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Finite Termination of Inexact Proximal Point Algorithms in Hilbert Spaces

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Abstract

In the present paper, we study the finite termination of sequences generated by inexact proximal point algorithms for finding zeroes of a maximal monotone (set-valued) operator \(T\) on a Hilbert space. Under some mild conditions, we get that a sequence generated by inexact proximal point algorithm stops after a finite number of iterations. Our results extend the corresponding results in Rockafellar (SIAM J Control Optim 14:877–898, 1976). In particular, for optimization problems, our results improve corresponding results in Ferris (Math Progr 50:359–366, 1991). As applications, we obtain finite termination of projected gradient method.

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Acknowledgments

The authors thank the referees and the editor for their valuable comments and constructive suggestions which improve the presentation of this manuscript. Research of the first author is supported in part by the National Natural Science Foundation of China (Grant 11371325) and by Zhejiang Provincial Natural Science Foundation of China (Grant LY13A010011). Research of the second author is supported in part by the National Natural Science Foundation of China (Grant 11171300). Research of the third author was partially supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-037-002-MY3.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Technology, Hangzhou, 310032, People’s Republic of China

    J. H. Wang

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    C. Li

  3. Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 80702, Taiwan

    J.-C. Yao

  4. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    J.-C. Yao

Authors
  1. J. H. Wang
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  2. C. Li
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  3. J.-C. Yao
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Correspondence to J. H. Wang.

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Communicated by Viorel Barbu.

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Wang, J.H., Li, C. & Yao, JC. Finite Termination of Inexact Proximal Point Algorithms in Hilbert Spaces. J Optim Theory Appl 166, 188–212 (2015). https://doi.org/10.1007/s10957-014-0689-1

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  • DOI: https://doi.org/10.1007/s10957-014-0689-1

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