Extragradient Method Research Articles

Continuous Projection Generalized Extra-Gradient Quasi-Newton Second-Order Method for Solving Saddle Point Problems

Continuous Projection Generalized Extra-Gradient Quasi-Newton Second-Order Method for Solving Saddle Point Problems

Inertial Extragradient Method for Solving Variational Inequality and Fixed Point Problems of a Bregman Demigeneralized Mapping in a Reflexive Banach Spaces

In this paper, by employing a Bregman distance approach, we introduce a self-adaptive inertial extragradient method for finding a common solution of variational inequality problem involving a pseudo-monotone operator and fixed point problem of a Bregman demigeneralized mapping in a reflexive Banach spaces. Using a Bregman distance approach, we establish a strong convergence result for approximating the common solution of the aforementioned problems under some mild assumptions. We display some numerical examples to show the performance of our iterative method. The result presented in this paper extends and complements many related results in literature.

A New Alternative Regularization Method for Solving Generalized Equilibrium Problems

The purpose of this paper is to present a numerical method for solving a generalized equilibrium problem involving a Lipschitz continuous and monotone mapping in a Hilbert space. The proposed method can be viewed as an improvement of the Tseng’s extragradient method and the regularization method. We show that the iterative process constructed by the proposed method converges strongly to the smallest norm solution of the generalized equilibrium problem. Several numerical experiments are also given to illustrate the performance of the proposed method. One of the advantages of the proposed method is that it requires no knowledge of Lipschitz-type constants.

Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

Abstract In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

An Accelerated Proximal Extragradient Method with Applications in Representative Selection

Proximal algorithms has the advantages of low iteration cost and fast convergence speed, and is a common method for dealing with sparse structures. In this paper, we present an accelerated proximal extra-gradient method for figuring out the representative selection problems. We proved the convergence of the algorithm we proposed. Moreover, we apply our method to solve representative selection problem. Numerical experiments illustrate the advantages of our method.

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

"Extragradient method with a new adaptive step size for solving non-Lipschitzian pseudo-monotone variational inequalities"

"The purpose of this work is to develop a new version of the extragradient method for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. First, we prove a sufficient condition for weak convergence of a proposed algorithm under relaxed assumptions. Next, under strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a Q-linear convergence rate of this algorithm. Our results improve some recent contributions in the literature on the extragradient method."

Variants of the A-HPE and large-step A-HPE algorithms for strongly convex problems with applications to accelerated high-order tensor methods

For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods, SIAM J. Optim. 23 (2013), pp. 1092–1125.]. We prove linear and the superlinear global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaining a new inexact (relative-error) tensor method for (smooth) strongly convex optimization with iteration-complexity . In particular, for p = 2, we obtain an inexact proximal-Newton algorithm with fast global convergence rate.

A New Construction and Convergence Analysis of Non-Monotonic Iterative Methods for Solving ρ-Demicontractive Fixed Point Problems and Variational Inequalities Involving Pseudomonotone Mapping

Two new inertial-type extragradient methods are proposed to find a numerical common solution to the variational inequality problem involving a pseudomonotone and Lipschitz continuous operator, as well as the fixed point problem in real Hilbert spaces with a ρ-demicontractive mapping. These inertial-type iterative methods use self-adaptive step size rules that do not require previous knowledge of the Lipschitz constant. We also show that the proposed methods strongly converge to a solution of the variational inequality and fixed point problems under appropriate standard test conditions. Finally, we present several numerical examples to show the effectiveness and validation of the proposed methods.

Improved variance reduction extragradient method with line search for stochastic variational inequalities

Improved variance reduction extragradient method with line search for stochastic variational inequalities

On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis

The main objective of this research is to find the numerical solution of variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces. The main advantage of these iterative schemes is that they allow the uncomplicated calculation of step size rules that depend on the knowledge of an operator explanation instead of the Lipschitz constant or some other line search method. The proposed iterative schemes follow a monotone and non-monotone step size procedure based on mapping (operator) information as a replacement for its Lipschitz constant or some other line search method. The strong convergences are well proven, analogous to the proposed methods, and impose certain control specification conditions. Finally, to verify the effectiveness of the iterative methods, we present some numerical experiments.

Strong Convergence of an Inertial Extragradient Method with an Adaptive Nondecreasing Step Size for Solving Variational Inequalities

Strong Convergence of an Inertial Extragradient Method with an Adaptive Nondecreasing Step Size for Solving Variational Inequalities

"A modified Halpern extragradient method for equilibrium and fixed point problems in CAT(0) space"

"In this paper, we introduce a modified Halpern extragradient-type algorithm for approximating an element in the intersection of the set of common solutions of equilibrium problems and common fixed points of family of nonexpansive mappings in a complete CAT(0) space. We establish strong convergence theorem that improve and generalize recently announced results in the literature."

Revisiting subgradient extragradient methods for solving variational inequalities

Revisiting subgradient extragradient methods for solving variational inequalities

A note on approximate accelerated forward-backward methods with absolute and relative errors, and possibly strongly convex objectives

In this short note, we provide a simple version of an accelerated forward-backward method (a.k.a. Nesterov's accelerated proximal gradient method) possibly relying on approximate proximal operators and allowing to exploit strong convexity of the objective function. The method supports both relative and absolute errors, and its behavior is illustrated on a set of standard numerical experiments. Using the same developments, we further provide a version of the accelerated proximal hybrid extragradient method of Monteiro and Svaiter (2013) possibly exploiting strong convexity of the objective function.

A Regularized Generalized Popov’s Method to Solve the Hierarchical Variational Inequality Problem with Generalized Lipschitzian Mappings

In this article, we introduce a new inertial multi-step regularized generalized Popov’s extra-gradient method to solve the hierarchical variational inequality problem (HVIP). We extend the previous Lipschitzian and strongly monotone mapping to a hemicontinuous, generalized Lipschitzian and strongly monotone mapping. We also obtain a strong convergence theorem about the new Popov’s algorithm. Furthermore, we utilize some numerical experiments to highlight the feasibility and effectiveness of our method.

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

Investigation on the performance of the expansion water seal by using isogeometric analysis with variational inequalities for the frictional contact

In this study, IGA is introduced into the performance analysis of the expansion water seal for an exact representation of its complex geometrical shape. Firstly, the incremental equilibrium equations used in the large deformation analysis are derived for the IGA discretization model based on the updated Lagrangian formulation. The unknown contact forces are directly involved in the equilibrium equations, which results in a simple expression. Then, through the investigation of the contact conditions, the normal and tangential contact forces are verified to be the solutions of two box-constrained variational inequalities representing the normal and tangential contact conditions, respectively. Furthermore, the incremental equilibrium equations for all components of the water seal system and the variational inequalities for all contacts between the components are assembled together and reformulated as a global variational inequality, which is resolved by using the Extra-gradient method. Ultimately, the new method is applied in the performance comparison of two optional expansion water seals, in which the capability and precision of the proposed method are investigated. Results verify that the proposed method is effective in the numerical simulation of the expansion water seal and has a higher precision than the traditional FEM under the same conditions.