On ergodic algorithms for equilibrium problems

Abstract

In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159-164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371-386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.

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

“…First, it is to illustrate our motivation of splitting the bifunction involved in (EP ). Second, it is to compare the performance Algorithm 1 to the method proposed by Santos and Scheimberg in [27] without using the ergodic strategy, and the algorithm by Anh et al in [2] for the ergodic sequence. For that we consider the following differentiated Cournot-Nash model.…”

“…In this subsection we compare the performance of ergodic version of Algorithm 1 to the algorithm proposed in [2]. For that we consider the following instance of equilibrium problem (EP ).…”

“…This section aims to evaluate the performance of Algorithm 1 on some numerical examples. We also present some experiments on comparing the performance of our proposed algorithm to the exact version of inexact projected subgradient method proposed in [27] and the ergodic algorithm in [2]. We used MATLAB R2016a for implementing the algorithms, and conducted all experiments on a computer with a Core i5 processor, 16 GB of RAM, and Windows 10.…”

“…In order to reduce the computational cost, several convergent algorithms that require solving only one strongly convex program or computing only one projection at each iteration have been proposed for some classes of bifunctions such as strongly pseudomonotone and paramonotone with or without using an ergodic sequence (see e.g. [2,7,27]). In another direction, also for the sake of reducing computational cost, some splitting algorithms have been developed (see e.g.…”

“…We apply the two versions of the algorithm (with and without ergodic sequence) for solving some versions of a differentiated Cournot-Nash model. Some preliminary computational results for comparing the proposed algorithms with the ones in [2] and [27] are reported.…”