Monotone Variational Inclusion Problem Research Articles

Self-adaptive Technique with Double Inertial Steps for Inclusion Problem on Hadamard Manifolds

AbstractIn this article, we investigate monotone and Lipschitz continuous variational inclusion problem in the settings of Hadamard manifolds. We propose a forward–backward method with a self-adaptive technique for solving variational inclusion problem. To increase the rate of convergence of our proposed method, we incorporate our iterative method with double inertial steps and establish a convergence result of our iterative method under some mild conditions. Finally, in order to illustrate the computational effectiveness of our method, some numerical examples are also discussed. The result present in this article is new in this space and extends many related results in the literature.

Relaxed double inertial Tseng's extragradient method for solving non-Lipschitz split monotone variational inclusion problems with fixed point constraints

Relaxed double inertial Tseng's extragradient method for solving non-Lipschitz split monotone variational inclusion problems with fixed point constraints

Iterative Algorithm of Split Monotone Variational Inclusion Problem for New Mappings

In this paper, we developed a new type iterative scheme to approximate a common solution of split monotone variational inclusion, variational inequality and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we proved that the sequence generated by the proposed iterative method converges strongly to a common solution of split monotone variational inclusion, variational inequality and fixed point problems. Furthermore, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm. The result presented in this paper unifies and extends some known results in this area.

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.

Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

Accelerated Krasnoselski-Mann type algorithm for hierarchical fixed point and split monotone variational inclusion problems in Hilbert spaces

In this paper, a new accelerated extrapolation Krasnoselski-Mann type algorithm for finding common element in the solution set of the hierarchical fixed point and split monotone variational inclusion problems are introduced in the setting of a real Hilbert space. We then prove that a sequence generated by the algorithm converges strongly to such common element which also approximates solution of some fixed point problem of demimetric mapping in the space. Finally, some applications and numerical experiment are given to show effectiveness of the proposed algorithm over the recently known related results in the literature. The established results extend and generalize many recent ones announced in the literature.

Split monotone variational inclusion with errors for image-feature extraction with multiple-image blends problem

In this paper, we introduce a new iterative forward–backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique.

An inertial viscosity algorithm for solving monotone variational inclusion and common fixed point problems of strict pseudocontractions

In this paper, we study the problem of finding the solution of monotone variational inclusion problem (MVIP) with constraint of common fixed point problem (CFPP) of strict pseudocontractions. We propose a new viscosity method, which combines the inertial technique with self-adaptive step size strategy for approximating the solution of the problem in the framework of Hilbert spaces. Unlike several of the existing results in the literature, our proposed method does not require the co-coerciveness and Lipschitz continuity assumptions of the associated single-valued operator. Also, our method does not involve any linesearch technique which could be time-consuming, rather we employ a self-adaptive step size technique that generates a nonmonotonic sequence of step sizes. Moreover, we prove strong convergence result for our algorithm under some mild conditions and apply our result to study other optimization problems. We present several numerical experiments to demonstrate the computational advantage of our proposed method over the existing methods in the literature. Our result complements several of the existing results in the current literature in this direction.

On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints

Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.

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.

Approximation of Solutions of Split Monotone Variational Inclusion Problems and Fixed Point Problems

Many researchers have incorporated an inertial term and will continue to involve it in iterative algorithms due to the fact that it speeds up the rate of convergence which is desirable in applications. In this paper, we propose a new inertial extrapolation algorithm for solving split monotone variational inclusion problems which turns out to solve fixed point problems in the framework of real Hilbert spaces. Our proposed algorithm which is the generalization of split feasibility problems among many others, does not involve the knowledge of operator norm which is sometimes difficult in practice. Furthermore, we prove under some mild assumptions that the sequence generated recursively by our algorithm converges strongly to a common solution of split monotone variational inclusion problems and fixed point of κ-demicontractive mapping. Finally, we apply our algorithm to solve other related problems, precisely linear inverse problems. Some numerical illustrations are provided to further demonstrate the efficiency and competitiveness of our algorithm.

