Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space

2020 ◽  
Vol 186 (1) ◽  
pp. 134-147 ◽  
Author(s):  
Behzad Djafari Rouhani ◽  
Vahid Mohebbi
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Inexact Proximal

scholarly journals Iterative methods for solving split feasibility problems and fixed point problems in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5345-5353
Author(s):  
Min Liu ◽  
Shih-Sen Changb ◽  
Ping Zuo ◽  
Xiaorong Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Recent Result ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Split Feasibility Problems ◽  
Split Feasibility

In this paper, we consider a class of split feasibility problems in Banach space. By using shrinking projective method and the modified proximal point algorithm, we propose an iterative algorithm. Under suitable conditions some strong convergence theorems are proved. Our results extend a recent result of Takahashi-Xu-Yao (Set-Valued Var. Anal. 23, 205-221 (2015)) from Hilbert spaces to Banach spaces. Moreover, the method of proof is also different.

MONOTONE OPERATORS AND THE PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) METRIC SPACES

2016 ◽  
Vol 103 (1) ◽  
pp. 70-90 ◽  
Author(s):  
HADI KHATIBZADEH ◽  
SAJAD RANJBAR
Keyword(s):  
Strong Convergence ◽  
Monotone Operator ◽  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Monotone Operators ◽  
Proximal Point ◽  
Range Condition ◽  
Special Cases ◽  
Inexact Proximal

In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

Another version of the proximal point algorithm in a Banach space

2009 ◽  
Vol 70 (9) ◽  
pp. 3159-3171 ◽  
Author(s):  
Ya.I. Alber ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Proximal Point Algorithm ◽  
Proximal Point

An inexact algorithm with proximal distances for variational inequalities

2018 ◽  
Vol 52 (1) ◽  
pp. 159-176 ◽  
Author(s):  
E.A. Papa Quiroz ◽  
L. Mallma Ramirez ◽  
P.R. Oliveira
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Proximal Point Algorithm ◽  
Convergence Properties ◽  
Proximal Point ◽  
Extra Condition ◽  
Bregman Distances ◽  
Optim Theory ◽  
Inexact Proximal ◽  
Inexact Algorithm

In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generated by the algorithm is convergent for the pseudomonotone case and assuming an extra condition on the solution set we prove the convergence for the quasimonotone case. This approach unifies the results obtained by Auslender et al. [Math Oper. Res. 24 (1999) 644–688] and Brito et al. [J. Optim. Theory Appl. 154 (2012) 217–234] and extends the convergence properties for the class of φ-divergence distances and Bregman distances.

scholarly journals Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

2009 ◽  
Vol 2009 ◽  
pp. 1-11
Author(s):  
Ram U. Verma
Keyword(s):  
Convergence Analysis ◽  
Proximal Point Algorithm ◽  
Evolution Equations ◽  
Real Hilbert Space ◽  
Linear Convergence ◽  
Proximal Point ◽  
Evolution Inclusions ◽  
Inclusion Problems ◽  
Set Valued Mapping ◽  
Inexact Proximal

Based on a notion ofrelatively maximal(m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.

scholarly journals On the proximal point algorithm and demimetric mappings in CAT(0) spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 277-294 ◽  
Author(s):  
Kazeem O. Aremu ◽  
Chinedu Izuchukwu ◽  
Godwin C. Ugwunnadi ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Finite Family ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Numerical Example ◽  
Common Solution ◽  
Minimization Problems

Abstract In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

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