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In this paper, we propose a reflected forward-backward splitting algorithm with two different inertial extrapolations to find a zero of the sum of three monotone operators consisting of the maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed algorithm is that both the Lipschitz continuous monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We then obtain weak and strong convergence results under some mild conditions. We finally give a numerical illustration to show that our proposed algorithm is effective and competitive with other related algorithms in the literature. [ABSTRACT FROM AUTHOR]

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results. [ABSTRACT FROM AUTHOR]

In this paper, we propose an alternated inertial general splitting method with linearization for a split feasibility problem. Four rules of inertial parameters and relaxation parameters are discussed, where the adaptive inertial parameters are firstly investigated. The convergence of the proposed method is established under standard conditions. The numerical examples are presented to test four choices of the parameters and illustrate the advantage of our methods. An application in signal processing is presented to illustrate the recovery of the proposed algorithm for the sparse signal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We propose one-step and two step inertial three-operator splitting algorithms by selecting the deviations along a momentum direction. A numerical experiment for DC regularized sparse recovery problems shows that the proposed algorithms have better performance than the original three-operator splitting algorithm. [ABSTRACT FROM AUTHOR]

In this paper we propose a new probability distribution for the randomized Kaczmarz (RK) algorithm where each row of the coefficient matrix is selected in the current iteration with the probability proportional to the square of the sine of the angle between it and the chosen row in the previous iteration. This probability distribution is helpful to accelerate the convergence of the RK algorithm. We obtain the linear convergence rate estimation in expectation for the RK algorithm with the new probability distribution (RKn), which is different from the existing results. In order to avoid the calculation and storage of probability matrix when solving the large-scale linear systems, one practical probability distribution is also proposed. Furthermore, an acceleration for the RKn algorithm and its simplified version are introduced respectively, whose core is to replace the projection onto one hyperplane with that onto the intersection of two hyperplanes. The accelerated schemes are proved to have faster convergence rates. Finally, numerical experiments are provided to illustrate the performance of the proposed algorithms by comparing with the RK algorithm and Motzkin's algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are obtained under some appropriate conditions on the parameters. Finally, we give some numerical experiments to show the advantages of our proposed algorithms. The results obtained in this paper extend and improve some recent works in the literature. [ABSTRACT FROM AUTHOR]

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapolation step with relaxation parameter to obtain a common fixed point of a countable family of averaged quasi-nonexpansive mappings in real Hilbert spaces. Our results bring a unification of several versions of fixed point methods for averaged quasi-nonexpansive mappings considered in the literature and give several implications of our results. We also give some applications to monotone inclusion problem with three-operator splitting method and composite convex and non-convex relaxed inertial proximal methods to solve both convex and nonconvex reweighted l Q regularization for recovering a sparse signal. Finally, some numerical experiments are drawn from sparse signal recovery to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

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 Krasnosel'ski–Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward–backward splitting methods and Douglas–Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski–Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others. [ABSTRACT FROM AUTHOR]

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 of a new variational inequality problem involving inverse-strongly monotone operators in Hilbert spaces. We prove some strong convergence theorems under some suitable assumptions imposed on the parameters by using a modified extragradient method. The results obtained in this paper may be an improvement of many recent ones in this fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special split feasibility problem (SFP): x ∈ C and A x ∈ Q where C is the solution set of an equilibrium problem, Q is a convex subset in R m , and A : R n → R m is a linear operator. We introduce two projection algorithms for solving the SFP by combining the projection method for the equilibrium problem and the gradient method for the inclusion A x ∈ Q. The proposed algorithms are shown to converge a solution of the SFP under weak conditions. We present a numerical example for a jointly constrained Nash equilibrium model in electricity production market to demonstrate the behaviour of the proposed algorithms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given. [ABSTRACT FROM AUTHOR]

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. (J. Optim. Theory Appl. 148, 318-335, 2011), replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant. [ABSTRACT FROM AUTHOR]

This paper reports a multiscale analysis and numerical algorithms for the elastic wave equations with rapidly oscillating coefficients. We mainly focus on the first-order and the second-order multiscale asymptotic expansions for the wave equations, which is proved to enjoy an explicit convergence rate. In our method, the homogenized equations are discretized by the finite element method in space and a symplectic geometric scheme in time. The multiscale solutions are then obtained efficiently by the standard multisclae asymptotic expansion framework. Several numerical simulations are carried out to validate the predicted convergence results. [ABSTRACT FROM AUTHOR]

In this article, we first present a projection method with outer perturbations for the split equality problem (SEP) and establish its weak convergence. Then combining the proposed projection method with the outer approximation method and Halpern iteration method, we introduce two approximation methods to solve the SEP, where the two closed convex sets are both level sets of convex functions. Finally, a preliminary experiment presented compares the approximation methods and a relaxed projection method. [ABSTRACT FROM AUTHOR]

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), where 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. [ABSTRACT FROM AUTHOR]

