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The Proximal Point Algorithm Revisited

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  • Yunda Dong

    (Zhengzhou University)

Abstract

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator’s value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem’s solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brézis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Güler’s classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an ϵ-approximation solution of the problem above.

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  • Yunda Dong, 2014. "The Proximal Point Algorithm Revisited," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 478-489, May.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:2:d:10.1007_s10957-013-0351-3
    DOI: 10.1007/s10957-013-0351-3
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    References listed on IDEAS

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    1. A. J. Zaslavski, 2011. "Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 20-32, July.
    2. Arkadi Nemirovski & Shmuel Onn & Uriel G. Rothblum, 2010. "Accuracy Certificates for Computational Problems with Convex Structure," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 52-78, February.
    3. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
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    Cited by:

    1. Yunda Dong, 2021. "Weak convergence of an extended splitting method for monotone inclusions," Journal of Global Optimization, Springer, vol. 79(1), pages 257-277, January.
    2. Yunda Dong, 2015. "Comments on “The Proximal Point Algorithm Revisited”," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 343-349, July.
    3. Eike Börgens & Christian Kanzow, 2019. "Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 73(3), pages 755-790, July.

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