Inexact Halpern-type proximal point algorithm

O. A. Boikanyo; G. Moroşanu

doi:10.1007/s10898-010-9616-7

Inexact Halpern-type proximal point algorithm

O. A. Boikanyo, G. Moroşanu

Mathematics & Statistical Sciences

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

Abstract

We present several strong convergence results for the modified, Halpern-type, proximal point algorithm x_n+1 = α_nu + (1 - α_n)J_βn x_n + e_n (n = 0, 1, . . .; u, x₀ ∈ H given, and J_βn = (I+β_nA)^-1, for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on α_n ∈ (0,1) and β_n ∈ (0,∞). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.

Original language	English
Pages (from-to)	11-26
Number of pages	16
Journal	Journal of Global Optimization
Volume	51
Issue number	1
DOIs	https://doi.org/10.1007/s10898-010-9616-7
Publication status	Published - Sept 2011

All Science Journal Classification (ASJC) codes

Control and Optimization
Applied Mathematics
Computer Science Applications
Management Science and Operations Research

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N2 - We present several strong convergence results for the modified, Halpern-type, proximal point algorithm xn+1 = αnu + (1 - αn)Jβn xn + en (n = 0, 1, . . .; u, x0 ∈ H given, and Jβn = (I+βnA)-1, for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on αn ∈ (0,1) and βn ∈ (0,∞). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.

AB - We present several strong convergence results for the modified, Halpern-type, proximal point algorithm xn+1 = αnu + (1 - αn)Jβn xn + en (n = 0, 1, . . .; u, x0 ∈ H given, and Jβn = (I+βnA)-1, for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on αn ∈ (0,1) and βn ∈ (0,∞). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.

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Inexact Halpern-type proximal point algorithm

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