An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces

Cholamjiak, Watcharaporn; Cholamjiak, Prasit; Suantai, Suthep

doi:10.1007/s11784-018-0526-5

Cited by 98 publications

(66 citation statements)

References 34 publications

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“…The introduction of the inertial viscosity splitting algorithms sheds new light on inclusion problem. Combined with recent research findings ( [4,13,19,20]), Theorem 1 can be further applied to the fixed-point problem, the split feasibility problem and the variational inequality problem. Indeed, it is an important but unsolved problem to choose the optimal inertia parameters α n in the acceleration algorithm.…”

Section: Resultsmentioning

confidence: 84%

“…The strong convergence theorems are established, and the numerical experiments are presented to support that the inertial extrapolation greatly improves the efficiency of the algorithm. In Theorem 1 and Corollary 1, if f (x n ) = u and A is an inverse strongly monotone operator in Hilbert space, it is the main results of Cholamjiak et al [20]. In Theorem 1, if α n = 0, f (x n ) = u and E is a uniformly convex and q-uniformly smooth Banach space, it is the main results of Pholasa et al [10].…”

Section: Resultsmentioning

confidence: 94%

“…converges weakly to a solution of the inclusion 0 ∈ (A + B)(x). In 2018, Cholamjiak et al [20] proposed a Halpern-type inertial iterative method for monotone operators in Hilbert spaces and they proved the strong convergence of the algorithm. Inspired and motivated by the above-mentioned works, we apply inertial extrapolation algorithms and viscosity approximation to give an extension, and then we study a modified splitting method for accretive operators in Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Pan

Wang

2019

Mathematics

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In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method.

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Section: Resultsmentioning

confidence: 84%

Section: Resultsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Pan

Wang

2019

Mathematics

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“…The forward‐backward method based on iteration has been studied by many authors (see, eg, previous studies)…”

Section: Introductionmentioning

confidence: 99%

“…The forward-backward method based on iteration (3) has been studied by many authors (see, eg, previous studies [10][11][12][13][14][15][16][17][18][19][20][21][22][23] In this work, we present a new modified forward-backward splitting method for solving (1) with a new linesearch. The convergence and the complexity are established under suitable conditions.…”

Section: Introductionmentioning

confidence: 99%

On convergence and complexity of the modified forward‐backward method involving new linesearches for convex minimization

Kankam

Pholasa

Cholamjiak

2019

Math Methods in App Sciences

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In optimization theory, convex minimization problems have been intensively investigated in the current literature due to its wide range in applications. A major and effective tool for solving such problem is the forward‐backward splitting algorithm. However, to guarantee the convergence, it is usually assumed that the gradient of functions is Lipschitz continuous and the stepsize depends on the Lipschitz constant, which is not an easy task in practice. In this work, we propose the modified forward‐backward splitting method using new linesearches for choosing suitable stepsizes and discuss the convergence analysis including its complexity without any Lipschitz continuity assumption on the gradient. Finally, we provide numerical experiments in signal recovery to demonstrate the computational performance of our algorithm in comparison to some well‐known methods. Our reports show that the proposed algorithm has a good convergence behavior and can outperform the compared methods.

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Tseng methods with inertial for solving inclusion problems and application to image deblurring and image recovery problems

Padcharoen

Kitkuan

Kumam

2020

Comp and Math Methods

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In this article, we offer two modifications of the modified forward‐backward splitting method based on inertial Tseng method and viscosity method for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak and strong convergence of the proposed algorithms. We give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for solving the image deblurring and image recovery problems. Our result extends some related works in the literature and the primary experiments might also suggest their potential applicability.

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An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces

Cited by 98 publications

References 34 publications

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

On convergence and complexity of the modified forward‐backward method involving new linesearches for convex minimization

Tseng methods with inertial for solving inclusion problems and application to image deblurring and image recovery problems

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