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Auxiliary Principle and Iterative Algorithms for Lions-Stampacchia Variational Inequalities

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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.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, 610066, People’s Republic of China

    F. Q. Xia

  2. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, People’s Republic of China

    N. J. Huang

Authors
  1. F. Q. Xia
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  2. N. J. Huang
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Corresponding author

Correspondence to N. J. Huang.

Additional information

Communicated by F. Giannessi.

This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005), the National Natural Science Foundation of Sichuan Education Department of China (07ZB068) and the Open Fund (PLN0703) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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Xia, F.Q., Huang, N.J. Auxiliary Principle and Iterative Algorithms for Lions-Stampacchia Variational Inequalities. J Optim Theory Appl 140, 377–389 (2009). https://doi.org/10.1007/s10957-008-9441-z

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