A Modified Extragradient Method for Infinite-Dimensional Variational Inequalities

Abstract

A modified form of the extragradient method for solving infinite-dimensional variational inequalities is considered. The weak convergence and the strong convergence for the iterative sequence generated by this method are studied. We also propose several examples to analyze the obtained results.

On the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities

“…The following theorem states that the sequence {u k } converges weakly to a solution of VI(K , F). This result extends the Extragradient Algorithm for solving monotone VIs [7,9] to pseudo-monotone VIs.…”

“…We recall an important property of the iterative sequence {u k } generated by the Extragradient Algorithm; see, e.g., [6,9].…”

“…Recently, the extragradient method has been considered for solving VIs in infinitedimensional Hilbert spaces [7][8][9]. Providing that the VI has solutions and the assigned mapping is monotone and Lipschitz continuous, it is proved that the iterative sequence generated by the extragradient method converges weakly to a solution.…”

A unified framework for three accelerated extragradient methods and further acceleration for variational inequality problems

“…where λ ∈ (0, 1/L). Recently, the extragradient methods have received great attention for solving pseudo-monotone variational inequality problems in infinite dimensional Hilbert spaces [9,19,21,37]. Censor et al [10] proposed the following subgradient extragradient method (SEM) for solving variational inequality problem (3) (3):…”

“…The relaxed normal S-iteration method ( 19) is new from normal S-iteration method (17) in the following contexts: (i) the iteration parameter {α n } is lying on interval (0, α max ), (ii) the given operator S is not involved in Algorithm (19).…”