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Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems

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Abstract

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.

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References

  1. Martinet, B., Regularization d’Inéquations Variationelles par Approximations Successives, Revue Francaise d»Informatique et de Recherche Opérationelle, Vol. 4, pp. 154–159, 1970.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  4. Varga, R.S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.

  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.

    Article  MATH  MathSciNet  Google Scholar 

  6. Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer Verlag, New York, NY, 1984.

  7. Glowinski, R. and Le Tallec, P., Augmented Lagrangian and Operator-Splitting Method in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, 1989.

  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.

    MathSciNet  Google Scholar 

  9. He, B.S., Inexact Implicit Methods for Monotone General Variational Inequalities, Mathematical Programming, Vol. 86, pp. 199–217, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  11. Noor, M.A., An Implicit Method for Mixed Variational Inequalities, Applied Mathematics Letters, Vol. 11, pp. 109–113, 1998.

    Article  MATH  Google Scholar 

  12. Noor, M.A., Algorithms for General Monotone Mixed Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 229, 330–343, 1999.

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  16. Rockafellar, R.T., Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  17. Bertsekas, D.P., and Tsitsiklis, J.N., Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.

  18. Eaves, B.C., Computing Stationary Points, Mathematical Programming Study, Vol. 7, pp. 1–14, 1978.

    MATH  MathSciNet  Google Scholar 

  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.

  20. Pang, J.S., Error Bounds in Mathematical Programming, Mathematical Programming, Vol. 79, pp. 299–332, 1997.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Nanjing Normal University, Nanjing, China

    D. Han (Associate Professor)

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  1. D. Han
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Correspondence to D. Han.

Additional information

Communicated by R. Glowinski

This work was supported by the NSFC grant 10501024.

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Han, D. Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems. J Optim Theory Appl 132, 227–243 (2007). https://doi.org/10.1007/s10957-006-9060-5

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  • DOI: https://doi.org/10.1007/s10957-006-9060-5

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