scholarly journals Forcing strong convergence of proximal point iterations in a Hilbert space

2000 ◽  
Vol 87 (1) ◽  
pp. 189-202 ◽  
Author(s):  
M.V. Solodov ◽  
B.F. Svaiter
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Proximal Point

scholarly journals On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

2021 ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Monotone Operators ◽  
Rates Of Convergence ◽  
Effective Rate ◽  
Regularity Assumption ◽  
Proximal Point ◽  
Compact Case ◽  
Quantitative Results

AbstractIn a recent paper, Bauschke et al. study $$\rho $$ ρ -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent $$J_A.$$ J A . In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for $$\rho $$ ρ -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. $$zer\, A$$ z e r A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for $$\rho $$ ρ -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

scholarly journals A New Approximation Method for Finding Zeros of Maximal Monotone Operators

2021 ◽  
Vol 31 (2) ◽  
pp. 117-124
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Point ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

scholarly journals A Modified Regularization Method for the Proximal Point Algorithm

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Shuang Wang
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Regularization Method ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Proximal Point ◽  
Weaker Conditions

Under some weaker conditions, we prove the strong convergence of the sequence generated by a modified regularization method of finding a zero for a maximal monotone operator in a Hilbert space. In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the corresponding results.

scholarly journals Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bin-Chao Deng ◽  
Tong Chen
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Mild Conditions ◽  
Strongly Monotone

LetHbe a real Hilbert space. LetT1,T2:H→Hbek1-,k2-strictly pseudononspreading mappings; letαnandβnbe two real sequences in (0,1). For givenx0∈H, the sequencexnis generated iteratively byxn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn,∀n∈N, whereTwi=1−wiI+wiTiwithi=1,2andB:H→His strongly monotone and Lipschitzian. Under some mild conditions on parametersαnandβn, we prove that the sequencexnconverges strongly to the setFixT1∩FixT2of fixed points of a pair of strictly pseudononspreading mappingsT1andT2.

Modified CQ-Algorithms for G-Nonexpansive Mappings in Hilbert Spaces Involving Graphs

2020 ◽  
Vol 16 (01) ◽  
pp. 89-103
Author(s):  
W. Cholamjiak ◽  
D. Yambangwai ◽  
H. Dutta ◽  
H. A. Hammad
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Rate Of Convergence ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Iterative Schemes ◽  
Cq Method

In this paper, we introduce four new iterative schemes by modifying the CQ-method with Ishikawa and [Formula: see text]-iterations. The strong convergence theorems are given by the CQ-projection method with our modified iterations for obtaining a common fixed point of two [Formula: see text]-nonexpansive mappings in a Hilbert space with a directed graph. Finally, to compare the rate of convergence and support our main theorems, we give some numerical experiments.

scholarly journals Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 480
Author(s):  
Manatchanok Khonchaliew ◽  
Ali Farajzadeh ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Constraint Qualifications ◽  
Solution Sets ◽  
New Algorithms

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm.

scholarly journals Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Baogui Xin
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings

Let{T}i=1NbeNquasi-nonexpansive mappings defined on a closed convex subsetCof a real Hilbert spaceH. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.

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