Variational Inequalities In Hilbert Spaces Research Articles

A reconsideration on convergence of the extra-gradient method for solving quasimonotone variational inequalities

ABSTRACT In this paper, we continue to investigate convergence of the extra-gradient method for solving quasimonotone variational inequalities in Hilbert spaces. Under some suitable conditions, we prove that the iterative sequence generated by the extra-gradient method converges weakly to a solution of quasimonotone variational inequalities. Our result corrects and improves the main result in Salahuddin [The extra-gradient method for quasimonotone variational inequalities. Optimization. 2022;71:2519–2528].

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Projection-type method with line-search process for solving variational inequalities

We propose an accelerated projection-type method for solving variational inequalities in Hilbert spaces. The method is suitable for pseudomonotone non-Lipschitz continuous operators and obtains strong convergence for iterative sequences under appropriate conditions. Based on the classical double-projection gradient method, we establish a new line-search rule and add an inertia term to improve the convergence speed of the proposed algorithm. Finally, some numerical examples are implemented to compare the proposed algorithm with the existing results, thereby verifying that our method has better performance.

A parallel algorithm for solving a pair of two variational inequalities in Hilbert spaces

The purpose of this paper is to introduce a new parallel algorithm for finding solutions to pairs of variational inequalities over the solution sets of systems of split equality null-point problems in Hilbert spaces. More precisely, we propose a new self-adaptive algorithm based on the hybrid steepest descent method and establish a strong convergence theorem for it. Moreover, we also give some applications of our proposed algorithm to solving some related problems, such as systems of split equality minimum point problems and systems of split equality feasibility problems.

A refined convergence analysis of Popov's algorithm for pseudo-monotone variational inequalities

In this work, we present a refined convergence analysis of the Popov's projection algorithm for solving pseudo-monotone variational inequalities in Hilbert spaces. Our analysis results in a larger range of stepsize, which is achieved by using a new Lyapunov function. Furthermore, when the operator is strongly pseudo-monotone and Lipschitz continuous, we establish the linear convergence of the sequence generated by the Popov's algorithm. As a by-product of our analysis, we extend the range of stepsize in the projected reflected gradient algorithm for solving unconstrained pseudo-monotone variational inequalities.

Infinite dimensional tensor variational inequalities with applications to an economic equilibrium problem

In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example.

A single projection algorithm with double inertial extrapolation steps for solving pseudomonotone variational inequalities in Hilbert space

In this work, we investigate a single projection method with double inertial extrapolation steps and self-adaptive step sizes for solving pseudomonotone variational inequality problems in a real Hilbert space. Weak and linear convergence are presented under suitable conditions. Some simple numerical examples are given to illustrate the performance of the proposed algorithm.

РЕГУЛЯРИЗОВАНИЙ АЛГОРИТМ ОПЕРАТОРНОЇ ЕКСТРАПОЛЯЦIЇ

This work is devoted to the study of new algorithm for solving variational inequalities in Hilbert spaces. The proposed algorithm is a variant of the operator extrapolation method regularized using the Halpern scheme. The algorithm has an advantage over the Korpelevich extragradient method and the method of extrapolation from the past in terms of the amount of calculations required for the iterative step. For variational inequalities with monotone, Lipschitz continuous operators acting in Hilbert space, a theorem on strong convergence of the method is proved.

A self-adaptive forward-backward-forward algorithm for solving split variational inequalities

In this paper, we consider an iterative approximation problem of split variational inequalities in Hilbert spaces. In order to solve this split problem, we construct an iterative algorithm which combines a forward-backward-forward method and a self-adaptive rule to update the step-sizes. We prove that the constructed algorithm converges strongly to a solution of the split variational inequalities under some mild assumptions.

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

Strong and linear convergence of projection-type method with an inertial term for finding minimum-norm solutions of pseudomonotone variational inequalities in Hilbert spaces

Strong and linear convergence of projection-type method with an inertial term for finding minimum-norm solutions of pseudomonotone variational inequalities in Hilbert spaces

An effective iterative projection method for variational inequalities in Hilbert spaces

An effective iterative projection method for variational inequalities in Hilbert spaces

A Forward-Backward-Forward Algorithm for Solving Quasimonotone Variational Inequalities

In this paper, we continue to investigate the convergence analysis of Tseng-type forward-backward-forward algorithms for solving quasimonotone variational inequalities in Hilbert spaces. We use a self-adaptive technique to update the step sizes without prior knowledge of the Lipschitz constant of quasimonotone operators. Furthermore, we weaken the sequential weak continuity of quasimonotone operators to a weaker condition. Under some mild assumptions, we prove that Tseng-type forward-backward-forward algorithm converges weakly to a solution of quasimonotone variational inequalities.

