Subgradient Extragradient Method Research Articles

S-Iteration inertial subgradient extragradient method for variational inequality and fixed point problems

In developing iterative methods for approximating solutions of optimization problems, one of the major goals of researchers is to construct efficient iterative schemes. Over the years, different techniques have been devised by authors to achieve this goal. In this paper, we study a non-Lipschitz pseudomonotone variational inequality problem with common fixed points constraint of Bregman quasi-nonexpansive mappings. We introduce a new iterative method for approximating the solution of this problem in a more general framework of reflexive Banach spaces. Our method employs several techniques to achieve high level of efficiency. One of the techniques employed is the S-iteration process, a method known to be highly efficient in comparison with several of the well-known methods. Moreover, our proposed algorithm utilizes the inertial technique and a non-monotonic self-adaptive step size to guarantee high rate of convergence and easy implementation. Unlike several of the existing results on variational inequality problem with non-Lipschitz operator, the design of our method does not involve any linesearch technique. We obtain strong convergence result for the proposed algorithm without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when solving pseudomonotone variational inequality problems. Furthermore, we apply our result to study utility-based bandwidth allocation problem and image restoration problem. Finally, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with existing state-of-the-art methods. Our result extends and improves several of the recently announced results in this direction in the literature.

Subgradient extragradient method with double inertial steps for quasi-monotone variational inequalities

In this paper, we present a modified subgradient extragradient method with double inertial extrapolation terms and a non-monotonic adaptive step size for solving quasi-monotone and Lipschitz continuous variational inequalities in real Hilbert spaces. Under some suitable conditions, we obtain the weak convergence theorem of our proposed algorithm. Moreover, strong convergence is obtained when the cost operator is strongly pseudo-monotone and Lipschitz continuous. Finally, several numerical results illustrate the effectiveness and competitiveness of our algorithm.

A new Tseng method for supply chain network equilibrium model.

To solve the equilibrium problem of the supply chain network, a new subgradient extragradient method is introduced. The proposal achieves adaptive parameter selection, and supports a one-step subgradient projection operator, which can theoretically reduce the computational complexity of the algorithm. The introduction of subgradient projection operators makes the calculation of algorithms easier, and transforms the projection difficulty problem into how to find suitable sub-differential function problems. The given convergence proof further shows the advantages of the proposed algorithm. Finally, the presented algorithm is operated to a concrete supply chain network model. The comparisons show the proposed algorithm is better than other methods in term of CPU running time and iteration steps.

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.

Inertial subgradient extragradient with projection method for solving variational inequality and fixed point problems

<abstract><p>In this paper, we introduce a new modified inertial Mann-type method that combines the subgradient extragradient method with the projection contraction method for solving quasimonotone variational inequality problems and fixed point problems in real Hilbert spaces. We establish strong convergence of the proposed method under some mild conditions without knowledge of the operator norm. Finally, we give numerical experiments to illustrate the efficiency of the method over the existing one in the literature.</p></abstract>

An inertial subgradient extragradient method with Armijo type step size for pseudomonotone variational inequalities with non-Lipschitz operators in Banach spaces

In this paper, we study the pseudomonotone variational inequality problems with non-Lipschitz operators. We propose an inertial subgradient extragradient method with Halpern technique and Armijo type step size for approximating the solution of the problem in the framework of 2-uniformly convex real Banach spaces. We prove that the sequence generated by our proposed method converges strongly to the solution of the problem under some mild conditions and without the weak sequential continuity condition often assumed by authors in solving pseudomonotone variational inequality problems. Finally, we provide some numerical experiments for the proposed method in comparison with other existing methods in the literature. Our result extends and improves several of the existing results in the current literature in this direction.

Adaptive inertial subgradient extragradient methods for finding minimum-norm solutions of pseudomonotone variational inequalities

In this paper, four modified inertial subgradient extragradient methods with a new non-monotonic step size criterion are investigated for pseudomonotone variational inequality problems in real Hilbert spaces. Our algorithms employ two different step sizes in each iteration to update the values of iterative sequences, and they work well without the prior information about the Lipschitz constant of the operator. Strong convergence theorems of the proposed iterative schemes are established under some suitable and mild conditions. Some numerical examples are provided to demonstrate the computational efficiency and advantages of the proposed methods over other known ones.

