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Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities

Author

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  • G. Salmon

    (Universitaires Notre Dame de la Paix)

  • V. H. Nguyen

    (Universitaires Notre Dame de la Paix)

  • J. J. Strodiot

    (Universitaires Notre Dame de la Paix)

Abstract

Many algorithms for solving variational inequality problems can be derived from the auxiliary problem principle introduced several years ago by Cohen. In recent years, the convergence of these algorithms has been established under weaker and weaker monotonicity assumptions: strong (pseudo) monotonicity has been replaced by the (pseudo) Dunn property. Moreover, well-suited assumptions have given rise to local versions of these results. In this paper, we combine the auxiliary problem principle with epiconvergence theory to present and study a basic family of perturbed methods for solving general variational inequalities. For example, this framework allows us to consider barrier functions and interior approximations of feasible domains. Our aim is to emphasize the global or local assumptions to be satisfied by the perturbed functions in order to derive convergence results similar to those without perturbations. In particular, we generalize previous results obtained by Makler-Scheimberg et al.

Suggested Citation

  • G. Salmon & V. H. Nguyen & J. J. Strodiot, 2000. "Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 629-657, March.
    • Handle: RePEc:spr:joptap:v:104:y:2000:i:3:d:10.1023_a:1004693710334
    DOI: 10.1023/A:1004693710334
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    References listed on IDEAS

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    1. J. H. Wu, 1997. "Modified Primal Path-Following Scheme for the Monotone Variational Inequality Problem," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 189-208, October.
    2. F. Sharifi-Mokhtarian & J. L. Goffin, 1998. "Long-Step Interior-Point Algorithms for a Class of Variational Inequalities with Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 181-210, April.
    3. J. H. Wu, 1998. "Long-Step Primal Path-Following Algorithm for Monotone Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 99(2), pages 509-531, November.
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    Cited by:

    1. Jie Shen & Ya-Li Gao & Fang-Fang Guo & Rui Zhao, 2018. "A Redistributed Bundle Algorithm for Generalized Variational Inequality Problems in Hilbert Spaces," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(04), pages 1-18, August.
    2. T. T. Hue & J. J. Strodiot & V. H. Nguyen, 2004. "Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 119-145, April.

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