Abstract
In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.
Highlights
Equilibrium problems involve many mathematical problems as a particular instance, such as minimization problems, complementarity problems, problems of fixed point, Non-cooperative games of Nash equilibrium problem, problems of saddle point and problem of vector minimization and the variational inequality problems (VIP)
We are presenting our first main algorithm and prove a weak convergence theorem to find a solution to the equilibrium problems (EP) involving pseudo-montone bifunction
We have provided an extra-gradient-like method to resolve pseudo-monotone equilibrium problems in real Hilbert space
Summary
Equilibrium problems involve many mathematical problems as a particular instance, such as minimization problems, complementarity problems, problems of fixed point, Non-cooperative games of Nash equilibrium problem, problems of saddle point and problem of vector minimization and the variational inequality problems (VIP) (for more details follow e.g., [1,2,3,4]). In the case that the bifunction is a more general particular pseudo-monotone, we are not in a position to solve the equilibrium problem Another important concept is the auxiliary problem principle, that is established on the understanding of forming a new problem that is analogous and generally simpler to carry out with respect to our initial problem. In the case of equilibrium problems, Moudafi initiated and proposed an inertial-type approach, the second-order differential proximal method [36]. Such inertial methods are basically used to accelerate the iterative process to the desired solution.
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