Approximating $G$-variational inequality problem by $G$-subgradient extragradient method in Hilbert space endowed with graphs

In this article, we introduce $G$-subgradient extragradient method for solving the $G$- variational inequality problem in Hilbert space endowed with a direct graph. Utilizing our mathematical tools, weak and strong convergence theorem are established for the proposed algorithm. In addition, we provide numerical experiments to illustrate the convergence behavior of our proposed algorithm.

Inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and fixed point problems in Hilbert spaces

This paper proposes a new inertial subgradient extragradient method for solving equilibrium problems with pseudomonotone and Lipschitz-type bifunctions and fixed point problems for nonexpansive mappings in real Hilbert spaces. Precisely, we prove that the sequence generated by proposed algorithm converges strongly to a common solution of equilibrium problems and fixed point problems. We use an effective self-adaptive step size rule to accelerate the convergence process of our proposed iterative algorithm. Moreover, some numerical results are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature.

A Relaxed Inertial Factor of the Modified Subgradient Extragradient Method for Solving Pseudo Monotone Variational Inequalities in Hilbert Spaces

A Relaxed Inertial Factor of the Modified Subgradient Extragradient Method for Solving Pseudo Monotone Variational Inequalities in Hilbert Spaces

Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces

Abstract The projection technique is a very important method and efficient for solving variational inequality problems. In this study, we developed the subgradient extragradient method for solving pseudomonotone variational inequality in real Hilbert spaces. Our first algorithm requires only computing one projection onto the feasible set per iteration and the strong convergence is proved without the prior knowledge of the Lipschitz constant as well as the sequentially weak continuity of the associated mapping. The second algorithm uses the linesearch procedure such that its convergence does not require the Lipschitz continuous condition of the variational inequality mapping. Finally, some numerical experiments are provided to demonstrate the advantages and efficiency of the proposed methods.

Modified Inertial Subgradient Extragradient Method with Regularization for Variational Inequality and Null Point Problems

The paper develops a modified inertial subgradient extragradient method to find a solution to the variational inequality problem over the set of common solutions to the variational inequality and null point problems. The proposed method adopts a nonmonotonic stepsize rule without any linesearch procedure. We describe how to incorporate the regularization technique and the subgradient extragradient method; then, we establish the strong convergence of the proposed method under some appropriate conditions. Several numerical experiments are also provided to verify the efficiency of the introduced method with respect to previous methods.

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.

A new subgradient extragradient method for solving the split modified system of variational inequality problems and fixed point problem

We introduce a new subgradient extragradient algorithm utilizing the concept of the set of solutions of the split modified system of variational inequality problems (SMSVIP). Our main theorem is weak convergence theorem for such an algorithm for approximating the fixed point problem in a real Hilbert space. We also apply these results to approximate the split minimization problem. In the last section, we provide an example to illustrate the potential of our main theorem.

ON THE STRONG CONVERGENCE FOR ITERATIVE METHODS TO FIND A SOLUTION OF A SPLIT PSEUDOMONOTONE VARIATIONAL INEQUALITY AND A SPLIT FEASIBILITY PROBLEM

In this paper, we introduce two different interative methods for finding a solution of a split pseudomonotone variational inequality and a split feasibility problem in Hilbert spaces. The proposed algorithm is generated based on the subgradient extragradient method which requires only two projections at each iteration step and the second projection is conducted onto the half-space containing the constrained set. The strong convergence is proven with some mild conditions imposed on the operators as well as the parameters. A numerical result is provided at the end of the paper with the use of Python for the convergence illustration purpose of the studied method.

Analysis of Subgradient Extragradient Method for Variational Inequality Problems and Null Point Problems

In this paper, we introduce a new numerical method for finding a common solution to variational inequality problems involving monotone mappings and null point problems involving a finite family of inverse-strongly monotone mappings. The method is inspired by the subgradient extragradient method and the regularization method. Strong convergence results of the proposed algorithms have been obtained under some suitable conditions.

A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems

A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems

Revisiting subgradient extragradient methods for solving variational inequalities

Revisiting subgradient extragradient methods for solving variational inequalities

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

<abstract><p>In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.</p></abstract>