An Extragradient Algorithm for Monotone Variational Inequalities

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

“…When E = H (a Hilbert space), c 1 = 1, Π C = P C , and J = I, then, Theorem 3.5 reduces to Theorem 1.5. That is to say that Theorem 3.5 absolutely generalizes the results of [17] from Hilbert spaces to Banach spaces. (1) The inverse-strong-monotonicity of A is relaxed to monotonicity and Lipschitz continuity.…”

“…Inspired by the results of [11,17], we propose the following Algorithm 3.1 to extend Algorithm 1.4 from Hilbert spaces to Banach spaces and prove a weak convergence theorem.…”

“…The weak convergence of this Algorithm is proved. Consequently, we generalize the results of [17] from Hilbert spaces to Banach spaces and solve simultaneously problems (P1) and (P2). Furthermore, we give two simple examples which show Algorithm 3.1 constructed by this paper has a competitive advantage over the time of execution and the iterative number being far less than those of Algorithm 1.3 constructed in [16] provided with the same initial point.…”

“…We modify the extragradient algorithm introduced by Malitsky and Semenov in [17] and construct Algorithm 3.1 to find an element of VI(C, A) for a Lipschitz-continuous, monotone mapping in a 2-uniformly convex and uniformly smooth Banach space. The weak convergence of this Algorithm is proved.…”

“…To avoid the drawback, Malitsky and Semenov [17] combined the Popov algorithm [21] and the subgradient extragradient method [6] to propose the following iterative algorithm: Algorithm 1.4.…”