Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings

Cyclic subgradient extragradient methods for equilibrium problems

“…Three algorithms in [35] used the extragradient method [30] for equilibrium problems while the idea of Algorithm 3.8 comes from the subgradient extragradient method. The hybrid step for finding projection x n+1 = P H n ∩W n (x 0 ) in Algorithm 3.8 is explicit, but that one for the algorithms in [35] still deals with the feasible set C. The approximation z n in Step 1 belongs to the halfspace T n and it, in general, is not in C. Thus, we assume here that S is defined on the whole space H .…”

“…The hybrid step for finding projection x n+1 = P H n ∩W n (x 0 ) in Algorithm 3.8 is explicit, but that one for the algorithms in [35] still deals with the feasible set C. The approximation z n in Step 1 belongs to the halfspace T n and it, in general, is not in C. Thus, we assume here that S is defined on the whole space H . For N = 1, the author in [1] proposed a strongly convergent hybrid extragradient algorithm for an equilibrium problem and a fixed point problem which does not use cutting-halfspaces.…”

“…, 1) T ∈ m . We compare Algorithm 3.1 with the parallel hybrid extragradient method (PHEGM) [35,Algorithm 1]. The advantage of the proposed algorithms is a computational modification of an optimization program over each iteration.…”

“…In fact, per each step Algorithm 3.1 only computes a bifunction while PHEGM computes simultaneously N bifunctions. Figures 3, 4 and 5 show the results for {D n } generated by Algorithm 3.1 and PHEGM [35] for the chosen stepsizes of λ n . In these figures, the y-axes represent the value of D n while the x-axes represent elapsed time (in second).…”

“…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).…”