Abstract
In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.
Highlights
Let K to be a nonempty convex, closed subset of a Hilbert space E and f : E × E → R be a bifunction with f (u, u) = 0 for each u ∈ K
Iterative methods are efficient tools to evaluate an approximate solution to an equilibrium problem
Several methods are well known for solving the problem (EP), for instance the proximal point method [17,18], projection methods [19], extragradient methods [20,21,22], the subgradient method [23,24,25], methods using the Bregman distance [26], and others [27,28,29,30]
Summary
Korpelevich proved the weak convergence of the generated sequence under the hypotheses of Lipschitz continuity and pseudo-monotonicity on the operator It involves determining two projections onto a closed convex set in each iteration of the method. In 2016, Lyashko et al [34] proposed an extension of the method (2) motivated by the result in [35] This iterative sequence {un } is defined as follows: u0 , v0 ∈ C, un+1 = arg min{λ f (vn , y) + 1 kun − yk2 },. Polyak [36] began by taking inertial extrapolation as an acceleration method to solve smooth convex optimization problems These methods are two-step iterative schemes, while the iteration is evaluated by the use of the previous two iterations, and it could be viewed as an accelerating step of the iterative sequence.
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