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Solution Methods for Pseudomonotone Variational Inequalities

Author

Listed:
  • N. N. Tam

    (Hanoi University of Pedagogy No. 2)

  • J. C. Yao

    (National Sun Yat-Sen University)

  • N. D. Yen

    (Vietnamese Academy of Science and Technology)

Abstract

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:2:d:10.1007_s10957-008-9376-4
    DOI: 10.1007/s10957-008-9376-4
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    References listed on IDEAS

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    1. M.A. Noor, 2002. "Proximal Methods for Mixed Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 447-452, November.
    2. M. A. Noor, 2004. "Auxiliary Principle Technique for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 371-386, August.
    3. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    4. I.V. Konnov, 2003. "Application of the Proximal Point Method to Nonmonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 317-333, November.
    5. N. El Farouq, 2002. "Errata Corrige: Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 477-477, August.
    6. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of Proximal Methods," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 311-326, May.
    7. 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.
    8. 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.
    9. 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.
    10. M.A. Noor, 2002. "Proximal Methods for Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 453-459, November.
    11. M. A. Noor, 2003. "Resolvent Algorithms for Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 137-149, October.
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    Cited by:

    1. Phan Tu Vuong, 2018. "On the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 399-409, February.
    2. N. T. T. Huong & P. D. Khanh & N. D. Yen, 2013. "Multivalued Tikhonov Trajectories of General Affine Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 85-96, July.
    3. Pham Viet Hai & Phan Tu Vuong, 2024. "Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 519-553, August.
    4. Giuseppe Marino & Hong-Kun Xu, 2011. "Explicit Hierarchical Fixed Point Approach to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 61-78, April.
    5. Nils Langenberg, 2010. "Pseudomonotone operators and the Bregman Proximal Point Algorithm," Journal of Global Optimization, Springer, vol. 47(4), pages 537-555, August.
    6. L. C. Ceng & A. Petruşel, 2010. "Krasnoselski-Mann Iterations for Hierarchical Fixed Point Problems for a Finite Family of Nonself Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 617-639, September.

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