On the Strong Convergence of Halpern Type Proximal Point Algorithm

2012 ◽  
Vol 158 (2) ◽  
pp. 385-396 ◽  
Author(s):  
Hadi Khatibzadeh ◽  
Sajad Ranjbar
Keyword(s):  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Proximal Point

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.

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.

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