An alternated inertial method for pseudomonotone variational inequalities in Hilbert spaces

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

This note is concerned with stability of linear variational inequalities (VIs) in Hilbert space. We prove an asymptotic global convergence result under appropriate conditions for perturbations of all data of a linear VI, that consists of a linear operator, a right hand side, and a convex constraint set. Here we employ Hausdorff set convergence to handle perturbations in arbitrary closed convex constraint sets what is a main novelty of the paper. To provide a simple illustration of our abstract stability theory we consider a VI of Volterra type with memory term and with unilateral constraints on some time interval and a box constrained variational problem involving Fourier series. We derive asymptotic global stability results for these variational problems.

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.

In this paper, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method, and the projection method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings and a strict pseudocontraction, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the solution set of a variational inequality problem (VIP), which is just a unique solution of a system of hierarchical variational inequalities (SHVI) in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.

In this paper we are focused on solving monotone and Lipschitz continuous variational inequalities in real Hilbert spaces. Motivated by several recent results related to the subgradient extragradient method (SEM), we propose two SEM extensions which do not require the knowledge of the Lipschitz constant associated with the variational inequality operator. Under mild and standard conditions, we establish the strong convergence of our schemes. Primary numerical examples demonstrate the potential of our algorithms as well as compare their performances to several related results.

Strong convergence property for Halpern-type iterative method with inertial terms for solving variational inequalities in real Hilbert spaces is investigated under mild assumptions in this paper. Our proposed method requires only one projection onto the feasible set per iteration, the underline operator is monotone and uniformly continuous which is more applicable than most existing methods for which strong convergence is achieved and our method includes the inertial extrapolation step which is believed to increase the rate of convergence. Numerical comparisons of our proposed method with some other related methods in the literature are given.

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

A Mann-type hybrid steepest-descent method for solving the variational inequality ⟨F(u*), v − u*⟩ ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473–504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185–201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51–69).

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.

Generalized system for relaxed cocoercive 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.

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

We consider a new algorithm for a generalized system for relaxed cocoercive nonlinear inequalities involving three different operators in Hilbert spaces by the convergence of projection methods. Our results include the previous results as special cases extend and improve the main results of R. U. Verma (2004), S. S. Chang et al. (2007), Z. Y. Huang and M. A. Noor (2007), and many others.

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.