Strong Convergence of Two Proximal Point Algorithms with Possible Unbounded Error Sequences

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• 作者：Behzad Djafari Rouhani ; Sirous Moradi
• 关键词：Maximal monotone operator ; Proximal point algorithm ; Resolvent operator ; Metric projection ; Hilbert space
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：172
• 期：1
• 页码：222-235
• 全文大小：
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878
• 卷排序：172

文摘

We consider a proximal point algorithm with errors for a maximal monotone operator in a real Hilbert space, previously studied by Boikanyo and Morosanu, where they assumed that the zero set of the operator is nonempty and the error sequence is bounded. In this paper, by using our own approach, we significantly improve the previous results by giving a necessary and sufficient condition for the zero set of the operator to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the zero set of the operator, without assuming the boundedness of the error sequence. We study also in a similar way the strong convergence of a new proximal point algorithm and present some applications of our results to optimization and variational inequalities.