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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]

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]

We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the Lipschitz constant of the variational inequality operator and that it only computes one projection onto the feasible set per iteration. In addition, we do not need the sequential weak continuity of the variational inequality operator in order to establish our strong convergence theorem. Next, we also obtain an R-linear convergence rate for a related relaxed inertial gradient method under strong pseudo-monotonicity and Lipschitz continuity assumptions on the variational inequality operator. Finally, we present several numerical examples which illustrate the performance and the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]