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An Inexact Proximal-Type Algorithm in Banach Spaces

Author

Listed:
  • L. C. Zeng

    (Shanghai Normal University)

  • J. C. Yao

    (National Sun Yat-Sen University)

Abstract

In this paper, we investigate the strong convergence of an inexact proximal-point algorithm. It is known that the proximal-point algorithm converges weakly to a solution of a maximal monotone operator, but fails to converge strongly. Solodov and Svaiter (Math. Program. 87:189–202, 2000) introduced a new proximal-type algorithm to generate a strongly convergent sequence and established a convergence result in Hilbert space. Subsequently, Kamimura and Takahashi (SIAM J. Optim. 13:938–945, 2003) extended the Solodov and Svaiter result to the setting of uniformly convex and uniformly smooth Banach space. On the other hand, Rockafellar (SIAM J. Control Optim. 14:877–898, 1976) gave an inexact proximal-point algorithm which is more practical than the exact one. Our purpose is to extend the Kamimura and Takahashi result to a new inexact proximal-type algorithm. Moreover, this result is applied to the problem of finding the minimizer of a convex function on a uniformly convex and uniformly smooth Banach space.

Suggested Citation

  • L. C. Zeng & J. C. Yao, 2007. "An Inexact Proximal-Type Algorithm in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 135(1), pages 145-161, October.
    • Handle: RePEc:spr:joptap:v:135:y:2007:i:1:d:10.1007_s10957-007-9261-6
    DOI: 10.1007/s10957-007-9261-6
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    Cited by:

    1. L. C. Ceng & G. Mastroeni & J. C. Yao, 2009. "Hybrid Proximal-Point Methods for Common Solutions of Equilibrium Problems and Zeros of Maximal Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 431-449, September.

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