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Pseudomonotone Variational Inequalities: Convergence of Proximal Methods

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Abstract

In this paper, we study the convergence of proximal methods for solving pseudomonotone (in the sense of Karamardian) variational inequalities. The main result is given in the finite-dimensional case, but we show that we still obtain convergence in an infinite-dimensional Hilbert space under a strong pseudomonotonicity or a pseudo-Dunn assumption on the operator involved in the variational inequality problem.

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References

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

    Google Scholar 

  2. Karamardian, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–455, 1976.

    Google Scholar 

  3. Mangasarian, O. L., Pseudoconvex Functions, SIAM Journal on Control, Vol. 3, pp. 281–290, 1965.

    Google Scholar 

  4. Mangasarian, O. L., Nonlinear Programming, SIAM, Philadelphia, Pennsylvania, 1994.

    Google Scholar 

  5. Karamardian, S., and Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–47, 1990.

    Google Scholar 

  6. Schaible, S., Generalized Monotonicity: A Survey, Generalized Convexity, Edited by S. Komlosi, R. Rapcsak, and S. Schaible, Springer Verlag, Heidelberg, Germany, pp. 229–249, 1994.

    Google Scholar 

  7. Karamardian, S., Schaible, S., and Crouzeix, J. P., Characterizations of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993.

    Google Scholar 

  8. Komlosi, S., Generalized Monotonicity and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 84, pp. 361–376, 1995.

    Google Scholar 

  9. Zhu, D., and Marcotte, P., New Classes of Generalized Monotonicity, Journal of Optimization Theory and Applications, Vol. 87, pp. 457–471, 1995.

    Google Scholar 

  10. Schaible, S., Generalized Monotonicity: Concepts and Uses, Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Publishing Corporation, New York, NY, pp. 289–299, 1995.

    Google Scholar 

  11. Yao, J. C., Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.

    Google Scholar 

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

    Google Scholar 

  13. Crouzeix, J. P., Pseudomonotone Variational Inequality Problems: Existence of Solutions, Mathematical Programming, Vol. 78, pp. 305–314, 1997.

    Google Scholar 

  14. Zhu, D. L., and Marcotte, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714–726, 1996.

    Google Scholar 

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

    Google Scholar 

  16. Mataoui, M. A., Contributions à la Décomposition et à l' Agrégation des Problèmes Variationnels, Thesis Dissertation, É cole des Mines de Paris, Fontainebleau, France, 1990.

    Google Scholar 

  17. Brezis, H., Equations et Inéquations Nonlinéaires dans les Espaces Vectoriels en Dualité, Annales de l'Institut Fourier, Vol. 18, pp. 115–175, 1968.

    Google Scholar 

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

    Google Scholar 

  19. El Farouq, N., Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method, Journal of Optimization Theory and Applications (to appear).

  20. Lemaire, B., The Proximal Algorithm, International Series of Numerical Mathematics, Birkhäuser, Basel, Switzerland, Vol.87, pp. 73–87, 1989.

    Google Scholar 

  21. Lemaire, B., About the Convergence of the Proximal Method, Advances in Optimization, Proceedings of Lambrecht 1991, Lecture in Economics and Mathematical Systems, Springer Verlag, Vol. 382, pp. 39–51, 1992.

    Google Scholar 

  22. Martinet, B., Régularisation d'Inéquations Variationnelles par Approximations Successives, Revue d'Automatique, d'Informatique et de Recherche Opèrationnelle, Sèrie Rouge, Vol. 3, pp. 154–159, 1970.

    Google Scholar 

  23. Martinet, B., Algorithmes pour la Résolution de Problèmes d'Optimisation et de Minmax, Thèse d'Etat, Université de Grenoble, 1972.

  24. Eckstein, J., Nonlinear Proximal Point Algorithms Using Bregman Functions, Mathematics of Operations Research, Vol. 18, pp. 202–226, 1993.

    Google Scholar 

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

    Google Scholar 

  26. 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, Vol. 7, pp. 200–217, 1967.

    Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Université Blaise Pascal, Clermont II and LAAS CNRS, Toulouse, France

    N. EL FAROUQ (Maître de Conférences)

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  1. N. EL FAROUQ
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EL FAROUQ, N. Pseudomonotone Variational Inequalities: Convergence of Proximal Methods. Journal of Optimization Theory and Applications 109, 311–326 (2001). https://doi.org/10.1023/A:1017562305308

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