The proximal point algorithm in metric spaces

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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Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2018-0026 ◽

2019 ◽

Vol 10 (4) ◽

pp. 437-446 ◽

Cited By ~ 4

Author(s):

Godwin C. Ugwunnadi ◽

Chinedu Izuchukwu ◽

Oluwatosin T. Mewomo

Keyword(s):

Fixed Point ◽

Nonexpansive Mapping ◽

Metric Space ◽

Proximal Point Algorithm ◽

Metric Spaces ◽

Uniformly Convex ◽

Convex Metric Space ◽

Proximal Point ◽

Common Solution ◽

Uniformly Convex Metric Space

AbstractWe prove some important properties of the p-resolvent mapping recently introduced by B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc. 31 2016, 4, 845–855, in p-uniformly convex metric space. Furthermore, we introduce a modified Mann-type PPA involving nonexpansive mapping and prove that the sequence generated by the algorithm converges to a common solution of a finite family of minimization problems, which is also a fixed point of a nonexpansive mapping in the framework of a complete p-uniformly convex metric space.

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THE PROXIMAL POINT ALGORITHM IN UNIFORMLY CONVEX METRIC SPACES

Communications of the Korean Mathematical Society ◽

10.4134/ckms.c150114 ◽

2016 ◽

Vol 31 (4) ◽

pp. 845-855 ◽

Cited By ~ 3

Author(s):

Byoung Jin Choi ◽

Un Cig Ji

Keyword(s):

Proximal Point Algorithm ◽

Metric Spaces ◽

Uniformly Convex ◽

Proximal Point ◽

Convex Metric Spaces

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Proximal Point Algorithm for Quasi-Convex Minimization Problems in Metric Spaces

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2017.1374970 ◽

2017 ◽

Vol 39 (4) ◽

pp. 438-448 ◽

Cited By ~ 2

Author(s):

Hadi Khatibzadeh ◽

Vahid Mohebbi

Keyword(s):

Proximal Point Algorithm ◽

Metric Spaces ◽

Convex Minimization ◽

Proximal Point ◽

Minimization Problems ◽

Convex Minimization Problems

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

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000446 ◽

2016 ◽

Vol 103 (1) ◽

pp. 70-90 ◽

Cited By ~ 20

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.

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