# Four parameter proximal point algorithms

O. A. Boikanyo, G. Moroşanu

Research output: Contribution to journalArticle

28 Citations (Scopus)

### Abstract

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008)  while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996)  and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

Original language English 544-555 12 Nonlinear Analysis, Theory, Methods and Applications 74 2 https://doi.org/10.1016/j.na.2010.09.008 Published - Jan 15 2011

### Fingerprint

Proximal Point Algorithm
Strong Convergence
Distinct
Regularization Method
Weak Convergence
Convergence Results
Rate of Convergence
Converge
Norm
Generalise
Zero

### All Science Journal Classification (ASJC) codes

• Analysis
• Applied Mathematics

### Cite this

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abstract = "Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008)  while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996)  and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.",
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Four parameter proximal point algorithms. / Boikanyo, O. A.; Moroşanu, G.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 74, No. 2, 15.01.2011, p. 544-555.

Research output: Contribution to journalArticle

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AU - Moroşanu, G.

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AB - Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008)  while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996)  and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

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