In this article, we first introduce two simultaneous projection algorithms for solving the split equality problem by using a new choice of the stepsize, and then propose two semi-alternating projection algorithms. The weak convergence of the proposed algorithms is analyzed under standard conditions. As applications, we extend the results to solve the split feasibility problem. Finally, a numerical example is presented to illustrate the efficiency and advantage of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A projection and contraction algorithm for solving multi-valued variational inequalities is proposed. The algorithm is proved to converge globally to a solution of a given multi-valued variational inequality under standard conditions. We present an analysis of the convergence rate. Finally, preliminary computational experiments illustrate the advantage of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we introduce an algorithms by incorporating inertial terms in the extragradient algorithm. A weak convergence theorem is established for the proposed algorithm. Numerical experiments show that the inertial algorithms speed up the original ones. [ABSTRACT FROM PUBLISHER]

In this article, we investigate the bounded perturbation resilience of the viscosity algorithm and propose the superiorized version of the viscosity algorithm. The convergence of the proposed algorithm is analyzed for a nonexpansive mapping. A modified viscosity algorithm and its convergence is presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose an alternated inertial subgradient extragradient algorithm for variational inequalities with self-adaptive step-sizes and obtain weak and linear convergence results. We also obtain linear convergence results using an alternated inertial projected gradient algorithm for which knowledge of the modulus of strong pseudo-monotonicity and Lipschitz constant of the cost function are not needed. We compare numerically our algorithms with other projection-type algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a pseudo-monotone , Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only compute one projection onto a feasible set per iteration as well as without using the sequentially weakly continuity of the associated mapping. Under additional strong pseudo-monotonicity and Lipschitz continuity assumptions, we obtain also an R -linear convergence rate of the proposed algorithm. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new hybrid algorithm which is not based on the modification to weak convergence algorithms. The strong convergence theorem of the proposed algorithm is presented. Finally, the numerical experiments suggest that the new algorithm could be faster than Nakajo and Takahashi's algorithm in J. Math. Anal. Appl. 279:372-379, 2003. [ABSTRACT FROM AUTHOR]

The split equality problem has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Moudafi proposed an alternating CQ algorithm and its relaxed variant to solve it. However, to employ Moudafi’s algorithms, one needs to know a priori norm (or at least an estimate of the norm) of the bounded linear operators (matrices in the finite-dimensional framework). To estimate the norm of an operator is very difficult, but not an impossible task. It is the purpose of this paper to introduce a projection algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any priori information about the operator norms. We also practise this way of selecting stepsizes for variants of the projection algorithm, including a relaxed projection algorithm where the two closed convex sets are both level sets of convex functions, and a viscosity algorithm. Both weak and strong convergence are investigated. [ABSTRACT FROM AUTHOR]

In the paper, first, we introduce a new hybrid projection algorithm and present its strong convergence theorem. Next, we analyze different hybrid algorithms in computing and conclude that our proposed algorithm has an advantage. Finally, the numerical experiments validate the efficiency and advantages of the new algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a new projection method for solving pseudomonotone variational inequality problems in real reflexive Banach spaces. The main algorithm is based on the self-adaptive method, subgradient extragradient method and Bregman projection method. Under some appropriate assumptions imposed on the parameters, we prove a strong convergence theorem of the proposed algorithm. Finally, several numerical examples are given to show that our method has better convergence performance than the known results in the literatures. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate pseudomonotone variational inequalities in real Hilbert spaces. For solving this problem, we introduce a new method. The proposed algorithm combines the advantages of the subgradient extragradient method and the projection and contraction method. We establish the strong convergence of the proposed algorithm under conditions pseudomonotonicity and Lipschitz continuity assumptions. Moreover, under additional strong pseudomonotonicity and Lipschitz continuity assumptions, the linear convergence of the sequence generated by the proposed algorithm is obtained. Numerical examples provide to illustrate the potential of our algorithms as well as compare their performances to several related results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the wave equations in perforated domains with a homogeneous Neumann condition on the boundary of the holes. The multiscale asymptotic expansion of the solution for the problem are constructed and associated explicit convergence rates are obtained. A multiscale numerical method is introduced. Finally, we present some numerical results which support strongly the convergence theorem. [Copyright &y& Elsevier]

In this article, we introduce a projection regularized Krasnoselski-Mann iteration for cutters. The proposed algorithm ensures the strong convergence of the generated sequence toward the least norm element of the set of fixed points of the cutter. It is verified that the projection regularized Krasnoselski-Mann iteration converges locally faster than the regularized Krasnoselski-Mann iteration introduced by Maingé and Maruster [11]. Furthermore, we present projection regularized Krasnoselski-Mann iterations for quasi-nonexpansive and nonexpansive mappings in Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modified self adaptive subgradient extragradient algorithm for solving pseudomonotone variational inequality problems and fixed point problems in Hilbert spaces. Precisely, we prove that the sequence generated by our algorithm is strongly convergent under some suitable conditions imposed on the parameters. Moreover, some numerical experiments are given to illustrate the efficiency of our proposed method. [ABSTRACT FROM AUTHOR]

Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a W-mapping for a finite family of relatively quasi-nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operator and set of common fixed points of a finite family of relatively quasi-nonexpansive mappings. Strong convergence theorems are presented in a 2-uniformly convex and uniformly smooth Banach space. Our results generalize and extend relative results. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new W-mapping and present an iterative algorithm for an infinite family of strict pseudo-contractions. Strong convergence theorems are proved in Banach spaces. Our results improve and extend the corresponding result announced by Cai and Hu [G. Cai, C. Hu, Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions, Nonlinear Anal. 71(12) (2009) 6044–6053]. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the multiscale analysis and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. The formal multiscale asymptotic expansions of the solutions for these problems in four specific cases are presented. Higher order corrector methods are constructed and associated explicit convergence rates are obtained in some cases. A multiscale numerical method and a symplectic geometric scheme are introduced. Finally, some numerical results and unsolved problems are presented, and these numerical results support strongly the convergence theorem of this paper. [Copyright &y& Elsevier]

In this paper, we mainly introduce a new and different self adaptive extragradient algorithm with an inertial technique for solving variational inequality problems of pseudomonotone operators and fixed point problems of relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of variational inequality problems and fixed point problems. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved to have a faster convergence rate. Numerical experiments support the theory results and illustrate that the multi-inertial randomized Kaczmarz algorithm significantly outperform the two-subspace Kaczmarz method in solving coherent linear systems. [ABSTRACT FROM AUTHOR]

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 algorithm requires only one projection onto the feasible set. Also, we prove strong convergence theorems of the proposed method under mild conditions, which improve some related results in the literature. Finally, we present some numerical experiments to show the efficiency and advantages of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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 Popov's extragradient method. We replaced the second minimization problem onto a closed convex set in the Popov's extragradient method, with a half-space minimization problem that is updated on each iteration and also formulates a useful method for determining the appropriate stepsize on each iteration. The weak convergence theorem of the first method and strong convergence theorem for the second method is well-established based on a standard assumption on a cost bifunction. We also consider various numerical examples to support our well-established convergence results, and we can see that the proposed methods depict a significant improvement in terms of the number of iterations and execution time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a real Hilbert space H with inner product 〈⋅,⋅〉, and let f:H→H be a nonlinear operator. Consider the inverse variational inequality (in short, IVI(C,f)) problem of finding a point ξ∗∈H such that f(ξ∗)∈C,〈ξ∗,v−f(ξ∗)〉≥0,∀v∈C. In this paper, we prove that IVI(C,f) has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261-1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator PC is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some x * ∈ C : = ∩ i = 1 m { x ∈ H | c i (x) ≤ 0 } , where m is a positive integer, H is a real Hilbert space, and { c i } i = 1 m are convex functions defined as H. The key of the CPM is that, for the current iterate x k , the CPM firstly constructs a new level set H k through a convex combination of some of { c i } i = 1 m in an appropriate way, and then updates the new iterate x k + 1 only by using the projection P H k . We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider an improvement of the extragradient method to figure out the numerical solution for pseudomonotone equilibrium problems in arbitrary real Hilbert space. A new method is proposed with an inertial scheme and a self adaptive step size rule that is revised on each iteration based on the previous three iterations. The weak convergence of the method is proved by assuming standard cost bifunction assumptions. We also consider the application of our results to solve different kinds of variational inequality problems and a particular class of fixed point problems. For a numerical part, we study the well‐known Nash‐Cournot equilibrium model and other test problems to support our well‐established convergence results and to ensure that our proposed method has a competitive edge over CPU time and a number of iterations. [ABSTRACT FROM AUTHOR]

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This article is concerned with a universal version of projected reflected gradient method with new step size for solving variational inequality problem in Hilbert spaces. Under appropriate assumptions controlled by the operators and parameters, we acquire the weak convergence of the proposed algorithm. Moreover, we establish an R-linear convergence rate of our method on the condition that the relevant mapping is strongly monotone. We rework our first algorithm so that it can be simplified to several generalized methods in the literature. The efficacy and availability of our proposed iterative scheme are demonstrated in numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we mainly introduce a new algorithm with a different line-search process for solving variational inequality problem of pseudomonotone and non-Lipschitz operators in real Hilbert spaces. Under some appropriate restrictions imposed on the parameters, we prove a strong convergence theorem for finding an element of solutions of variational inequality problem. At the same time, we give some numerical examples to illustrate the effectiveness of our proposed algorithm. The main results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a modification of a recently studied Malitsky–Tam forward-reflected-backward splitting method with two-step inertial extrapolation to solve nonconvex mixed variational inequalities. We give global convergence results and nonasymptotic O (1 / k) convergence rate of our proposed method under some appropriate conditions and present some mathematical justifications. [ABSTRACT FROM AUTHOR]