An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems

Accelerated inertial subgradient extragradient algorithms with non-monotonic step sizes for equilibrium problems and fixed point problems

“…( 5) We can obtain three new iterative schemes for solving the equilibrium problem when S = I, where I is the identity operator. These results improve and generalize many algorithms used in the literature for solving equilibrium problems (see, e.g., [11,12,13,14,15,19,22,26]), based on the following five facts: (i) our algorithms improve the computational efficiency of extragradient-type algorithms [11,14,26] due to the fact that only one optimization problem in the feasible set needs to be computed in each iteration; (ii) our algorithms include a pseudomonotone bifunction, which extends the results used in [13,22,26] for solving monotone or strongly pseudomonotone equilibrium problems; (iii) our algorithms apply a new non-monotonic step size criterion, which is different from the non-summable and non-increasing step sizes used in [12,15,19,22,26]; (iv) our algorithms embed inertial terms to accelerate the convergence speed of the algorithms used; and (v) our algorithms obtain strong convergence theorems in infinite-dimensional Hilbert spaces, which is more preferable to the weakly convergent results proposed in [12,14,15].…”

“…Since the bifunctions in realistic equilibrium problems may be of pseudomonotone. Recently, some efforts have been put into solving pseudomonotone equilibrium problems (see, e.g., [12,13,14,15]), which extend some of the results obtained in [13,22,26] for solving monotone (or strongly pseudomonotone) equilibrium problems due to the fact that the pseudomonotone bifunctions contain the monotone bifunctions and strongly pseudomonotone bifunctions. On the other hand, we observe that the fixed point mapping S in [9,20] is demicontractive while the mapping S in [7,21,25] is quasi-nonexpansive or nonexpansive.…”

“…The extragradient method was proposed by Korpelevich [10] and further investigated by Quoc et al [11] for solving equilibrium problems in Euclidean spaces. Recently, based on the ideas of [11], a large number of extragradient-type methods have been proposed for solving equilibrium problems in infinite-dimensional Hilbert spaces; see, e.g., [12,13,14,15] and the references therein. Note that the method introduced by Quoc et al [11] is a two-step iterative method and requires solving the strongly convex programming problem on the feasible set C twice in each iteration, which affects the computational efficiency of such algorithms when the structure of the feasible set C is complex.…”

“…Note that the method introduced by Quoc et al [11] is a two-step iterative method and requires solving the strongly convex programming problem on the feasible set C twice in each iteration, which affects the computational efficiency of such algorithms when the structure of the feasible set C is complex. To overcome this difficulty, many authors extended the subgradient extragradient method proposed by Censor, Gibali and Reich [16,17,18] to solve equilibrium problems in infinite-dimensional Hilbert spaces (see, e.g., [12,15,19]). These methods convert the optimization problem on the feasible set in the second step to an optimization problem on a special half-space in each iteration.…”

“…Recall that the inertial idea is based on a discrete version of the second-order dissipative dynamical system (see, e.g., [27,28] for more details). Recently, many researchers proposed a large number of iterative algorithms to solve equilibrium problems, variational inequalities, splitting problems, monotone inclusion problems, and fixed point problems; see, e.g., [13,15,29,30,31] and the references therein. A common characteristic of these inertial-type algorithms is that the next iteration depends on the combination of the previous two (or more) iterations.…”