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In this paper, we design a low-cost feasible projection algorithm for variational inequalities by replacing the projection onto the feasible set with the projection onto a ball. In each iteration, it only needs to calculate the value of the mapping once, and the projection onto the ball contained in the feasible set (which has an explicit expression), so the algorithm is easier to implement and feasible. The convergence of the algorithm is proved when the Slater condition holds for the feasible set and the mapping is pseudomonotone, Lipschitz continuous. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of hybrid inertial and contraction proximal point algorithm for the variational inclusion problem of the sum of two mappings in Hilbert spaces. We prove that the proposed algorithm converges strongly to a solution of the variational inclusion problem whenever its solution set is nonempty and the single-valued mapping f is Lipschitz continuous, monotone, and the set-valued mapping A is maximal monotone in infinite-dimensional real Hilbert spaces. Our work generalize and extend some related existing results in the literature. Finally, we illustrate the numerical performance of our Algorithm 1 and we give an application to the split feasibility problem. [ABSTRACT FROM AUTHOR]

It is well known that the monotonicity of the underlying mapping of variational inequalities plays a central role in the convergence analysis. In this paper, we propose an infeasible projection algorithm (IPA for short) for nonmonotone variational inequalities. The next iteration point of IPA is generated by projecting a vector onto a half-space. Hence, the computational cost of computing the next iteration point of IPA is much less than the algorithm of Ye and He (Comput. Optim. Appl. 60, 141–150, 2015) (YH for short). Moreover, if the underlying mapping is Lipschitz continuous with its modulus is known, by taking suitable parameters, IPA requires only one projection onto the feasible set per iteration. The global convergence of IPA is obtained when the solution set of its dual variational inequalities is nonempty. Moreover, if in addition error bound holds, the convergence rate of IPA is Q-linear. IPA can be used for a class of quasimonotone variational inequality problems and a class of quasiconvex minimization problems. Comparing with YH and Algorithm 2 in Deng, Hu and Fang (Numer. Algor. 86, 191–221, 2021) (DHF for short) by solving high-dimensional nonmonotone variational inequalities, numerical experiments show that IPA is much more efficient than YH and DHF from CPU time point of view. Moreover, IPA is less dependent on the initial value than YH and DHF. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two new algorithms for solving bilevel pseudomonotone variational inequality problems in real Hilbert spaces. The advantages of our algorithms are that they only need to calculate one projection on the feasible set in each iteration, and do not require the prior information of the Lipschitz constant of the cost operator. Furthermore, two new algorithms are derived to solve variational inequality problems. We establish the strong convergence of the proposed algorithms under some suitable conditions imposed on parameters. Finally, several numerical results and applications in optimal control problems are reported to illustrate the efficiency and advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized viscosity algorithm for finding a fixed point of an asymptotically nonexpansive mapping in the intermediate sense which is also a solution to a variational inequality problem of two inverse-strongly monotone operators in 2-uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are given under suitable assumptions imposed on the parameters. The results obtained in this paper improve and extend many recent ones in the literature. Three numerical examples are also given to show the efficiency and implementation of our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new extragradient-like algorithm for solving a variational inequality problem with a pseudo-monotone and Lipschitz continuous mapping in a Hilbert space. The iterative algorithm combines inertial ideas and hybrid extragradient ideas with the Armijo-like step size rule. Strong convergence of the algorithm is obtained and numerical experiments are provided. [ABSTRACT FROM AUTHOR]

In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive mappings and two nonlinear operators. It is shown that the proposed algorithm converges strongly under mild conditions. The rate of convergence of the proposed iterative algorithm is also studied. An equivalence of convergence between the normal S-iterative algorithm and Algorithm 2.6 of Noor (J. Math. Anal. Appl. 331, 810–822, 2007) is established and a comparison between the two is also discussed. As an application, a modified algorithm is employed to solve convex minimization problems. Numerical examples are given to validate the theoretical findings. The results obtained herein improve and complement the corresponding results in Noor (J. Math. Anal. Appl. 331, 810–822, 2007). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator and the set of fixed points of a demicontractive mapping in real Hilbert spaces. Although the problem can be translated to a common fixed point problem, the algorithm's structure is not derived from algorithms in this field but from the field of variational inequalities and hence can be computed quite easily. We extend several results in the literature from weak to strong convergence in real Hilbert spaces and moreover, the prior knowledge of the Lipschitz constant of cost operator is not needed. We prove two strong convergence theorems for the sequences generated by these new methods. Primary numerical examples illustrate the validity and potential applicability of the proposed schemes. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose Linesearch methods for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a self-adaptive inertial gradient projection algorithm for solving monotone or strongly pseudomonotone variational inequalities in real Hilbert spaces. The algorithm is designed such that the stepsizes are dynamically chosen and its convergence is guaranteed without the Lipschitz continuity and the paramonotonicity of the underlying operator. We will show that the proposed algorithm yields strong convergence without being combined with the hybrid/viscosity or linesearch methods. Our results improve and develop previously discussed gradient projection-type algorithms by Khanh and Vuong (J. Global Optim. 58, 341–350 2014). [ABSTRACT FROM AUTHOR]

In this paper, basing on the subgradient extragradient method and inertial method with line-search process, we introduce two new algorithms for finding a common element of the solution set of a variational inequality and the fixed point set of a quasi-nonexpansive mapping with a demiclosedness property. The weak convergence of the algorithms are established under standard assumptions imposed on cost operators. The proposed algorithms can be considered as an improvement of the previously known inertial extragradient method over each computational step. Finally, for supporting the convergence of the proposed algorithms, we also consider several preliminary numerical experiments on a test problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial subgradient-type algorithm to find the common element of fixed point set of a family of nonexpansive mappings and the solution set of the single-valued variational inequality problem. Under the assumption that the mapping is monotone and Lipschitz continuous, we show that the sequence generated by our algorithm converges strongly to some common element of the fixed set and the solution set. Moreover, preliminary numerical experiments are also reported. [ABSTRACT FROM AUTHOR]

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ϕ∗-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we improve the convergence theorem in the paper by Yang (Journal of Industrial and Management Optimization 1, 211-217, 2005), and propose a new modified convergence theorem. The theorem and the proof presented in the present paper are interesting improvements on the convergence theorem of Yang. [ABSTRACT FROM AUTHOR]

A nonsmooth Levenberg-Marquard (LM) method with double parameter adjusting strategies is presented for solving vertical complementarity problems based on the computation of an element of a vextor-valued minimum function's B-differential in this paper. At each iteration, the LM parameter is adjusted based on the norm of the vector-valued minimum function and the ratio between the actual reduction and the predicted reduction. Under the local error bound condition, which is strictly weaker than nonsingular assumption, the local convergence rate is discussed. Finally, the numerical tests indicate that the present algorithm is effective. [ABSTRACT FROM AUTHOR]