Variational Inequalities In Hilbert Spaces Research Articles

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.

Improved subgradient extragradient methods for solving pseudomonotone variational inequalities in Hilbert spaces

The purpose of this work to investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. For solving this problem, we propose two new methods which combine advantages of the subgradient extragradient method and the projection contraction method. Similar to some recent developments, the proposed methods do not require the knowledge of the Lipschitz constant associated with the variational inequality mapping. Under suitable mild conditions, we establish the weak and strong convergence of the proposed algorithms. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions. Numerical examples in fractional programming and optimal control problems demonstrate the potential of our algorithms as well as compare their performances to several related results.

A Second Order Dynamical System and Its Discretization for Strongly Pseudo-monotone Variational Inequalities

We consider a second order dynamical system for solving variational inequalities in Hilbert spaces. Under standard conditions, we prove the existence and uniqueness of strong global solution of the proposed dynamical system. The exponential convergence of trajectories is established under strong pseudo-monotonicity and Lipschitz continuity assumptions. A discrete version of the proposed dynamical system leads to a relaxed inertial projection algorithm whose linear convergence is proved under suitable conditions on parameters. We discuss the possibility of extension to general monotone inclusion problems. Finally some numerical experiments are reported demonstrating the theoretical results.

An Adaptive Algorithm for the Variational Inequality Over the Set of Solutions of the Equilibrium Problem

In the paper, we consider bilevel problems: variational inequality problems over the set of solutions of the equilibrium problem. Finding normal Nash equilibrium is an example of such a problem. To solve these problems, an iterative algorithm is proposed that combines the ideas of the two-stage proximal method, adaptability, and iterative regularization. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of information on bifunction’s Lipschitz constants and operator’s Lipschitz and strong monotonicity constants. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators, the theorem on strong convergence of sequences generated by the algorithm is proved. The proposed algorithm is shown to be applicable to monotone bilevel variational inequalities in Hilbert spaces.

ДВОРIВНЕВI ЗАДАЧI ТА ДВОЕТАПНИЙ ПРОКСИМАЛЬНИЙ АЛГОРИТМ

In this paper, a two-level problem is considered: a variational inequality on the set of solutions to the equilibrium problem. An example of such a problem is the search for the normal Nash equilibrium. To solve this problem, two algorithms are proposed. The first combines the ideas of a two-step proximal method and iterative regularization. And the second algorithm is an adaptive version of the first with a parameter update rule that does not use the values of the Lipschitz constants of the bifunction. Theorems on strong convergence of algorithms are proved for monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators. It is shown that the proposed algorithms can be applied to monotone two-level variational inequalities in Hilbert spaces.

An Adaptive Two-Stage Proximal Algorithm for Equilibrium Problems in Hadamard Spaces

Equilibrium problems in Hadamard metric spaces are considered in the paper. For approximate solution of problems, a new iterative adaptive two-stage proximal algorithm is proposed and analyzed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of the value of bifunction’s Lipschitz constants. For pseudo-monotone bifunctions of Lipschitz type, the theorem on weak convergence of the sequences generated by the algorithm is proved. It is shown that the proposed algorithm is applicable to pseudo-monotone variational inequalities in Hilbert spaces.

An iterative regularization method for variational inequalities in Hilbert spaces

We consider an iterative method for regularization of a variational inequality (VI) defined by a Lipschitz continuous monotone operator in the case where the set of feasible solutions is decomposed to the intersection of finitely many closed convex subsets of a Hilbert space. We prove the strong convergence of the sequence generated by our algorithm. It seems that this is the first time in the literature to handle iterative solution of ill-posed VIs in the domain decomposition case.

Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces

In this paper, the isotonicity of the proximity operator and its applications are discussed. We first establish a few new conditions of the mappings such that their proximity operators are isotone with respect to orders induced different minihedral cones. Some properties and examples for these conditions are then introduced. We especially consider the isotonicity of the proximity operator with respect to one order induced by a subdual cone and two orders. To estimate the convergence rate of the iterative algorithms, some other inequality characterizations of the proximity operator with respect to the orders are then proved. As applications, some solvability and approximation theorems for the mixed variational inequality and optimization problems are established by order approaches, in which the mappings need not to be continuous and the solutions are optimal with respect to the orders. By using the isotonicity of the proximity operator with respect to two orders, we overcome the absence of the regularity of the order. The convergence rate of forward–backward algorithms is finally estimated by order approaches.

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line‐search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

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.

The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces

Tseng’s forward–backward–forward algorithm is a valuable alternative for Korpelevich’s extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich’s method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward–backward–forward dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng’s forward–backward–forward algorithm with relaxation parameters, which we prove to converge also when it is applied to pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. Numerical experiments are carried out for pseudo-monotone variational inequalities over polyhedral sets and fractional programming problems.

Weak Sharpness and Finite Convergence for Solutions of Nonsmooth Variational Inequalities in Hilbert Spaces

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates.

A parallel iterative method for solving a class of variational inequalities in Hilbert spaces

A parallel iterative method for solving a class of variational inequalities in Hilbert spaces

Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces

In this paper, we introduce a new algorithm for solving a variational inequality problem in a Hilbert space. The algorithm originates from an explicit discretization of a dynamical system in time. We establish the convergence of the algorithm for a class of non-monotone and Lipschitz continuous operators, provided by the sequentially weak-to-weak continuity of cost operators. The rate of convergence of the algorithm is also proved under some standard hypotheses. Moreover, the new algorithm uses variable step-sizes which are updated at each iteration by a cheap computation without linesearch. This step-size rule allows the resulting algorithm to work more easily without the prior knowledge of Lipschitz constant of operator. Also, it is particularly interesting in the case where the Lipschitz constant is unknown or difficult to approximate. Several numerical experiments are implemented to illustrate the theoretical results and also to compare with existing algorithms.

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.

An efficient projection-type method for monotone variational inequalities in Hilbert spaces

We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising.

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.