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

Behzad Djafari Rouhani () and Sirous Moradi ()
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Behzad Djafari Rouhani: University of Texas at El Paso
Sirous Moradi: Arak University

Journal of Optimization Theory and Applications, 2017, vol. 172, issue 1, No 13, 222-235

Abstract: Abstract 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.

Keywords: Maximal monotone operator; Proximal point algorithm; Resolvent operator; Metric projection; Hilbert space; 47J25; 47H05; 47H09; 90C29; 90C90 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-016-1028-5

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