Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems

Abstract

In this paper, two splitting extragradient-like algorithms for solving strongly pseudomonotone equilibrium problems given by a sum of two bifunctions are proposed. The convergence of the proposed methods is analyzed and the R-linear rate of convergence under suitable assumptions on bifunctions is established. Moreover, a noisy data case, when a part of the bifunction is contaminated by errors, is studied. Finally, some numerical experiments are given to demonstrate the efficiency of our algorithms.

Contraction-Mapping Algorithm for the Equilibrium Problem Over the Fixed Point Set of a Nonexpansive Semigroup

“…We note that there exist several definitions of the Lipschitz-type continuity of bifunctions [1,4,8,15]. The Lipschitz-type condition used in this paper is a relaxation of the one introduced in [8].…”

“…Remark 1. (a) One example of {λ n } and {s n } satisfying the conditions in Algorithm 1 is λ n = 1/n ρ and s n = n δ with ρ ∈ (0, 1 2 ) and δ > 0. (b) To implement Algorithm 1, we do not have to calculate the constants of the strong monotonicity and the Lipschitz-type continuity of f .…”

“…where the mapping Φ : R m × R m → R m is bounded on a bounded set and ϕ is a L-Lipschitz continuous mapping on R m . The bifunction f in this example is a generalized form of the cost bifunctions in Nash-Cournot models considered in [1,3,8,19,20]. For any bounded sequences {x n } and {y n }, we have…”

“…It is well known that under the assumptions that f is L-Lipschitz type continuous, γ-strongly monotone and λ ∈ (0; 2γ L 2 ), Algorithm (5.1) linearly converges to the unique solution of EP (f, C) (see [1]).…”

“…Let C be a nonempty, closed, convex subset of a real Hilbert space H and f : C × C → R be a bifunction such that f (x, x) = 0 for all x ∈ C. Such a bifunction is called an equilibrium bifunction. The equilibrium problem for f on C can be formulated as others; see, for instance, [1,9,10,11,12,14,19,20,25] and references therein. Equilibrium problems have been generalized and extensively studied in many directions due to its importance.…”