scholarly journals The strong convergence of a proximal point algorithm in complete CAT(0) metric spaces

2019 ◽  
pp. 1-10
Author(s):  
Sirous Moradi ◽  
Mohsen Tahernia ◽  
Somayeh Jafari
Keyword(s):  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Proximal Point

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.

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.

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.

New results on the proximal point algorithm in non-positive curvature metric spaces

Optimization ◽  
2017 ◽  
Vol 66 (7) ◽  
pp. 1191-1199 ◽  
Author(s):  
Hadi Khatibzadeh ◽  
Vahid Mohebbi ◽  
Sajad Ranjbar
Keyword(s):  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Positive Curvature ◽  
Proximal Point

scholarly journals Halpern-type proximal point algorithm in complete CAT(0) metric spaces

2016 ◽  
Vol 24 (3) ◽  
pp. 141-159 ◽  
Author(s):  
Mohammad Taghi Heydari ◽  
Sajad Ranjbar
Keyword(s):  
Convergence Theorem ◽  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Proximal Point

AbstractFirst, Halpern-type proximal point algorithm is introduced in complete CAT(0) metric spaces. Then, Browder convergence theorem is considered for this algorithm and also we prove that Halpern-type proximal point algorithm converges strongly to a zero of the operator.

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