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Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

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Abstract

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization introduced in Refs. 1, 2.

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References

  1. El Farouq, N., Algorithmes de Résolution d'Inéquations Variationnelles, Thesis Dissertation, École des Mines de Paris, Fontainebleu, France, 1993.

    Google Scholar 

  2. El Farouq, N., and Cohen, G., Progressive Regularization of Variational Inequalities and Decomposition Algorithms, Journal of Optimization Theory and Applications, 97, 407-433, 1998.

    Google Scholar 

  3. Cohen, G., Optimization by Decomposition and Coordination: A Unified Approach, IEEE Transactions on Automatic Control, 23, 222-232, 1978.

    Google Scholar 

  4. Cohen, G., Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, 32, 277-305, 1980.

    Google Scholar 

  5. Cohen, G., Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, 59, 325-333, 1988.

    Google Scholar 

  6. Cohen, G., and Zhu, D. L., Decomposition Coordination Methods in Large-Scale Optimization Problems: The Nondifferentiable Case and the Use of Augmented Lagrangians, Advances in Large-Scale Systems Theory and Applications, J. B. Cruz, JAI Press, Greenwich, Connecticut, 1, 203-266, 1983.

    Google Scholar 

  7. Bregman, L. M., The Relaxation Method of Finding the Common Point of Convex Sets and Its Applications to the Solution of Problems in Convex Programming, USSR Computational Mathematics and Mathematical Physics, 7, 200-217, 1967.

    Google Scholar 

  8. Eckstein, J., Nonlinear Proximal Point Algorithms Using Bregman Functions, Mathematics of Operations Research, 18, 202-226, 1993.

    Google Scholar 

  9. Pang, J. S., and Chang, D., Iterative Methods for Variational and Complementary Problems, Mathematical Programming, 24, 284-313, 1982.

    Google Scholar 

  10. Harker, P. T., and Pang, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, 48, 161-220, 1990.

    Google Scholar 

  11. Mataoui, M. A., Contributions à la Décomposition et à l'Agrégation des Problèmes Variationnels, Thesis Dissertation, École des Mines de Paris, Fontainbleau, France, 1990.

    Google Scholar 

  12. Zhu, D. L., and Marcotte, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, 6, 714-426, 1996.

    Google Scholar 

  13. Eckstein, J., and Bertsekas, D. P., On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators, Mathematical Programming, 55, 293-318, 1992.

    Google Scholar 

  14. Vial, J. P., Strong and Weak Convexity of Sets and Functions, Mathematics of Operations Research, 8, 231-258, 1983.

    Google Scholar 

  15. Spingarn, J. E., Submonotone Mappings and the Proximal Point Algorithm, Numerical Functional Analysis and Optimization, 4, 123-150, 1981-1982.

    Google Scholar 

  16. Crouzeix, J. P., Pseudomonotone Variational Inequality Problems: Existence of Solutions, Mathematical Programming, 78, 305-314, 1997.

    Google Scholar 

  17. Karamardian, S., Complementary Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, 18, 445-455, 1976.

    Google Scholar 

  18. Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, 83, 391-403, 1994.

    Google Scholar 

  19. El Farouq, N., Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method, Journal of Optimization Theory and Applications, 111, 305-326, 2001.

    Google Scholar 

  20. Zeidler, E., Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer Verlag, Berlin, Germany, 1990.

    Google Scholar 

  21. Brezis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies, North-Holland, Amsterdam, Holland, 5, 1973.

  22. El Farouq, N., Sequential Progressive Regularization for Solving Pseudomonotone Variational Inequalities, Report D01215, LIMOS, Clermont-Ferrand, France, 2001.

    Google Scholar 

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Authors and Affiliations

  1. Maître de Conférences, Université Blaise Pascal, Clermont II and LAAS-CNRS, Toulouse, France

    N. El Farouq

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  1. N. El Farouq
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Farouq, N.E. Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities. Journal of Optimization Theory and Applications 120, 455–485 (2004). https://doi.org/10.1023/B:JOTA.0000025706.49562.08

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  • DOI: https://doi.org/10.1023/B:JOTA.0000025706.49562.08

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