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On the Proximal Point Algorithm

Author

Listed:
  • B. Djafari Rouhani

    (University of Texas at El Paso)

  • H. Khatibzadeh

    (Tarbiat Modarres University)

Abstract

Let A be a maximal monotone operator in a real Hilbert space H and let {u n } be the sequence in H given by the proximal point algorithm, defined by u n =(I+c n A)−1(u n−1−f n ), ∀ n≥1, with u 0=z, where c n >0 and f n ∈H. We show, among other things, that under suitable conditions, u n converges weakly or strongly to a zero of A if and only if lim inf n→+∞|w n |

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:137:y:2008:i:2:d:10.1007_s10957-007-9329-3
    DOI: 10.1007/s10957-007-9329-3
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    Citations

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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.
    2. Vahid Dadashi & Mihai Postolache, 2017. "Hybrid Proximal Point Algorithm and Applications to Equilibrium Problems and Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 518-529, August.
    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.
    4. 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.
    5. Hadi Khatibzadeh, 2012. "Some Remarks on the Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 769-778, June.
    6. Hadi Khatibzadeh & Sajad Ranjbar, 2013. "On the Strong Convergence of Halpern Type Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 385-396, August.

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