A splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions and application to cournot-nash model

¹ Ho Chi Minh University of Natural Resources and Environment, Ho Chi Minh City, Vietnam
² Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

^* Corresponding author: lxthanh@math.ac.vn

Received: 17 June 2019
Accepted: 18 March 2020

Abstract

In this paper we propose a splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each component bifunction. In contrast to the splitting algorithms previously proposed in Anh and Hai (Numer. Algorithms 76 (2017) 67–91) and Hai and Vinh (Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2017) 1051–1069), our algorithm is convergent for paramonotone and strongly pseudomonotone bifunctions without any Lipschitz type as well as Hölder continuity condition of the bifunctions involved. Furthermore, we show that the ergodic sequence defined by the algorithm iterates converges to a solution without paramonotonicity property. Some numerical experiments on differentiated Cournot-Nash models are presented to show the behavior of our proposed algorithm with and without ergodic.