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Auxiliary Principle and Iterative Algorithms for Lions-Stampacchia Variational Inequalities

Author

Listed:
  • F. Q. Xia

    (Sichuan Normal University)

  • N. J. Huang

    (Sichuan University)

Abstract

In this paper, we extend the auxiliary principle (Cohen in J. Optim. Theory Appl. 49:325–333, 1988) to study a class of Lions-Stampacchia variational inequalities in Hilbert spaces. Our method consists in approximating, in the subproblems, the nonsmooth convex function by a sequence of piecewise linear and convex functions, as in the bundle method for nonsmooth optimization. This makes the subproblems more tractable. We show the existence of a solution for this Lions-Stampacchia variational inequality and explain how to build a new iterative scheme and a new stopping criterion. This iterative scheme and criterion are different from those commonly used in the special case of nonsmooth optimization. We study also the convergence of iterative sequences generated by the algorithm.

Suggested Citation

  • F. Q. Xia & N. J. Huang, 2009. "Auxiliary Principle and Iterative Algorithms for Lions-Stampacchia Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 377-389, February.
    • Handle: RePEc:spr:joptap:v:140:y:2009:i:2:d:10.1007_s10957-008-9441-z
    DOI: 10.1007/s10957-008-9441-z
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    Cited by:

    1. Ren-you Zhong & Nan-jing Huang, 2012. "Strict Feasibility for Generalized Mixed Variational Inequality in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 696-709, March.
    2. Xue-ping Luo, 2018. "Quasi-Strict Feasibility of Generalized Mixed Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 439-454, August.

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