Viscosity Approximation Method Research Articles

A novel double inertial viscosity algorithm for convex bilevel optimization problems applied to image restoration problems

Convex bilevel optimization holds a crucial role in the decision-making process in a hierarchical setting, while image restoration plays a critical role in signal processing and computer vision. In this paper, a new double inertial viscosity approximation method with a linesearch technique is introduced for solving convex bilevel optimization problems and then its strong convergence is analyzed under moderate conditions. As an application, our proposed algorithm is employed for solving image restoration problems. The experiment results highlight the specific advantages and potential infect of the algorithm over the existing algorithms in the literature, particularly in the domain of image restoration.

Quantitative asymptotic regularity of the VAM iteration with error terms for $m$-accretive operators in Banach spaces

In this paper, we obtain, by using proof mining methods, quantitative results on the asymptotic regularity of the viscosity approximation method (VAM) with error terms for m -accretive operators in Banach spaces. For concrete instances of the parameter sequences, linear rates are computed by applying a lemma due to Sabach and Shtern.

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Viscosity approximation method for attractive points of widely more generalized hybrid mappings

Viscosity approximation method for attractive points of widely more generalized hybrid mappings

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

On the convergence of the trajectories of the dynamical Moudafi’s viscosity approximation system

We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241:46–55, 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_{n})$$\\end{document} generated by the discrete viscosity approximation (DDS)

Viscosity approximation method for a variational problem

Viscosity approximation method for a variational problem

Viscosity approximation method for split best proximity point and monotone variational inclusion problem

ABSTRACT To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper.

Inertial Viscosity Approximation Methods for General Split Variational Inclusion and Fixed Point Problems in Hilbert Spaces

The purpose of this paper is to find a common element of the fixed point set of a nonexpansive mapping and the set of solutions of the general split variational inclusion problem in symmetric Hilbert spaces by using the inertial viscosity iterative method. Some strong convergence theorems of the proposed algorithm are demonstrated. As applications, we use our results to study the split feasibility problem and the split minimization problem. Finally, the numerical experiments are presented to illustrate the feasibility and effectiveness of our theoretical findings, and our results extend and improve many recent ones.

A Novel Inertial Viscosity Algorithm for Bilevel Optimization Problems Applied to Classification Problems

Fixed-point theory plays many important roles in real-world problems, such as image processing, classification problem, etc. This paper introduces and analyzes a new, accelerated common-fixed-point algorithm using the viscosity approximation method and then employs it to solve convex bilevel optimization problems. The proposed method was applied to data classification with the Diabetes, Heart Disease UCI and Iris datasets. According to the data classification experiment results, the proposed algorithm outperformed the others in the literature.

A new inertial algorithm for equilibrium problem and a family of nonexpansive operators in Hilbert space

In this article, we suggest and study a new inertial algorithm combining inertial extrapolation, S‐iteration process and viscosity approximation method for computing a common solution of an equilibrium problem and a family of nonexpansive operators in real Hilbert space. We provide a strong convergence result of the suggested algorithm under some simple assumptions on control sequences. Further, we compare our suggested algorithm with some existing well‐known algorithms by an analytical example. Some prominent recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

Mann Hybrid Deepest-Descent Extragradient Method with Line-Search Process for Hierarchical Variational Inequalities for Countable Nonexpansive Mappings

In real Hilbert spaces, let the CFPP indicate a common fixed-point problem of asymptotically nonexpansive operator and countably many nonexpansive operators, and suppose that the HVI and VIP represent a hierarchical variational inequality and a variational inequality problem, respectively. We put forward Mann hybrid deepest-descent extragradient approach for solving the HVI with the CFPP and VIP constraints. The proposed algorithms are on the basis of Mann’s iterative technique, viscosity approximation method, subgradient extragradient rule with linear-search process, and hybrid deepest-descent rule. Under suitable restrictions, it is shown that the sequences constructed by the algorithms converge strongly to a solution of the HVI with the CFPP and VIP constraints.

Strong convergence of new split general system of monotone variational inclusion problem

The aim of this research is to introduce a split general system of monotone variational inclusion problem to generalize several known problems in the literature. We propose a viscosity approximation method to prove strong convergence of the suggested algorithm to a common solution of split general system of monotone variational inclusion problem, variational inequality problem involving pseudomonotone operator and fixed point problem for a finite family of quasi-pseudocontractive mappings. Our result can be viewed as significant extension of the previously known results. Furthermore, we provide numerical experiments to validate the performance of our algorithm and compare it with other existing methods.

Viscosity approximation method for fixed point of pseudocontraction mapping in Hadamard spaces

Viscosity approximation method for fixed point of pseudocontraction mapping in Hadamard spaces

Subgradient-like extragradient algorithms for systems of variational inequalities with constraints

In this paper, we introduce a modified viscosity subgradient-like extragradient implicit rule with line-search process for finding a solution of a general system of variational inequalities (GSVI) with a variational inequality (VIP) and a fixed-point (FPP) constraints in Hilbert spaces. The suggested algorithms are based on the subgradient extragradient method with line-search process, hybrid Mann implicit iteration method, and composite viscosity approximation method. Under suitable restrictions, we demonstrate the strong convergence of the suggested algorithm to a solution of the GSVI with the VIP and FPP constraints, which is a unique solution of a certain hierarchical variational inequality.

Generalized viscosity approximation method for solving split generalized mixed equilibrium problem with application to compressed sensing

<abstract><p>In this study, we establish a new inertial generalized viscosity approximation method and prove that the resulting sequence strongly converges to a common solution of a split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. As an application, we demonstrate the use of our main finding in compressed sensing in signal processing. Additionally, we include numerical examples to evaluate the efficiency of the suggested method and then conduct a comparative analysis of its efficiency with different methods. Our findings can be used in a variety of contexts to improve results.</p></abstract>

Generalized Viscosity Approximation Methods of Ishikawa Type for Nonexpansive Mappings in Hilbert Spaces

In this paper, by using generalized viscosity mappings, we prove two strong convergence theorems for finding fixed points of a nonexpansive mapping which is also a unique solution of the variational inequality. Our results extend and improve the recent ones announced by some authors.

A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z)=zn+mz+r; n≥2 where m,r∈ℂ, and biomorphs for any complex function through a viscosity approximation method which is among the most widely used iterative methods for finding fixed points of non-linear operators. We derive novel escape criterion for generating Julia and Mandelbrot sets via proposed viscosity approximation method. Moreover, we visualize the sets using the escape time algorithm and the proposed iteration. Then, we discuss the shape change of the obtained sets depending on the parameters of the iteration using graphical and numerical experiments. The presented examples reveal that this change can be very complex, and we are able to obtain a great variety of shapes.

On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set.