Abstract
In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others.
Highlights
An equilibrium problem (EP) consider various mathematical problems as a special case i.e., the variational inequality problems (VIP), the optimization problems, the fixed point problems, the complementarity problems, Nash equilibrium of non-cooperative games, the saddle point and the vector minimization problem
Konnov [31] proposed another modification of the proximal point method, with weaker assumptions to deal with equilibrium problems involving monotone bifunction
It is noted that Tn represent a half-space and C ⊂ Tn
Summary
An equilibrium problem (EP) consider various mathematical problems as a special case i.e., the variational inequality problems (VIP), the optimization problems, the fixed point problems, the complementarity problems, Nash equilibrium of non-cooperative games, the saddle point and the vector minimization problem (see [1,2,3] for more details). Konnov [31] proposed another modification of the proximal point method, with weaker assumptions to deal with equilibrium problems involving monotone bifunction. Each regularized sub-problem converts into strongly monotone and its unique solution exists Another well-known technique is the auxiliary problem principle, i.e., formulate a new equivalent problem that is usually simpler and easier to figure out compared to the initial problem. We are introducing a new algorithm to solve the pseudomonotone equilibrium problem (1) incorporating the Lipschitz-type condition (2) on a bifunction in a real Hilbert space.
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