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Zbl 1217.65102
He, Bingsheng; Liao, Lizhi; Yang, Zhenhua
A new approximate proximal point algorithm for maximal monotone operator.
(English)
[J] Sci. China, Ser. A 46, No. 2, 200-206 (2003). ISSN 1006-9283; ISSN 1862-2763/e

Summary: The problem concerned in this paper is the set-valued equation $0 \in T(z)$ where $T$ is a maximal monotone operator. For given $x^{k}$ and $\beta_{k}> 0$, some existing approximate proximal point algorithms take $x^{k + 1} = \tilde{x}^{k}$ such that $x^{k} + e^{k}\in \tilde{x}^{k} + b_{k} T(\tilde{x}^{k})$ and $\Vert e^{k}\Vert\leq \eta_{k} \Vert x^{k} - \tilde{x}^{k}\Vert$, where $\{\eta_k\}$ is a non-negative summable sequence. Instead of $x^{k + 1} = \tilde{x}^{k}$, the new iterate of the proposing method is given by $x^{k + 1} = P_{\Omega} [\tilde{x}^{k} - e^{k}]$, where $\Omega$ is the domain of $T$ and $P_{\Omega} (\cdot )$ denotes the projection on $\Omega$. The convergence is proved under a significantly relaxed restriction $\sup K>0 \eta K\eta 1$.
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47J25 Methods for solving nonlinear operator equations (general)
90C47 Minimax problems

Keywords: proximal point algorithms; monotone operators; approximate methods

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