An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems
- PDF / 1,367,815 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 69 Downloads / 223 Views
An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems Yekini Shehu1,2 · Olaniyi S. Iyiola3 · Duong Viet Thong4 · Nguyen Thi Cam Van5 Received: 27 January 2020 / Revised: 17 September 2020 / Accepted: 15 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The paper introduces an inertial extragradient subgradient method with self-adaptive step sizes for solving equilibrium problems in real Hilbert spaces. Weak convergence of the proposed method is obtained under the condition that the bifunction is pseudomonotone and Lipchitz continuous. Linear convergence is also given when the bifunction is strongly pseudomonotone and Lipchitz continuous. Numerical implementations and comparisons with other related inertial methods are given using test problems including a real-world application to Nash–Cournot oligopolistic electricity market equilibrium model. Keywords Equilibrium problem · Variational inequality problem · Extragradient method Mathematics Subject Classification 90C33 · 47J20
B
Duong Viet Thong [email protected] Yekini Shehu [email protected] Olaniyi S. Iyiola [email protected]; [email protected] Nguyen Thi Cam Van [email protected]
1
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China
2
Institute of Science and Technology (IST), Am Campus 1, 3400 Klosterneuburg, Vienna, Austria
3
Department of Mathematics and Physical Sciences, California University of Pennsylvania, California, PA, USA
4
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
5
Faculty of Economics Mathematics, National Economics University, Hanoi City, Vietnam
123
Y. Shehu et al.
1 Introduction Let C be a nonempty closed convex subset of a real Hilbert space H . Let f : H × H → R be a bifunction with f (x, x) = 0 for all x ∈ C. The equilibrium problem (EP) for the bifunction f on C is stated as follows: find x ∗ ∈ C such that f (x ∗ , y) ≥ 0, ∀y ∈ C.
(1)
Let us denote E P( f , C) the solution set of EP (1). Mathematically, EP (1) is a generalization of many mathematical models including variational inequality problems, optimization problems and fixed point problems, see Blum and Oettli (1994), Konnov (2000), Konnov (2007) and Muu and Oettli (1992). EPs have been considered by many authors in recent years, see, e.g., Combettes and Hirstoaga (2005), Hieu et al. (2018), Lyashko et al. (2011), Moudafi (2003), Vinh and Muu (2019), Iusem et al. (2009), Tran et al. (2008), Nguyen et al. (2014), Vuong et al. (2015) and the references therein. Some notable methods for EPs have been proposed such as: proximal point methods (PPM) (Moudafi 1999), auxiliary problem principle methods (Mastroeni 2000) and gap function methods (Mastroeni 2003). The PPM is often used for monotone EPs and this method is based on a regularized equilibrium problem which is strongly monotone and therefore the solution is unique and can be found more easily than solutions of the original problem. Soluti
Data Loading...
