### 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 language | English |
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Title of host publication | Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics |

Pages | 893-896 |

Number of pages | 4 |

Volume | 1389 |

DOIs | |

Publication status | Published - Nov 28 2011 |

Event | International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011 - Halkidiki, Greece Duration: Sep 19 2011 → Sep 25 2011 |

### Other

Other | International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011 |
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Country | Greece |

City | Halkidiki |

Period | 9/19/11 → 9/25/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Proximal point methods revisited

AU - Boikanyo, Oganeditse A.

AU - Moroşanu, Gheorghe

PY - 2011/11/28

Y1 - 2011/11/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=81855211947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81855211947&partnerID=8YFLogxK

U2 - 10.1063/1.3636878

DO - 10.1063/1.3636878

M3 - Conference contribution

SN - 9780735409569

VL - 1389

SP - 893

EP - 896

BT - Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics

ER -