A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

Abstract

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

Modified Popov's Subgradient Extragradient Algorithm With Inertial Technique for Equilibrium Problems and Its Applications

“…One of the most interesting and effective areas of research in equilibrium problem theory is the development of new iterative methods, the improvement of existing methods, and the examination of their convergence analysis. Several methods have already been used in recent years to estimate the solution of the problem of equilibrium in both finite and infinite-dimensional spaces, i.e., the extragradient methods [6,7,8,9,9,10,11,12,13,14,15,16] and others in [17,18,19,20,21,22,23,24,25,26].…”

A Parallel Hybrid Bregman Subgradient Extragradient Method for a System of Pseudomonotone Equilibrium and Fixed Point Problems

“…Furthermore, related to our work are several methods such as [26], where the stepsize is the variable stepsize formula, that is the bifunction has a Lipschitz-like condition defined on it, and the algorithm also operates without the prior estimation of Lipschitz-type constants. The authors in [27,28] considered algorithms for solving the mixed equilibrium problem, split variational inclusion and fixed point theorems. Lastly, we also mention in passing that the authors in [29] considered a convex feasibility problem, that is finding a common element in the finite intersection of a finite family of convex sets C i ; whereas, in our work, we consider a finite family of pseudomonotone bifunctions, and our C is fixed.…”

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

“…Many authors established and generalized several results on the existence and nature of the solution of the equilibrium problems (see for more detail [1,4,5]). Due to the importance of this problem (EP) in both pure and applied sciences, many researchers studied it in recent years [6][7][8][9][10][11][12][13][14][15][16][17] and other in [18][19][20][21][22]. Tran et al in [23] introduced iterative sequence {u n } in the following way:…”