scholarly journals A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 42
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Hilbert Spaces ◽  
Minimization Problem ◽  
Line Search ◽  
Optimization Problem ◽  
Splitting Method ◽  
Optimal Choice ◽  
Convex Minimization ◽  
Signal Recovery ◽  
Objective Functions ◽  
Convex Minimization Problem

In this work, we aim to investigate the convex minimization problem of the sum of two objective functions. This optimization problem includes, in particular, image reconstruction and signal recovery. We then propose a new modified forward-backward splitting method without the assumption of the Lipschitz continuity of the gradient of functions by using the line search procedures. It is shown that the sequence generated by the proposed algorithm weakly converges to minimizers of the sum of two convex functions. We also provide some applications of the proposed method to compressed sensing in the frequency domain. The numerical reports show that our method has a better convergence behavior than other methods in terms of the number of iterations and CPU time. Moreover, the numerical results of the comparative analysis are also discussed to show the optimal choice of parameters in the line search.

scholarly journals The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

2021 ◽  
Vol 1 (1) ◽  
pp. 19-33
Author(s):  
Sang B Mendy ◽  
John T Mendy ◽  
Alieu Jobe
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Minimization Problem ◽  
Nonexpansive Mappings ◽  
Convex Minimization ◽  
Convex Minimization Problem ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

scholarly journals General Iterative Methods for System of Equilibrium Problems and Constrained Convex Minimization Problem in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Peichao Duan
Keyword(s):  
Iterative Methods ◽  
Hilbert Spaces ◽  
Minimization Problem ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Convex Minimization ◽  
Convex Minimization Problem ◽  
Constrained Convex Minimization ◽  
Constrained Convex Minimization Problem ◽  
System Of Equilibrium Problems

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by Tian and Liu (2012) and many others. Furthermore, we give numerical example to demonstrate the effectiveness of our iterative scheme.

scholarly journals Novel forward–backward algorithms for optimization and applications to compressive sensing and image inpainting

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Suthep Suantai ◽  
Muhammad Aslam Noor ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Compressive Sensing ◽  
Splitting Method ◽  
Image Inpainting ◽  
Optimal Choice ◽  
Convex Minimization ◽  
Objective Functions ◽  
Classification Problems ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Continuity Assumption

AbstractThe forward–backward algorithm is a splitting method for solving convex minimization problems of the sum of two objective functions. It has a great attention in optimization due to its broad application to many disciplines, such as image and signal processing, optimal control, regression, and classification problems. In this work, we aim to introduce new forward–backward algorithms for solving both unconstrained and constrained convex minimization problems by using linesearch technique. We discuss the convergence under mild conditions that do not depend on the Lipschitz continuity assumption of the gradient. Finally, we provide some applications to solving compressive sensing and image inpainting problems. Numerical results show that the proposed algorithm is more efficient than some algorithms in the literature. We also discuss the optimal choice of parameters in algorithms via numerical experiments.

scholarly journals An intermixed iteration for constrained convex minimization problem and split feasibility problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Kanyanee Saechou ◽  
Atid Kangtunyakarn
Keyword(s):  
Strong Convergence ◽  
Minimization Problem ◽  
Convergence Theorem ◽  
Split Feasibility Problem ◽  
Convex Minimization ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Convex Minimization Problem ◽  
Constrained Convex Minimization Problem ◽  
Split Feasibility

Abstract In this paper, we first introduce the two-step intermixed iteration for finding the common solution of a constrained convex minimization problem, and also we prove a strong convergence theorem for the intermixed algorithm. By using our main theorem, we prove a strong convergence theorem for the split feasibility problem. Finally, we apply our main theorem for the numerical example.

On an application of the complex nonlinear complementarity problem

1976 ◽  
Vol 15 (1) ◽  
pp. 141-148 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Minimization Problem ◽  
Complementarity Problem ◽  
Complex Space ◽  
Nonlinear Complementarity Problem ◽  
Polyhedral Cone ◽  
Convex Minimization ◽  
Convex Minimization Problem ◽  
Polyhedral Cones ◽  
Existence Of A Solution ◽  
Nonlinear Complementarity

A theorem on the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving a convex minimization program naturally leads to the consideration of the following nonlinear complementarity problem: given g: Cn → Cn, find z such that g(z) ∈ S*, z ∈ S, and Re〈g(z), z〉 = 0, where S is a polyhedral cone and S* its polar.

scholarly journals Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 156 ◽  
Author(s):  
Chanjuan Pan ◽  
Yuanheng Wang
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Splitting Method ◽  
Splitting Methods ◽  
Strong Convergence Theorem ◽  
Accretive Operators ◽  
Convex Minimization Problem ◽  
Inertial Extrapolation

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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