solve than the variational inequalities. However, solving the subproblems exactly may be prohibitively difficult or even impossible. In this paper, we propose an inexact operator splitting method, where the subproblems are solved approximately with some relative error tolerance. Another contribution is that we adjust the scalar parameter automatically at each iteration and the adjustment parameter can be a positive constant, which makes the methods more practical and efficient. We prove the convergence of the method and present some preliminary computational results, showing that the proposed method is promising.


Variational inequality problems operator splitting methods inexact methods self-adaptive algorithms strongly monotone mappings 


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  1. 1.
    Martinet, B., Regularization d’Inéquations Variationelles par Approximations Successives, Revue Francaise d»Informatique et de Recherche Opérationelle, Vol. 4, pp. 154–159, 1970.MathSciNetGoogle Scholar
  2. 2.
    Peaceman, D.H., and Rachford, H.H., The Numerical Solution of Parabolic Elliptic Differential Equations, SIAM Journal on Applied Mathematics, Vol. 3, pp. 28–41, 1955.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Douglas, J., and Rachford, H.H., On the Numerical Solution of the Heat Conduction Problem in 2 and 3 Space Variables, Transactions of American Mathematical Society, Vol. 82, pp. 421–439, 1956.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Varga, R.S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.Google Scholar
  5. 5.
    Lions, P.L., and Mercier, B., Splitting Algorithms for the Sum of Two Nonlinear Operators, SIAM Journal on Numerical Analysis, Vol. 16, pp. 964–979, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer Verlag, New York, NY, 1984.Google Scholar
  7. 7.
    Glowinski, R. and Le Tallec, P., Augmented Lagrangian and Operator-Splitting Method in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, 1989.Google Scholar
  8. 8.
    Fukushima, M., The Primal Douglas-Rachford Splitting Algorithm for a Class of Monotone Mappings with Application to the Traffic Equilibrium Problem, Mathematical Programming, Vol. 72, pp. 1–15, 1996.MathSciNetGoogle Scholar
  9. 9.
    He, B.S., Inexact Implicit Methods for Monotone General Variational Inequalities, Mathematical Programming, Vol. 86, pp. 199–217, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    He, B.S., Liao L.Z., and Wang, S.L., Selfadaptive Operator Splitting Methods for Monotone Variational Inequalities, Numerische Mathematik, Vol. 94, pp. 715–737, 2003.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Noor, M.A., An Implicit Method for Mixed Variational Inequalities, Applied Mathematics Letters, Vol. 11, pp. 109–113, 1998.zbMATHCrossRefGoogle Scholar
  12. 12.
    Noor, M.A., Algorithms for General Monotone Mixed Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 229, 330–343, 1999.Google Scholar
  13. 13.
    Huang, N.J., A New Method for a Class of Nonlinear Variational Inequalities with Fuzzy Mapping, Applied Mathematics Letters, Vol. 10, pp. 129–133, 1997.zbMATHCrossRefGoogle Scholar
  14. 14.
    Huang, N.J., A New Method for a Class of Set-Valued Nonlinear Variational Inequalities, Zeitschrift fuuml;r Angewandte Mathematik und Mechanik, Vol. 78, pp. 427–430, 1998.zbMATHCrossRefGoogle Scholar
  15. 15.
    Han,D.R., and He, B.S., A New Accuracy Criterion for Approximate Proximal Point Methods, Journal of Mathematical Analysis and Applications, Vol. 263, pp. 343–354, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rockafellar, R.T., Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Bertsekas, D.P., and Tsitsiklis, J.N., Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.Google Scholar
  18. 18.
    Eaves, B.C., Computing Stationary Points, Mathematical Programming Study, Vol. 7, pp. 1–14, 1978.zbMATHMathSciNetGoogle Scholar
  19. 19.
    B.S. He, Some Predictor-Corrector Projection Methods for Monotone Variational Inequalities, Report 95–68, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, Holland, 1995.Google Scholar
  20. 20.
    Pang, J.S., Error Bounds in Mathematical Programming, Mathematical Programming, Vol. 79, pp. 299–332, 1997.MathSciNetGoogle Scholar
  21. 21.
    Solodov, M.V., and Svaiter, B.F., A Hybrid Approximate Extragradient. Proximal Point Algorithm Using the Enlargement of a Maximal Monotone Operator, Set-Valued Analysis, Vol. 7, pp. 323–345, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Harker, P.T., and Pang, J.S., A Damped-Newton Method for the Linear Complementarity Problem, Lectures in Applied Mathematics, Vol. 26, pp. 265–284, 1990.MathSciNetGoogle Scholar
  23. 23.
    Marcotee, P., and Dussault, J.P., A Note on a Globally Convergent Newton Method for Solving Variational Inequalities, Operation Research Letters, Vol. 6, pp. 35–42, 1987.CrossRefGoogle Scholar
  24. 24.
    Taji, K., Fukishima, M., and Ibaraki, T., A Globally Convergent Newton Method for Solving Strongly Monotone Variational Inequalities, Mathematical Programming, Vol. 58, pp. 369–383, 1993.CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceNanjing Normal UniversityNanjingChina

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