Proximal point methods revisited

Oganeditse A. Boikanyo, Gheorghe Moroşanu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The proximal point methods have been widely used in the last decades to approximate the solutions of nonlinear equations associated with monotone operators. Inspired by the iterative procedure defined by B. Martinet (1970), R.T. Rockafellar introduced in 1976 the so-called proximal point algorithm (PPA) for a general maximal monotone operator. The sequence generated by this iterative method is weakly convergent under appropriate conditions, but not necessarily strongly convergent, as proved by O. Güler (1991). This fact explains the introduction of different modified versions of the PPA which generate strongly convergent sequences under appropriate conditions, including the contraction-PPA defined by H.K. Xu in 2002. Here we discuss Xu's modified PPA as well as some of its generalizations. Special attention is paid to the computational errors, in particular the original Rockafellar summability assumption is replaced by the condition that the error sequence converges to zero strongly.

Original languageEnglish
Title of host publicationNumerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics
Pages893-896
Number of pages4
Volume1389
DOIs
Publication statusPublished - Nov 28 2011
EventInternational Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011 - Halkidiki, Greece
Duration: Sep 19 2011Sep 25 2011

Other

OtherInternational Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011
CountryGreece
CityHalkidiki
Period9/19/119/25/11

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operators
nonlinear equations
contraction

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

Cite this

Boikanyo, O. A., & Moroşanu, G. (2011). Proximal point methods revisited. In Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics (Vol. 1389, pp. 893-896) https://doi.org/10.1063/1.3636878
Boikanyo, Oganeditse A. ; Moroşanu, Gheorghe. / Proximal point methods revisited. Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics. Vol. 1389 2011. pp. 893-896
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Boikanyo, OA & Moroşanu, G 2011, Proximal point methods revisited. in Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics. vol. 1389, pp. 893-896, International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011, Halkidiki, Greece, 9/19/11. https://doi.org/10.1063/1.3636878

Proximal point methods revisited. / Boikanyo, Oganeditse A.; Moroşanu, Gheorghe.

Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics. Vol. 1389 2011. p. 893-896.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Boikanyo OA, Moroşanu G. Proximal point methods revisited. In Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics. Vol. 1389. 2011. p. 893-896 https://doi.org/10.1063/1.3636878