Algorithmic approach to the solution of pseudomonotone equilibrium problems and generalized variational inequalities

In this paper, we consider pseudomonotone equilibrium problems and generalized variational inequalities in Hilbert spaces. We suggest an iterative procedure for solving pseudomonotone equilibrium problems and generalized variational inequalities. Strong convergence result is proved under some mild assumptions.

ЗБIЖНIСТЬ АДАПТИВНИХ ЕКСТРАПРОКСИМАЛЬНИХ АЛГОРИТМIВ ДЛЯ ЗАДАЧ ПРО РIВНОВАГУ В ПРОСТОРАХ АДАМАРА

New iterative extra-proximal algorithms have been pro\-posed and investigated for approximate solution of problems of equilibrium in Hadamard metric spaces. The para\-meter update rule does not use the values of the Lipschitz constants of the bifunction. In contrast to the rules of the linear search type, it does not require calculations of the bifunction values at additional points. In addition, at the initial stages of the algorithms, the step size parameter can increase from iteration to iteration. For pseudo-monotone bifunctions of the Lipschitz type we proved convergence theorems. It is shown that the proposed algorithms are applicable to pseudo-monotone variational inequalities in Hilbert spaces.

Adaptive operator extrapolation method

This paper is devoted to the study of nоvel algorithm with Bregman projection for solving variational inequalities in Hilbert space. Proposed algorithm is an adaptive version of the operator extrapolation method, where the used rule for updating the step size does not require knowledge of Lipschitz constants and the calculation of operator values at additional points. An attractive feature of the algorithm is only one computation at the iterative step of the Bregman projection onto the feasible set.

A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces

In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included.

An alternated inertial method for pseudomonotone variational inequalities in Hilbert spaces

In this paper, we introduce a new relaxed extrgadient algorithm with alternated inertial extrapolation step and self adaptive variable stepsizes for solving variational inequality problems whose cost operator is pseudomonotone operator in Hilbert spaces. We establish the weak convergence of the proposed algorithm and linear convergence under some standard assumptions. Numerical experiments are given to support theoretical results and comparison with recent related methods.

Сходимость метода экстраполяции из прошлого и метода операторной экстраполяции

Одним из популярных направлений современного прикладного нелинейного анализа является исследование вариационных неравенств. Многие актуальные проблемы исследования операций и математической физики можно записать в форме вариационных неравенств. С появлением генерирующих соревновательных нейронных сетей интерес к алгоритмам решения вариационных неравенств возник и в среде специалистов машинного обучения. Данная работа посвящена исследованию трех новых алгоритмов с брегмановской проекцией для решения вариационных неравенств в гильбертовом пространстве. Первый алгоритм — результат модификации двухэтапного брегмановского метода с помощью экономного регулирования величины шага, не требующего знания лепшицевой константы оператора. Второй алгоритм — алгоритм операторной экстраполяции, полученный заменой в методе Малицкого-Тама евклидовой метрики на дивергенцию Брегмана. Привлекательная черта алгоритма — всего одно вычисление на итерационном шаге проекции Брэгмана на допустимое множество. Третий алгоритм — адаптивный вариант второго, где используется правило обновления величины шага, не требующее знания лепшицевых констант и вычислений значений оператора в дополнительных точках. Для вариационных неравенств с псевдомонотонными, лепшицевыми и секвенционно слабо непрерывными операторами, действующими в гильбертовом пространстве, доказаны теоремы о сходимости методов.

A Strong Convergence Theorem for an Iterative Method for Solving the Split Variational Inequalities in Hilbert Spaces

There are many iterative methods for solving the split variational inequality problems involving step sizes that depend on the norm of a bounded linear operator $\mathcal F$ . We know that the implementation of such algorithms is usually difficult to handle, because we have to compute the norm of the operator $\mathcal F$ . In this paper, we introduce a new iterative algorithm for approximating a solution of a class of multiple-sets split variational inequality problems, without prior knowledge of operator norms. Strong convergence of the iterative process is proved. As an application, we obtain a strong convergence result for a class of multiple-sets split feasibility problem. Two numerical examples are given to illustrate the proposed iterative algorithm.