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Strong Convergence of Regularized New Proximal Point Algorithms

Author

Listed:
  • Behzad Djafari Rouhani

    (University of Texas at El Paso)

  • Sirous Moradi

    (Arak University)

Abstract

We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu’s theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence results to optimization and variational inequalities.

Suggested Citation

  • Behzad Djafari Rouhani & Sirous Moradi, 2019. "Strong Convergence of Regularized New Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 864-882, June.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:3:d:10.1007_s10957-019-01497-9
    DOI: 10.1007/s10957-019-01497-9
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    References listed on IDEAS

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    1. B. Djafari Rouhani & H. Khatibzadeh, 2008. "On the Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 411-417, May.
    2. Behzad Djafari Rouhani & Sirous Moradi, 2017. "Strong Convergence of Two Proximal Point Algorithms with Possible Unbounded Error Sequences," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 222-235, January.
    3. Fenghui Wang, 2011. "A note on the regularized proximal point algorithm," Journal of Global Optimization, Springer, vol. 50(3), pages 531-535, July.
    4. Yamin Wang & Fenghui Wang & Hong-Kun Xu, 2016. "Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 901-916, March.
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    Cited by:

    1. Behzad Djafari Rouhani & Vahid Mohebbi, 2020. "Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 134-147, July.

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