Subgradient Extragradient Method Research Articles

Subgradient Extragradient Method for Finite Lipschitzian Demicontractions and Variational Inequality Problems in a Hilbert Space

In this research, the modified subgradient extragradient method and K-mapping generated by a finite family of finite Lipschitzian demicontractions are introduced. Then, a strong convergence theorem for finding a common element of the common fixed point set of finite Lipschitzian demicontraction mappings and the common solution set of variational inequality problems is established. Furthermore, numerical examples are given to support the main theorem.

A new inertial condition on the subgradient extragradient method for solving pseudomonotone equilibrium problem

In this paper, we study the pseudomonotone equilibrium problem. We consider a new inertial condition for the subgradient extragradient method with self-adaptive step size for approximating a solution of the equilibrium problem in a real Hilbert space. Our proposed method contains inertial factor with new conditions that only depend on the iteration coefficient. We obtain a weak convergence result of the proposed method under weaker conditions on the inertial factor than many existing conditions in the literature. Finally, we present some numerical experiments for our proposed method and compared it to existing methods in the literature. Our result improves, extends, and generalizes several existing results in the literature.

A modified subgradient extragradient method with non-monotonic step sizes for solving quasimonotone variational inequalities

A modified subgradient extragradient method with non-monotonic step sizes for solving quasimonotone variational inequalities

A new double inertial subgradient extragradient algorithm for solving split pseudomonotone equilibrium problems and fixed point problems

AbstractThe purpose of this article is to suggest a modified subgradient extragradient method that includes double inertial extrapolations and viscosity approach for finding the common solution of split equilibrium problem and fixed point problem. The strong convergence result of the suggested method is obtained under some standard assumptions on the control parameters. Our method does not require solving two strongly convex optimization problems in the feasible sets per iteration, and the step-sizes do not depend on bifunctional Lipschitz-type constants. Furthermore, unlike several methods in the literature, our method does not depend on the prior knowledge of the operator norm of the bounded linear operator. Instead, the step-sizes are self adaptively updated. We apply our method to solve split variational inequality problem. Lastly, we conduct some numerical test to compare our method with some well known methods in the literature.

Projection and contraction method with double inertial steps for quasi-monotone variational inequalities

In this paper, we present a modified projection and contraction method for solving quasi-monotone variational inequalities in real Hilbert spaces. Our proposed method is a combination of double inertial extrapolation steps, the subgradient extragradient method and the projection contraction method, which can effectively accelerate the convergence rate. The weak and linear convergence have been obtained under some suitable conditions. Some numerical experiments are given to show that our proposed method outperforms the related methods.

Two subgradient extragradient methods based on the golden ratio technique for solving variational inequality problems

Two subgradient extragradient methods based on the golden ratio technique for solving variational inequality problems

A New Inertial-Projection Method for Solving Split Generalized Mixed Equilibrium and Hierarchical Fixed Point Problems

In this paper, we introduce a new iterative algorithm of inertial form for approximating the common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Hierarchical Fixed Point Problem (HFPP) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP and HFPP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature.

Modified inertial subgradient extragradient algorithms for generalized equilibria systems with constraints of variational inequalities and fixed points

<abstract><p>In this research, we studied modified inertial composite subgradient extragradient implicit rules for finding solutions of a system of generalized equilibrium problems with a common fixed-point problem and pseudomonotone variational inequality constraints. The suggested methods consisted of an inertial iterative algorithm, a hybrid deepest-descent technique, and a subgradient extragradient method. We proved that the constructed algorithms converge to a solution of the considered problem, which also solved some hierarchical variational inequality.</p></abstract>

A Totally Relaxed Self-Adaptive Subgradient Extragradient Scheme for Equilibrium and Fixed Point Problems in a Banach Space

The goal of this paper is to introduce a Totally Relaxed Self adaptive Subgradient Extragradient Method (TRSSEM) together with an Halpern iterative method for approximating a common solution of Fixed Point Problem (FPP) and Equilibrium Problem (EP) in 2-uniformly convex and uniformly smooth Banach space. We prove the strong convergence of the sequence generated by our proposed method. The proposed method does not require the computation of a projection onto a feasible set, it instead requires a projection onto a finite intersection of sub-level sets of convex functions. Our result generalizes, unifies and extends some related results in the literature.

Modified mildly inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and nonexpansive fixed point problems

<abstract><p>This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments.</p></abstract>

Inertial subgradient extragradient method for solving pseudomonotone variational inequality problems in Banach spaces

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures.

Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method.

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint.

A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities

Abstract The primary goal of this research is to investigate the approximate numerical solution of variational inequalities using quasimonotone operators in infinite-dimensional real Hilbert spaces. In this study, the sequence obtained by the proposed iterative technique for solving quasimonotone variational inequalities converges strongly toward a solution due to the viscosity-type iterative scheme. Furthermore, a new technique is proposed that uses an inertial mechanism to obtain strong convergence iteratively without the requirement for a hybrid version. The fundamental benefit of the suggested iterative strategy is that it substitutes a monotone and non-monotone step size rule based on mapping (operator) information for its Lipschitz constant or another line search method. This article also provides a numerical example to demonstrate how each method works.

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature.

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.

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.

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.