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Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space

Author

Listed:
  • Behzad Djafari Rouhani

    (University of Texas at El Paso)

  • Vahid Mohebbi

    (University of Texas at El Paso)

Abstract

By using our own approach, we study the strong convergence of an inexact proximal point algorithm with possible unbounded errors for a maximal monotone operator in a Banach space. We give a necessary and sufficient condition for the zero set of the operator to be nonempty and show that, in this case, this iterative sequence converges strongly to a zero of the operator. We present also some applications of our results to equilibrium problems and optimization. Our proximal point algorithm contains, as a special case, the one considered in Hilbert space by Djafari Rouhani and Moradi in (J Optim Theory Appl 172:222–235, 2017) and solves the open problem of extending it to a Banach space, which was stated in that paper and in Djafari Rouhani and Moradi in (J Optim Theory Appl 181:864–882, 2019) . Since the nonexpansiveness of the resolvent operator, which holds in Hilbert space, is not valid anymore in Banach space, our results require new methods of proofs, and significantly improve upon the previous results, both in theory and in applications.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:1:d:10.1007_s10957-020-01695-w
    DOI: 10.1007/s10957-020-01695-w
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
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