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A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method

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  • Benar F. Svaiter

    (IMPA)

Abstract

A new framework for analyzing Fejér convergent algorithms is presented. Using this framework, we define a very general class of Fejér convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents, which preserves some useful features of the exact resolvent and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng’s Modified Forward-Backward splitting method, and Korpelevich’s method. We show that methods, based on families of approximate resolvents, fall within the aforementioned class of Fejér convergent methods. We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method, which is a generalization of the classical Proximal Point Algorithm.

Suggested Citation

  • Benar F. Svaiter, 2014. "A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 133-153, July.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0449-7
    DOI: 10.1007/s10957-013-0449-7
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    References listed on IDEAS

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    1. Alfredo N. Iusem & B. F. Svaiter & Marc Teboulle, 1994. "Entropy-Like Proximal Methods in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 790-814, November.
    2. R. S. Burachik & S. Scheimberg & B. F. Svaiter, 2001. "Robustness of the Hybrid Extragradient Proximal-Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 117-136, October.
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    2. Yonghong Yao & Mihai Postolache & Jen-Chih Yao, 2019. "An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems," Mathematics, MDPI, vol. 7(1), pages 1-15, January.

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