On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators

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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Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/174796 ◽

2011 ◽

Vol 2011 ◽

pp. 1-31 ◽

Cited By ~ 3

Author(s):

Kriengsak Wattanawitoon ◽

Poom Kumam

Keyword(s):

Equilibrium Problems ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Zero Point ◽

Monotone Operators ◽

Common Element ◽

Proximal Point ◽

Strong And Weak Convergence ◽

Mixed Equilibrium Problems ◽

Inverse Strongly Monotone

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors.

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Quantitative results on Fejér monotone sequences

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500158 ◽

2017 ◽

Vol 20 (02) ◽

pp. 1750015 ◽

Cited By ~ 2

Author(s):

Ulrich Kohlenbach ◽

Laurenţiu Leuştean ◽

Adriana Nicolae

Keyword(s):

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Proximal Point ◽

Convergence Results ◽

Asymptotically Nonexpansive ◽

Firmly Nonexpansive ◽

Quantitative Results ◽

Iterative Procedures ◽

Fixed Point Iterations

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations [Formula: see text] for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called [Formula: see text]-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.

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