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Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

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  • N. El Farouq

    (Université Blaise Pascal)

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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  • N. El Farouq, 2004. "Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 455-485, March.
    • Handle: RePEc:spr:joptap:v:120:y:2004:i:3:d:10.1023_b:jota.0000025706.49562.08
    DOI: 10.1023/B:JOTA.0000025706.49562.08
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    References listed on IDEAS

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    1. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 305-322, November.
    2. Jean-Philippe Vial, 1983. "Strong and Weak Convexity of Sets and Functions," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 231-259, May.
    3. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. N. El Farouq & G. Cohen, 1998. "Progressive Regularization of Variational Inequalities and Decomposition Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 97(2), pages 407-433, May.
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    Cited by:

    1. N. N. Tam & J. C. Yao & N. D. Yen, 2008. "Solution Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 253-273, August.
    2. M. R. Bai & N. Hadjisavvas, 2008. "Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 329-339, September.
    3. Pham Khanh & Phan Vuong, 2014. "Modified projection method for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 58(2), pages 341-350, February.

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