Subgradient Extragradient Method Research Articles

Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models

This manuscript aims to incorporate an inertial scheme with Popov’s subgradient extragradient method to solve equilibrium problems that involve two different classes of bifunction. The novelty of our paper is that methods can also be used to solve problems in many fields, such as economics, mathematical finance, image reconstruction, transport, elasticity, networking, and optimization. We have established a weak convergence result based on the assumption of the pseudomonotone property and a certain Lipschitz-type cost bifunctional condition. The stepsize, in this case, depends upon on the Lipschitz-type constants and the extrapolation factor. The bifunction is strongly pseudomonotone in the second method, but stepsize does not depend on the strongly pseudomonotone and Lipschitz-type constants. In contrast, the first convergence result, we set up strong convergence with the use of a variable stepsize sequence, which is decreasing and non-summable. As the application, the variational inequality problems that involve pseudomonotone and strongly pseudomonotone operator are considered. Finally, two well-known Nash–Cournot equilibrium models for the numerical experiment are reviewed to examine our convergence results and show the competitive advantage of our suggested methods.

New hybrid projection methods for variational inequalities involving pseudomonotone mappings

In this paper, we combine the subgradient extragradient method and the hybrid method with inertial extrapolation step to introduce two strongly convergent methods for solving variational inequalities for which the underlying cost function is a pseudo-monotone operator in real Hilbert spaces. The first proposed method involves step-sizes bounded by the inverse of the Lipschitz constant of the cost function and the second proposed method involves line search approach when the cost function is uniformly continuous. Our strong convergence results are obtained under some relaxed conditions on the inertial factor and the iterative parameters. Our numerical implementations show that our proposed methods are competitive and efficient.

Versions of the Subgradient Extragradient Method for Pseudomonotone Variational Inequalities

We develop versions of the subgradient extragradient method for variational inequalities in Hilbert spaces and establish sufficient conditions for their convergence. First we prove a sufficient condition for a weak convergence of a recent existing algorithm under relaxed assumptions. Then, we propose two other algorithms. Both weak and strong convergence of the considered algorithms are studied. Under additional strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a $Q$ -linear convergence rate of these algorithms. Our results improve some recent contributions in the literature. Illustrative numerical experiments are also provided by the end of the paper.

Inertial Subgradient Extragradient Methods for Solving Variational Inequality Problems and Fixed Point Problems

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems

In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.

The Ishikawa Subgradient Extragradient Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

In this paper, we introduce a novel algorithm for finding a common point of the solution set of a class of equilibrium problems involving pseudo-monotone bifunctions and satisfying the Lipschitz-type condition and the set of fixed points of a quasi-nonexpansive in a real Hilbert space. This algorithm can be considered as a combination of the subgradient extragradient method for equilibrium problems and the Ishikawa method for fixed point problems. The strong convergence of the iterates generated by the proposed method is obtained under the main assumptions that the fixed-point mapping is demiclosed at 0 and Lipschitz-type constant of the cost bifunction is unknown. Some numerical examples are implemented to show the computational efficiency of the proposed algorithm.

Inertial Haugazeau's hybrid subgradient extragradient algorithm for variational inequality problems in Banach spaces

The variational inequality problem plays an important role in nonlinear analysis and optimization. It is a generalization of the nonlinear complementarity problem. For a variational inequality problem in a Hilbert space, the extragradient algorithm with inertial effects has been studied. For a variational inequality problem in a Banach space, Nakajo introduced Haugazeau's hybrid method and Liu introduced the Halpern subgradient extragradient method. In this paper, we construct a new inertial iterative method for solving variational inequality problems in Banach spaces based on the work we mentioned above. We propose a strong convergence theorem. As applications, our result can be used to solve constrained convex minimization problems.

A new low-cost double projection method for solving variational inequalities

In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of Gibali et al. and Popov’s method. The proposed scheme combines some of the advantages of the methods mentioned above, first it requires only one orthogonal projection onto the feasible set of the problem while the next computation has a closed formula. Second, only one mapping evaluation is required per each iteration and there is also a usage of an adaptive step size rule that avoids the need to know the Lipschitz constant of the associated mapping. We present two convergence theorems of the proposed method, weak convergence result which requires pseudomonotonicity, Lipschitz and sequentially weakly continuity of the associated mapping and strong convergence theorem with rate of convergence which requires Lipschitz continuity and strongly pseudomonotone only. Primary numerical experiments and comparisons demonstrate the advantages and potential applicability of the new scheme.

Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the setting of Hilbert space. We propose a modified inertial viscosity subgradient extragradient algorithm with self-adaptive stepsize in which the two projections are made onto some half-spaces. Moreover, we obtain a strong convergence result for approximating a common solution of the variational inequality and fixed point of quasi-nonexpansive mappings under some mild conditions. The main advantages of our method are: the self adaptive step-size which avoids the need to know apriori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which speeds up the rate of convergence of the algorithm. Numerical experiments are presented to demonstrate the efficiency of our algorithm in comparison with other existing algorithms in literature.

A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces

In this paper, we introduce an algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in Banach space. We modify the subgradient extragradient methods with a new and simple iterative step size, the strong convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, a numerical experiment is presented to show the efficiency and advantage of the proposed algorithm. Our results generalize some of the work in Hilbert spaces to Banach spaces.

A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems

A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems

A Strongly Convergent Modified Halpern Subgradient Extragradient Method for Solving the Split Variational Inequality Problem

We propose a method for solving the split variational inequality problem (SVIP) involving Lipschitz continuous and pseudomonotone mappings. The proposed method is inspired by the Halpern subgradient extragradient method for solving the monotone variational inequality problem with a simple step size. A strong convergence theorem for an algorithm for solving such a SVIP is proved without the knowledge of the Lipschitz constants of the mappings. As a consequence, we get a strongly convergent algorithm for finding the solution of the split feasibility problem (SFP), which requires only two projections at each iteration step. A simple numerical example is given to illustrate the proposed algorithm.

Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems

In this paper, two new algorithms are introduced for solving a pseudomontone variational inequality problem with a Lipschitz condition in a Hilbert space. The algorithms are constructed around three methods: the subgradient extragradient method, the inertial method and the viscosity method. With a new stepsize rule is incorporated, the algorithms work without any information of Lipschitz constant of operator. The weak convergence of the first algorithm is established, while the second one is strongly convergent which comes from the viscosity method. In order to show the computational effectiveness of our algorithms, some numerical results are provided.

A strong convergence of modified subgradient extragradient method for solving bilevel pseudomonotone variational inequality problems

ABSTRACTIn this work, we investigate a new extragradient method for finding an element of the set of solutions of the bilevel pseudomonotone variational inequality problem in real Hilbert spaces. We only use one projection to construct the algorithm and the strong convergence theorem proved under suitable conditions. Numerical experiments illustrate the performances of our new algorithm and present a comparison with related algorithms.

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

A new inertial double-projection method for solving variational inequalities

In this paper, we introduce a new algorithm of inertial form for solving monotone variational inequalities (VI) 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 Lipschitz continuity of the VI associated mapping, we establish the weak convergence of the scheme. Several numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature.

Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

ABSTRACTIn this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.

Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings

ABSTRACT In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.