Parallel Extragradient-Proximal Methods for Split Equilibrium Problems

Abstract

In this paper, we introduce two parallel extragradient-proximal methods for solving split equilibrium problems. The algorithms combine the extragradient method, the proximal method and the shrinking projection method. The weak and strong convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for equilibrium bifunctions. We also present an application to split variational inequality problems and a numerical example to illustrate the convergence of the proposed algorithms.Keywords: equilibrium problem, split equilibrium problem, extragradient method, proximal method, parallel algorithm.

Modified hybrid projection methods for finding common solutions to variational inequality problems

“…However, in infinite dimensional Hilbert spaces, the extragradient method only converges weakly. In recent years, the extragradient method has received a lot of attention, see, for example, [10,14,15,24,30,31] and the references therein. Nadezhkina and Takahashi [32] introduced the following hybrid extragradient method ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y n = P K (x n − λA(x n )), z n = P K (x n − λA(y n )), C n = {z ∈ C : ||z − z n || ≤ ||z − x n ||} , Q n = {z ∈ C : x 0 − x n , z − x n ≤ 0} , x n+1 = P C n ∩Q n (x 0 ), (4) where λ ∈ (0, 1 L ).…”

Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

“…Since x * ∈ T n , (16) λ n w, x * − x n+1 ≥ x n − x n+1 , x * − x n+1 . By w ∈ ∂ 2 f (y n , x n+1 ), we also obtain…”

“…The EP includes, as special cases, many mathematical models such as: variational inequality problems, fixed point problems, optimization problems, Nash equilirium problems, complementarity problems, etc., see [8,21] and the references therein. In recent years, many algorithms have been proposed for solving EPs [1,2,3,4,5,6,13,16,17,18,20,25,28]. In the case, the bifunction f is monotone, solution approximations of EPs are based on a regularization equilibrium problem, i.e., at the step n, known x n , the next approximation x n+1 is the solution of the following problem: (2) Find x ∈ C such that: f (x, y) + 1 r n y − x, x − x n ≥ 0, ∀y ∈ C, where r n is a suitable parameter.…”

Cyclic subgradient extragradient methods for equilibrium problems

“…CSEP (1) is very general in the sense that it includes, as special cases, many mathematical models: common solutions to variational inequalities, convex feasibility problems, common fixed point problems, see for instance [2,8,10,11,14,21,34,37]. These problems have been widely studied both theoretically and algorithmically over the past decades due to their applications to other fields [5,10,15,29].…”

“…In 2005, Combettes and Hirstoaga [14] introduced a general procedure for solving CSEPs. After that, many methods were also proposed for solving CSVIPs and CSEPs, see for instance [4,21,30,[32][33][34][35] and the references therein. However, the general procedure in [14] and the most existing methods are frequently based on the proximal point method (PPM) [22,28], i.e., at the current step, given x n , the next approximation x n+1 is the solution of the following regularized equilibrium problem (REP).…”