Strong convergence of regularized new proximal point algorithms. (English) Zbl 1481.47084
Summary: 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 H.-K. Xu [J. Glob. Optim. 36, No. 1, 115–125 (2006; Zbl 1131.90062)] and by O. A. Boikanyo and G. Moroşanu [Optim. Lett. 4, No. 4, 635–641 (2010; Zbl 1202.90271)], 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 [loc. cit.] 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 [loc. cit.]. Finally, we present some applications of our new convergence results to optimization and variational inequalities.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 90C29 | Multi-objective and goal programming |
| 90C90 | Applications of mathematical programming |
Keywords:
maximal monotone operator; proximal point algorithm; resolvent operator; metric projection; Hilbert space; strong convergenceReferences:
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