Proximal point algorithms based on S-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications

Sahu, D. R.; Kumar, Ajeet; Kang, Shin Min

doi:10.1007/s11075-020-00945-2

Cited by 10 publications

(4 citation statements)

References 34 publications

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“…Table 3 shows the computative result for the proposed iteration and Sahu's iteration with starting point (3, 1, -2), and it can be seen directly that both iteration processes converge to the point (-0.5, -2.5, 0) of P. Remark 4 From data in Tables 1-3, we see that computative results of our proposed iteration process have the lower number of iterations than the modified PPA of Sahu et al [34], 35% on average. This shows that our proposed iteration has better convergence rate than the modified PPA iteration.…”

Section: Applicationmentioning

confidence: 62%

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A modified proximal point algorithm involving nearly asymptotically quasi-nonexpansive mappings

2021

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In this paper, we propose a modified proximal point algorithm based on the Thakur iteration process to approximate the common element of the set of solutions of convex minimization problems and the fixed points of two nearly asymptotically quasi-nonexpansive mappings in the framework of $\operatorname{CAT}(0)$ CAT ( 0 ) spaces. We also prove the Δ-convergence of the proposed algorithm. We also provide an application and numerical result based on our proposed algorithm as well as the computational result by comparing our modified iteration with previously known Sahu’s modified iteration.

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Section: Applicationmentioning

confidence: 62%

“…Recently, Sahu et al [34] introduced a modified PPA based on the S-iterative scheme to approximate a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the setting of CAT(0) spaces, where the sequence {χ n } is generated as follows:…”

Section: Introductionmentioning

confidence: 99%

A modified proximal point algorithm involving nearly asymptotically quasi-nonexpansive mappings

2021

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“…□ Let C = {(x 1 , x 2 , x 3 ) ∈ M : 1 ≤ x i ≤ 2 for i = 1, 2, 3}. Then, C is a closed, geodesic convex subset of M. Note that h is a proper continuous convex function on R 3 (see [36]). Since g is an isometry, it follows from Proposition 2 that f is a proper continuous geodesic convex function on M. From Proposition 6, grad f is a monotone vector field.…”

Section: Algorithmmentioning

confidence: 99%

A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

D. R.,

Feeroz,

Shikher

2023

Numer Algor

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The objective of this work is to design a new iterative method based on Armijo's type-modified extragradient method for solving the inclusion problem (A + B) −1 (0), where A is a maximal monotone vector field and B is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the rate of convergence. A convergence theorem is established for the proposed extragradient method, which significantly improves existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field's monotonicity can be removed.

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“…Since last few years, these properties of normal S-iteration make it popular among research community to find fixed point. Several research articles related to S-iteration and normal S-iteration can be found in [14,15,18,42,43]. The weak convergence of the fixed point algorithms has reduced its applicability in infinite dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

Tikhonov Regularized Iterative Methods for Nonlinear Problems

Dixit¹,

R.²,

Gautam³

et al. 2021

Preprint

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We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of nonexpansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward-backward-type primal-dual algorithm and a Douglas-Rachford-type primal-dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.

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Proximal point algorithms based on S-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications

Cited by 10 publications

References 34 publications

A modified proximal point algorithm involving nearly asymptotically quasi-nonexpansive mappings

A modified proximal point algorithm involving nearly asymptotically quasi-nonexpansive mappings

A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

Tikhonov Regularized Iterative Methods for Nonlinear Problems

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