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Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities

Author

Listed:
  • T. T. Hue

    (University of Hue)

  • J. J. Strodiot

    (Facultés Universitaires Notre Dame de la Paix)

  • V. H. Nguyen

    (Facultés Universitaires Notre Dame de la Paix)

Abstract

We consider an extension of the auxiliary problem principle for solving a general variational inequality problem. This problem consists in finding a zero of the sum of two operators defined on a real Hilbert space H: the first is a monotone single-valued operator; the second is the subdifferential of a lower semicontinuous proper convex function ϕ. To make the subproblems easier to solve, we consider two kinds of lower approximations for the function ϕ: a smooth approximation and a piecewise linear convex approximation. We explain how to construct these approximations and we prove the weak convergence and the strong convergence of the sequence generated by the corresponding algorithms under a pseudo Dunn condition on the single-valued operator. Finally, we report some numerical experiences to illustrate the behavior of the two algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:1:d:10.1023_b:jota.0000026134.57920.e1
    DOI: 10.1023/B:JOTA.0000026134.57920.e1
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    References listed on IDEAS

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    1. Correa Romar, 2014. "Mathematical Foci," Mathematical Economics Letters, De Gruyter, vol. 2(1-2), pages 1-7, August.
    2. 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.
    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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