An Inexact Proximal-Type Algorithm in Banach Spaces

L. C. Zeng and J. C. Yao ()
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L. C. Zeng: Shanghai Normal University
J. C. Yao: National Sun Yat-Sen University

Journal of Optimization Theory and Applications, 2007, vol. 135, issue 1, No 10, 145-161

Abstract: 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.

Keywords: Inexact proximal-point algorithms; Maximal monotone operators; Banach spaces (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-007-9261-6

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