The Iterative Methods for Solving Pseudomontone Equilibrium Problems
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The Iterative Methods for Solving Pseudomontone Equilibrium Problems Jun Yang1 Received: 3 October 2019 / Revised: 13 July 2020 / Accepted: 12 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article, we introduce two new algorithms for solving equilibrium problems involving pseudomontone and Lipschitz-type bifunctions in real Hilbert space. The algorithms use a nonmonotonic step size. We establish weak convergence theorems without the knowledge of the Lipschitz-type constants of bifunction. Keywords Equilibrium problems · Pseudomonotone bifunction · Subgradient extragradient method · Convex set Mathematics Subject Classification 65J15 · 47J20 · 90C25 · 90C52
1 Introduction In this article, we are interested in the equilibrium problems (E P) of finding x ∗ ∈ C such that f (x ∗ , y) ≥ 0, ∀ y ∈ C,
(1)
where C is a nonempty closed convex subset in a real Hilbert space H , f is a bifunction from H × H to R. The set of solutions of (1) is denoted by E P( f ). This problem is also called Ky Fan inequality because he established the existence of the solution to problem (1). This problem unifies many important mathematical problems, such as fixed point problems, variational inequality problems, complementary problems, saddle point (minimax) problems, convex differentiable optimization, Nash equilibrium problem [1–7]. In recent decades, many methods have been proposed and analyzed for approximating solution of equilibrium problems [8–16]. Among them, proximity algorithms [8,9] is one of the important methods. The main idea of this method is to displace the initial problem by a sequence of regularization equilibrium subproblems which can be solved more easily.
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Jun Yang [email protected] School of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, Shaanxi, China 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 84:50
Another fundamental method is proximal-like method [10] which is also called the extragradient method [11]. It is worth noting that the method [11] is a requirement to know Lipschitz-type constants of the bifunction. Unfortunately, Lipschitz-type constants are often unknown or difficult to approximate. To overcome this problem, some new algorithms [14– 16] for solving equilibrium problems have been proposed after the step size of the variational inequality [17]. Note that the methods in [15,16] need at each iteration to solve two optimization programs which are performed on C and the step sizes in [14–16] are non-increasing monotone. If f (x, y) = F(x), y − x for all x, y ∈ H , where F : H → H is a mapping. Then the equilibrium problem reduces to variational inequality. That is, find x ∗ ∈ C such that F(x ∗ ), y − x ∗ ≥ 0, ∀ y ∈ C.
(2)
Recently, Yang et al. [18] combined the Popov method [19], the subgradient extragradient methods [20,21] and the selfadaptive methods [17,22] to propose the an algorithm for solving variational inequalities (2). Inspired by the aforementioned resu
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