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We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least O ( 1 / k ) O(1/k) , where k k is the number of iterations. Numerical experiments are also provided to show advantages and outperformance of our adaptive Halpern algorithm over the standard Halpern algorithm.

The focus of this paper is to obtain weak and linear convergence analysis of the subgradient extragradient method with alternated inertial step for solving equilibrium problems in real Hilbert spaces. The proposed method uses self-adaptive step sizes. Weak convergence is established without Lipschitz constant of the bifunction as an input parameter. Linear convergence is obtained without the modulus of strong pseudomonotonicity and Lipschitz constant as input parameters. We report some priori and posteriori error estimates and some numerical experiments to illustrate the behavior of our proposed method with related methods.

The purpose of this paper is to study a new Tseng’s extragradient method with two different stepsize rules for solving pseudomonotone variational inequalities in real Hilbert spaces. We prove a strong convergence theorem of the proposed algorithm under some suitable conditions imposed on the parameters. Moreover, we also give some numerical experiments to demonstrate the performance of our algorithm.

In this paper, we introduce a Totally Relaxed Self-adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of a Variational Inequality Problem (VIP) and the fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation of projection onto the feasible set of the VIP; instead, it uses a projection onto a finite intersection of sub-level sets of convex functions. The advantage of this is that any general convex feasible set can be involved in the VIP. We also introduce a modified TRSSEM which involves the projection onto the set of a convex combination of some convex functions. Under some mild conditions, we prove a strong convergence theorem for our algorithm and also present an application of our theorem to the approximation of a solution of nonlinear integral equations of Hammerstein’s type. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature. Our algorithm is simple and easy to implement for computation.

The projection technique is a very important method and efficient for solving variational inequality problems. In this study, we developed the subgradient extragradient method for solving pseudomonotone variational inequality in real Hilbert spaces. Our first algorithm requires only computing one projection onto the feasible set per iteration and the strong convergence is proved without the prior knowledge of the Lipschitz constant as well as the sequentially weak continuity of the associated mapping. The second algorithm uses the linesearch procedure such that its convergence does not require the Lipschitz continuous condition of the variational inequality mapping. Finally, some numerical experiments are provided to demonstrate the advantages and efficiency of the proposed methods.

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [ 0 , 1 ] with θ = 1 possible. Weak and linear convergence are obtained for this second proposed method. As far as we know, the choice θ = 1 has not been considered before in the literature for inertial projection and contraction method. Finally, we give some numerical illustrations of our methods.

In this paper, we revisit the subgradient extragradient method for solving a pseudomonotone variational inequality problem with the Lipschitz condition in real Hilbert spaces. A new algorithm based...

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate the advantages of our proposed algorithms. The main results obtained in this paper extend and improve some related works in the literature.

The purpose of this paper is to study the convergence analysis of an iterative algorithm with inertial extrapolation step for finding an approximate solution of split monotone inclusion problem in ...

In this paper, we propose a single projection method for finding a solution of the bilevel pseudo-monotone variational inequality problem in real Hilbert spaces. The advantage of the proposed algor...

In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is “optimal” in theory. These ways of selecting stepsizes and relaxation parameters are also practised to the relaxed splitting method with linearization where the two closed convex sets are both level sets of convex functions. The weak convergence of two proposed methods is established under standard conditions and the linear convergence of the general splitting method with linearization is analyzed. The numerical examples are presented to illustrate the advantage of our methods by comparing with other methods.

The aim of this paper is to introduce a new inertial Tseng’s extragradient algorithm for solving variational inequality problems with pseudo-monotone and Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in real Hilbert spaces. We prove a strong convergence theorem for the proposed algorithm under suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results.

The split feasibility problem is to find a point x∗ with the property that x∗∈ C and Ax∗∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem.

In this paper, we proposed two different methods for solving pseudomonotone and strongly pseudomonotone equilibrium problems. We can examine these methods as an extension and improvement of the Pop...

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis.

We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the cost operators. We prove weak and strong convergence theorems for the sequences generated by these new methods. The numerical behavior of the proposed algorithms when applied to several test problems is compared with that of several previously known algorithms.

The purpose of this paper is to study a new viscosity iterative algorithm for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping and the set of solutions ...

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski–Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnose...

In this work, we propose a new modified Popov’s method by using inertial effect for solving the variational inequality problem in real Hilbert spaces. The advantage of the proposed algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration as well as it does not need to the prior knowledge of the Lipschitz constants of the variational inequality mapping. We present weak convergence theorem of the proposed algorithm under pseudomonotonicity and Lipschitz continuity of the associated mapping. Our results generalize and extend some related results in the literature and primary numerical experiments demonstrate the applicability of the scheme.

In this paper, we consider a special split feasibility problem (SFP): x∈CandAx∈Q where C is the solution set of an equilibrium problem, Q is a convex subset in Rm, and A:Rn→Rm is a linear operator....

"In this article, we introduce the multi-step inertial proximal contraction algorithms (MiPCA) to approximate a zero of the sum of two monotone operators, with one of the two operators being monotone and Lipschitz continuous. The weak convergence of the MiPCA is shown under the summability condition formulated in terms of the iterative sequence in a Hilbert space setting. We also investigate the unconditional convergence of the one-step inertial proximal contraction algorithm. Finally, numerical experiments are given to illustrate the advantage of the multi-step inertial proximal contraction algorithms."