TY - GEN

T1 - Proximal point methods revisited

AU - Boikanyo, Oganeditse A.

AU - Moroşanu, Gheorghe

PY - 2011

Y1 - 2011

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

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U2 - 10.1063/1.3636878

DO - 10.1063/1.3636878

M3 - Conference contribution

AN - SCOPUS:81855211947

SN - 9780735409569

T3 - AIP Conference Proceedings

SP - 893

EP - 896

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

T2 - International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011

Y2 - 19 September 2011 through 25 September 2011

ER -