Strong convergence of extragradient methods with a new step size for solving variational inequality problems

Abstract

In this paper, we propose two different kinds of extragradient methods with a new step size for finding an element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator in real Hilbert spaces. We only use one projection to design the algorithms and the strong convergence theorems proved without the prior knowledge of the Lipschitz constant of cost operator. Numerical experiments illustrate the performances of our new algorithms and provide a comparison with related algorithms. Keywords Extragradient method • Subgradient extragradient method • Tseng's extragradient method • Mann-type method • Variational inequality problem Mathematics Subject Classification 65Y05 • 65K15 • 68W10 • 47H09 • 47J25 Communicated by Gabriel Haeser. Dedicated to Professor Pham Ky Anh on the Occasion of his 70th Birthday.

Improved inertial projection and contraction method for solving pseudomonotone variational inequality problems

“…Figures 1, 2, 3 have conformed that the proposed algorithm has the competitive advantage over existing Algorithm 2[11] and Algorithm 3.2[12].…”

“…From Tables 1, 2, we can easily observe that Algorithm 1 converges for a shorter iterate number than the previously studied Algorithm 2 [11] and Algorithm 3.2 [12]. Obviously, the operator A is monotone and Lipschitz continuous.…”

“…In 2019, Thong and Hieu [12] proposed a self-adaptive algorithm which was based on Mann-type Tseng's extragradient method, the algorithm is as follows.…”

“…In this paper, motivated and inspired by the results in the literature Tseng [7], Thong and Hieu [11,12], and by the ongoing research in these directions, we introduce a new algorithm for solving the (VIP) involving pseudomonotone and Lipschitz continuous operator. The algorithm combines the inertial technique with the projection and contraction method (PCM), it uses a new step size rule which allows the introduced algorithm to work without depending on the Lipschitz constant of cost operator, the step size is updated over each iteration.…”

“…In this section, we consider some numerical examples to evaluate the efficiency and advantages of our proposed algorithm in comparison with the well-known Algorithm 2 [11], Algorithm 3.2 [12]. The projections over are computed effectively by the function quadprog in Matlab 7.0 Optimization Toolbox.…”