Subgradient Extragradient Algorithm Research Articles

Accelerated Subgradient Extragradient Methods for Variational Inequality Problems

In this paper, we introduce two new iterative algorithms for solving monotone variational inequality problems in real Hilbert spaces, which are based on the inertial subgradient extragradient algorithm, the viscosity approximation method and the Mann type method, and prove some strong convergence theorems for the proposed algorithms under suitable conditions. The main results in this paper improve and extend some recent works given by some authors. Finally, the performances and comparisons with some existing methods are presented through several preliminary numerical experiments.

A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities

ABSTRACT In this paper, we introduce a new algorithm by incorporating inertial terms in a subgradient extragradient algorithm for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The strong convergence of the algorithm is obtained under mild assumptions. We also provide some numerical examples to illustrate that the acceleration of our algorithm is effective.

A new modified subgradient extragradient method for solving variational inequalities

The goal of the note is to introduce a new modified subgradient extragradient algorithm for solving variational inequalities in Hilbert spaces. A result on the strong convergence of the algorithm is proved without the knowledge of Lipschitz constant of the operator. Several numerical experiments for the proposed algorithm are presented.

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces

Abstract We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.

Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space

In this paper, we introduce a modified inertial subgradient extragradient algorithm in a 2-uniformly convex and uniformly smooth real Banach space and prove a strong convergence theorem for approximating a common solution of fixed point equation with a demigeneralized mapping and a variational inequality problem of a monotone and Lipschitz mapping. We present an example to validate our new findings. This work substantially improves and generalizes some well-known results in the literature.

Modified subgradient extragradient algorithms for solving monotone variational inequalities

ABSTRACTIn this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a new step size, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithms.

Convergence theorems of subgradient extragradient algorithm for solving variational inequalities and a convex feasibility problem

Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex real Banach space E with dual space E^{*}. In this paper, a Krasnoselskii-type subgradient extragradient iterative algorithm is constructed and used to approximate a common element of solutions of variational inequality problems and fixed points of a countable family of relatively nonexpansive maps. The theorems proved are improvement of the results of Censor et al. (J. Optim. Theory Appl. 148:318–335, 2011).

An improved projection method for solving generalized variational inequality problems

ABSTRACTIn this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is obtained by projecting the current iterate point onto a half-space. This half-space can separate strictly the current iterate point from the solution set of GVIP. Moreover, this method works without needing the current point belongs to the feasible set. Comparing with methods in Konnov [A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998;80(2):239–252] and Fang and Chen [Subgradient extragradient algorithm for solving multi-valued variational inequality. Appl Math Comput. 2014;229(3-4):123–130], our method can get rid of an auxiliary procedure in each iteration which is used to ensure the current iterate point belongs to feasible set. Consequently, our method is more simpler than those algorithms. The global convergence is proved under mild assumptions. Numerical results show that this method is much more efficient than the method in Li and He [An algorithm for generalized variational inequality with pseudomonotone mapping. J Comput Appl Math. 2009;228:212–218].

Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems

In this paper, basing on the subgradient extragradient method and inertial method with line-search process, we introduce two new algorithms for finding a common element of the solution set of a variational inequality and the fixed point set of a quasi-nonexpansive mapping with a demiclosedness property. The weak convergence of the algorithms are established under standard assumptions imposed on cost operators. The proposed algorithms can be considered as an improvement of the previously known inertial extragradient method over each computational step. Finally, for supporting the convergence of the proposed algorithms, we also consider several preliminary numerical experiments on a test problem.

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

ABSTRACTIn this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.

New subgradient extragradient methods for common solutions to equilibrium problems

In this paper, three parallel hybrid subgradient extragradient algorithms are proposed for finding a common solution of a finite family of equilibrium problems in Hilbert spaces. The proposed algorithms originate from previously known results for variational inequalities and can be considered as modifications of extragradient methods for equilibrium problems. Theorems of strong convergence are established under the standard assumptions imposed on bifunctions. Some numerical experiments are given to illustrate the convergence of the new algorithms and to compare their behavior with other algorithms.

WEAK AND STRONG CONVERGENCE OF SUBGRADIENT EXTRAGRADIENT METHODS FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

In this paper, we introduce three subgradient extragradient algorithms for solving pseudomonotone equilibrium problems. The paper originates from the subgradient extragradient algorithm for variational inequalities and the extragradient method for pseudomonotone equilibrium problems in which we have to solve two optimization programs onto feasible set. The main idea of the proposed algorithms is that at every iterative step, we have replaced the second optimization program by that one on a specific half-space which can be performed more easily. The weakly and strongly convergent theorems are established under widely used assumptions for bifunctions.

Cyclic subgradient extragradient methods for equilibrium problems

In this paper, we introduce a cyclic subgradient extragradient algorithm and its modified form for finding a solution of a system of equilibrium problems for a class of pseudomonotone and Lipschitz-type continuous bifunctions. The main idea of these algorithms originates from several previously known results for variational inequalities. The proposed algorithms are extensions of the subgradient extragradient method for variational inequalities to equilibrium problems and the hybrid (outer approximation) method. The paper can help in the design and analysis of practical algorithms and gives us a generalization of the most convex feasibility problems.

An Extragradient Algorithm for Monotone Variational Inequalities

A new iterative algorithm is proposed for solving the variational inequality problem with a monotone and Lipschitz continuous mapping in a Hilbert space. The algorithm is based on the following two well-known methods: the Popov algorithm and so-called subgradient extragradient algorithm. An advantage of the algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration. The weak convergence of sequences generated by the proposed algorithm is proved.

A subgradient extragradient algorithm for solving multi-valued variational inequality

In this paper, we propose a subgradient extragradient method for solving multi-valued variational inequality. It is showed that the method converges globally to a solution of multi-valued variational inequality, provided the multi-valued mapping is pseudomonotone with nonempty compact convex values. Convergence rate is analyzed. Preliminary computational experience is also reported.