Strong convergence of inertial algorithms for solving equilibrium problems
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Strong convergence of inertial algorithms for solving equilibrium problems Dang Van Hieu1 · Aviv Gibali2,3 Received: 21 August 2018 / Accepted: 10 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we introduce several inertial-like algorithms for solving equilibrium problems (EP) in real Hilbert spaces. The algorithms are constructed using the resolvent of the EP associated bifunction and combines the inertial and the Mann-type technique. Under mild and standard conditions imposed on the cost bifunction and control parameters strong convergence of the algorithms is established. We present several numerical examples to illustrate the behavior of our schemes and emphasize their convergence advantages compared with some related methods. Keywords Proximal point algorithm · Inertial-like algorithm · Monotone bifunction
1 Introduction Let H be a real Hilbert space and C be a nonempty, closed and convex subset of H . Given a bifunction f : C × C → , the equilibrium problem with respect to f and C, in the sense of Blum and Oettli [7,31], is formulated as follows: Find x ∗ ∈ C such that f (x ∗ , y) ≥ 0, ∀y ∈ C.
(EP)
The solution set of problem (EP) is denoted by E P( f , C). This problem is also known as the Ky Fan inequality [10] because of his early contributions to this field.
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Aviv Gibali [email protected] Dang Van Hieu [email protected]
1
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel
3
The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel
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D. Van Hieu, A. Gibali
Mathematically, problem (EP) unifies various fields of optimization problems, variational inequalities, fixed point problems, Nash equilibrium problem and many others, see e.g., [7,11,18,19,22]. Due to the applicability of EPs, during the last 20 years, problem (EP) has received a great interest by many authors who study the existence of solutions as well as developing iterative schemes for solving it, see for example [4,12,16,17,23,26,28,30,32,35,38]. One of the most popular method for solving problem (EP) is the proximal point algorithm (PPA). The algorithm was originally introduced by Martinet [25] for solving monotone variational inequalities, and later extended by Rockafellar [33] to monotone operators. Other researchers proposed generalizations and extensions to the PPA. For example, Moudafi [26] proposed a PPA-type method for solving EPs with monotone bifunctions and in [21], non-monotone bifunctions are considered. Let us recall the proximal point algorithm for solving problem (EP). Choose a starting point x0 ∈ H . Given the current iterate xn , find the next iterate xn+1 by solving the subproblem 1 xn+1 − xn , y − xn+1 ≥ 0, ∀y ∈ C. λ By introducing the resolvent of the bifunction f (see, [9,26]), defined by 1 f Tλ (x) = z ∈ C : f (z, y) + z − x, y − z ≥ 0, ∀y ∈ C λ f
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