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Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

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Abstract

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    L. C. Ceng

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China

    L. C. Ceng

  3. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    M. Teboulle

  4. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, 804, Republic of China

    J. C. Yao

Authors
  1. L. C. Ceng
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  2. M. Teboulle
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

Communicated by S. Schaible.

L.C. Ceng’s research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), PhD Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).

M. Teboulle thanks the National Sun Yat-sen University, Kaohsiung, Taiwan, where he was visiting while this research was conducted.

J.C. Yao’s research was partially supported by Grant NSC 98-2923-E-110-003-MY3.

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Ceng, L.C., Teboulle, M. & Yao, J.C. Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems. J Optim Theory Appl 146, 19–31 (2010). https://doi.org/10.1007/s10957-010-9650-0

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  • DOI: https://doi.org/10.1007/s10957-010-9650-0

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