scholarly journals A Fast Fixed-Point Algorithm for Convex Minimization Problems and Its Application in Image Restoration Problems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2619
Author(s):  
Panadda Thongpaen ◽  
Rattanakorn Wattanataweekul
Keyword(s):  
Fixed Point ◽  
Image Restoration ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Minimization ◽  
Fixed Point Algorithm ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Weak Convergence Theorem ◽  
Inertial Technique

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method’s weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems.

scholarly journals An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Suthep Suantai ◽  
Pachara Jailoka ◽  
Adisak Hanjing
Keyword(s):  
Lipschitz Constant ◽  
Optimization Methods ◽  
Convergence Result ◽  
Convex Minimization ◽  
Signal Recovery ◽  
Splitting Algorithm ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Continuity Assumption ◽  
Inertial Technique

AbstractIn this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

scholarly journals A fast viscosity forward-backward algorithm for convex minimization problems with an application in image recovery

2021 ◽  
Vol 37 (3) ◽  
pp. 449-461
Author(s):  
PACHARA JAILOKA ◽  
◽  
SUTHEP SUANTAI ◽  
ADISAK HANJING ◽  
◽  
...  
Keyword(s):  
Image Restoration ◽  
Minimization Problem ◽  
Convex Minimization ◽  
Strong Convergence Theorem ◽  
Convex Minimization Problem ◽  
Splitting Algorithm ◽  
Convex Minimization Problems ◽  
Nonexpansive Operators ◽  
Thresholding Algorithm ◽  
Inertial Technique

The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem. Theoretically, we first introduce an algorithm based on viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive operators. Under some suitable assumptions, a strong convergence theorem of the proposed algorithm is established. Subsequently, we utilize our proposed algorithm to solving a convex minimization problem of the sum of two convex functions. As an application, we apply and analyze our algorithm to image restoration problems. Moreover, we compare convergence behavior and efficiency of our algorithm with other well-known methods such as the forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm. By using image quality metrics, numerical experiments show that our algorithm has a higher efficiency than the mentioned algorithms.

Approximating the minimum-norm fixed point of pseudocontractive mappings

2015 ◽  
Vol 08 (02) ◽  
pp. 1550036
Author(s):  
H. Zegeye ◽  
O. A. Daman
Keyword(s):  
Fixed Point ◽  
Iterative Process ◽  
Convergence Result ◽  
Monotone Mappings ◽  
Minimum Norm ◽  
Nonlinear Operators ◽  
Linear Inverse Problems ◽  
Minimization Problems ◽  
Pseudocontractive Mappings ◽  
Convex Minimization Problems

We introduce an iterative process which converges strongly to the minimum-norm fixed point of Lipschitzian pseudocontractive mapping. As a consequence, convergence result to the minimum-norm zero of monotone mappings is proved. In addition, applications to convexly constrained linear inverse problems and convex minimization problems are included. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

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