An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

Mathematical Problems in Engineering ◽

10.1155/2011/262073 ◽

2011 ◽

Vol 2011 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Lingling Huang ◽

Sanyang Liu ◽

Weifeng Gao

Keyword(s):

Hilbert Spaces ◽

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Computational Results ◽

Monotone Inclusion ◽

Proximal Point ◽

Inclusion Problems ◽

Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

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Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces

Journal of Inequalities and Applications ◽

10.1186/1029-242x-2012-137 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 2

Author(s):

Zhiqiang Wei ◽

Guohong Shi

Keyword(s):

Hilbert Spaces ◽

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point

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Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space

SIAM Journal on Optimization ◽

10.1137/s1052623403427859 ◽

2004 ◽

Vol 14 (3) ◽

pp. 773-782 ◽

Cited By ~ 110

Author(s):

Felipe Alvarez

Keyword(s):

Hilbert Space ◽

Weak Convergence ◽

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point

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Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors

Journal of Optimization Theory and Applications ◽

10.1007/s10957-011-9820-8 ◽

2011 ◽

Vol 150 (1) ◽

pp. 20-32 ◽

Cited By ~ 10

Author(s):

A. J. Zaslavski

Keyword(s):

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point ◽

Computational Errors

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Tight Sublinear Convergence Rate of the Proximal Point Algorithm for Maximal Monotone Inclusion Problems

SIAM Journal on Optimization ◽

10.1137/19m1299049 ◽

2020 ◽

Vol 30 (3) ◽

pp. 1905-1921

Author(s):

Guoyong Gu ◽

Junfeng Yang

Keyword(s):

Convergence Rate ◽

Proximal Point Algorithm ◽

Maximal Monotone ◽

Monotone Inclusion ◽

Proximal Point ◽

Inclusion Problems ◽

Sublinear Convergence Rate

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Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

Journal of Applied Mathematics ◽

10.1155/2013/280909 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Huan-chun Wu ◽

Cao-zong Cheng

Keyword(s):

Hilbert Spaces ◽

Equilibrium Problems ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Common Element ◽

Inclusion Problems ◽

Minimization Problems ◽

Convex Minimization Problems ◽

New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

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Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

Journal of Inequalities and Applications ◽

10.1155/2008/598191 ◽

2008 ◽

Vol 2008 ◽

pp. 1-10 ◽

Cited By ~ 6

Author(s):

Yeol Je Cho ◽

Shin Min Kang ◽

Haiyun Zhou

Keyword(s):

Hilbert Spaces ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point ◽

Proximal Point Algorithms

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Maximal Monotone Operators and the Proximal Point Algorithm

Springer Optimization and Its Applications - Numerical Optimization with Computational Errors ◽

10.1007/978-3-319-30921-7_11 ◽

2016 ◽

pp. 169-181

Author(s):

Alexander J. Zaslavski

Keyword(s):

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point

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A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces

Nonlinear Analysis Hybrid Systems ◽

10.1016/j.nahs.2006.08.003 ◽

2007 ◽

Vol 1 (3) ◽

pp. 398-413 ◽

Cited By ~ 5

Author(s):

Liwei Li ◽

W. Song

Keyword(s):

Banach Spaces ◽

Proximal Point Algorithm ◽

Extragradient Method ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point ◽

Strongly Monotone ◽

Inverse Strongly Monotone ◽

Strongly Monotone Operators

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A Proximal Point Method Involving Two Resolvent Operators

Abstract and Applied Analysis ◽

10.1155/2012/892980 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Oganeditse A. Boikanyo

Keyword(s):

Hilbert Space ◽

Proximal Point Algorithm ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point Method ◽

Resolvent Operators ◽

Proximal Point ◽

Proximal Point Algorithms ◽

Frame Work

We construct a sequence of proximal iterates that converges strongly (under minimal assumptions) to a common zero of two maximal monotone operators in a Hilbert space. The algorithm introduced in this paper puts together several proximal point algorithms under one frame work. Therefore, the results presented here generalize and improve many results related to the proximal point algorithm which were announced recently in the literature.

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