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 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.

SHRINKING PROJECTION ALGORITHMS FOR THE SPLIT COMMON NULL POINT PROBLEM

2017 ◽  
Vol 96 (2) ◽  
pp. 299-306 ◽  
Author(s):  
VAHID DADASHI
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Convergence Theorem ◽  
Maximal Monotone ◽  
Null Point ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Projection Algorithms

We consider the split common null point problem in Hilbert space. We introduce and study a shrinking projection method for finding a solution using the resolvent of a maximal monotone operator and prove a strong convergence theorem for the algorithm.

scholarly journals Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Suthep Suantai ◽  
Wataru Takahashi
Keyword(s):  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Zero Point ◽  
Common Element ◽  
Fixed Point Set ◽  
Nonlinear Operators ◽  
Point Set

We establish strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

scholarly journals A Proximal Point Method Involving Two Resolvent Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
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.

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 Strong Convergence Theorems for Zeros of Bounded Maximal Monotone Nonlinear Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
C. E. Chidume ◽  
N. Djitté
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Iteration Process ◽  
Accretive Operator ◽  
Initial Vector ◽  
Proximal Point ◽  
Nonlinear Operators ◽  
Uniformly Continuous

An iteration process studied by Chidume and Zegeye 2002 is proved to convergestronglyto a solution of the equationAu=0whereAis a boundedm-accretive operator on certain real Banach spacesEthat includeLpspaces2≤p<∞.The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets ofE, setbacks associated with the classicalproximal point algorithmof Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.

scholarly journals A METHOD OF APPROXIMATION FOR A ZERO OF MAXIMAL MONOTONE OPERATOR IN HILBERT SPACE

2020 ◽  
Vol 225 (02) ◽  
pp. 82-90
Author(s):  
Phạm Thị Thu Hoài ◽  
Nguyễn Thị Thúy Hoa ◽  
Nguyễn Tất Thắng
Keyword(s):  
Hilbert Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone

Trong bài báo này chúng tôi đưa ra một phương pháp lặp hiện mới giải bài toán bất đẳng thức biến phân trên tập không điểm của toán tử đơn điệu cực đại trong không gian Hilbert. Bằng việc sử dụng hai toán tử giải của một toán tử đơn điệu tại mỗi bước lặp, chúng tôi chứng minh sự hội tụ mạnh của phương pháp dưới điều kiện suy rộng đặt lên tham số.

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