Strong Convergence of Regularized New Proximal Point Algorithms

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, 2019, vol. 181, issue 3, No 9, 864-882

Abstract: Abstract We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu’s theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence 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: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-019-01497-9

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