关于极大强单调算子的不精确邻近点算法的收敛性分析

摘要/Abstract

摘要：

该文研究集值映象方程0∈T(z)的解的迭代逼近，其中T是极大强单调算子．设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列，满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了，{x^k}收敛到T的一个根当且仅当
lim inf_{k→+∞} d(x^k,Z)=0，其中Z是方程0∈T(z)的解集

关键词: 邻近点算法,极大强单调算子,不精确方法

Abstract:

The purpose of this paper is to study the iterative approximation of solutions to the set valued mapping equation0∈T(z) where T is a maximal strongly monotone operator. Suppose that {x^k} and {e^k} are the sequences generated by the inexact proximal point algorithm
x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1} such that ‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞ and inf_(k≥0) η_k=μ≥1 Under suitable restrictions the author proves that{x^k} converges to a root of T if and only if liminf_{k→+∞} d(x^k,Z)=0

Key words: Proximal point algorithm, Maximal strongly monotone operator,Inexact method.

中图分类号:

47H09

曾六川. 关于极大强单调算子的不精确邻近点算法的收敛性分析[J]. 数学物理学报, 2005, 25(2): 281-288.

CENG Liu-Chuan. On the Convergence Analysis of Inexact Proximal Point Algorithms for Maximal Strongly Monotone Operators [J]. Acta mathematica scientia,Series A, 2005, 25(2): 281-288.

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