SOLVING MULTIPLE-SETS SPLIT MONOTONE VARIATIONAL INCLUSION PROBLEM IN REAL HILBERT SPACES.

In this paper, we study and introduce a self adaptive method to­gether with a Halpern iterative algorithm for approximating solutions of multiple-sets split monotone variational inclusion problem which in­cludes the multiple-sets split feasibility problem, split feasibility prob­lem, split monotone variational inclusion problem and split variational inclusion problem, to mention a few. Using our iterative algorithm, we prove a strong convergence result for approximating the solution of the aforementioned problems. Numerical examples on finite-dimensional and infinite-dimensional spaces are displayed to illustrate the perfor­mance of our iterative method. The result discussed in this article extends and complements many related results in literature.

Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications

<p style='text-indent:20px;'>In this paper, we study a modified relaxed inertial Mann-type iterative algorithm for solving split monotone variational inclusion problem without prior knowledge of the bounded linear operator norm in real Hilbert spaces. Under some appropriate assumptions on the parameters, we established a strong convergence result of the proposed algorithm. As applications, some special cases of the general problem are given. Finally, we also give some numerical illustrations of the proposed method in comparison with other methods in the literature to show the applicability and advantage of our results.</p>

Variational inequality problem over the solution set of split monotone variational inclusion problem with application to bilevel programming problem

The purpose of this paper is to study variational inequality problem over the solution set of multiple-set split monotone variational inclusion problem. We propose an iterative algorithm with inertial method for finding an approximate solution of this problem in real Hilbert spaces. Strong convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. The iterative scheme does not require prior knowledge of operator norm. Also we present some applications of our main result to solve the bilevel programming problem, the bilevel monotone variational inequalities, the split minimization problem, the multiple-set split feasibility problem and the multiple set split variational inequality problem.

An iterative technique for solving split equality monotone variational inclusion and fixed point problems

Abstract The purpose of this paper is to introduce an iterative algorithm for approximating the solution of the split equality monotone variational inclusion problem (SEMVIP) for monotone operators, which is also a solution of the split equality fixed point problem (SEFPP) for strictly pseudocontractive maps in real Hilbert spaces. We establish the strong convergence of the sequence generated by our iterative algorithm. Our result complements and extends some related results in literature.

Split monotone variational inclusion problem involving Cayley operators

Abstract In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.

A Modified inertial Halpern method for solving split monotone variational inclusion problems in Banach Spaces

A Modified inertial Halpern method for solving split monotone variational inclusion problems in Banach Spaces

A modified forward‐backward splitting methods for the sum of two monotone operators with applications to breast cancer prediction

This work proposes a modified forward‐backward splitting algorithm combining an inertial technique for solving the monotone variational inclusion problem. The weak convergence theorem is established under some suitable conditions in Hilbert space, and a new step size is presented for our algorithm to speed up the convergence. We give an example and numerical results for supporting our main theorem in infinite dimensional spaces. We also provide an application to predict breast cancer by using our proposed algorithm for updating the optimal weight in machine learning. Moreover, we use the Wisconsin original breast cancer data set as a training set to show efficiency comparing with the other three algorithms in terms of three key parameters, namely, accuracy, recall, and precision.

New self-adaptive methods with double inertial steps for solving splitting monotone variational inclusion problems with applications

In this paper, without the co-coerciveness assumption of the associated mappings, we show the iterative sequences generated by our new algorithms converge strongly to a solution of the splitting monotone variational inclusion problem in infinite dimensional real Hilbert spaces. The step sizes of the new algorithms are updated per iteration by a simple calculation without knowing the prior information of the operator norm. Some parameters are relaxed to enlarge the value range of the corresponding step sizes. Double inertial extrapolation steps are incorporated in the new algorithms to accelerate their convergence speed. As applications, the split variational inclusion problem, the monotone inclusion problem and the split variational inequality problem are studied. Some numerical experiments are performed to illustrate the computation efficiency of the new algorithms.

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Abstract In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.