Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space

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.

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

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.

We introduce a two-step direct method, like Korpelevich's, for solving monotone variational inequalities. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, whereas only weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem that lies closest to the initial iterate.

Generalized projection method for a variational inequality system with different mapping in Banach spaces

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.

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results.

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.

We study the existence theory for parabolic variational inequalities in weighted L 2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L 2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

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.

Chapter 4 Variational Inequalities in Hilbert Spaces

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.

Approximate proximal algorithms for generalized variational inequalities with paramonotonicity and pseudomonotonicity

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.