Weak convergence of explicit extragradient algorithms for solving equilibirum problems

Habib ur Rehman,Pasakorn Yordsorn,Poom Kumam,Yeol Je Cho

doi:10.1186/s13660-019-2233-1

Abstract

This paper aims to propose two new algorithms that are developed by implementing inertial and subgradient techniques to solve the problem of pseudomonotone equilibrium problems. The weak convergence of these algorithms is well established based on standard assumptions of a cost bi-function. The advantage of these algorithms was that they did not need a line search procedure or any information on Lipschitz-type bifunction constants for step-size evaluation. A practical explanation for this is that they use a sequence of step-sizes that are updated at each iteration based on some previous iterations. For numerical examples, we discuss two well-known equilibrium models that assist our well-established convergence results, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.

Highlights

Equilibrium problem can be considered as a general problem in the sense that it comprises many mathematical models such as variational inequality problems, optimization problems, fixed point problems, complementarity problems, Nash equilibrium of noncooperative games, saddle point, vector minimization problem and the Kirszbraun problem
This paper proposes two modifications of Algorithm 1 for a class of pseudomonotone equilibrium problems motivated from some recent results
3 Subgradient explicit iterative algorithm for a class of pseudomonotone equilibrium problem (EP) we suggest our first algorithm for finding a solution to a pseudomonotone problem (EP)

Summary

Introduction

Equilibrium problem (shortly, EP) can be considered as a general problem in the sense that it comprises many mathematical models such as variational inequality problems (shortly, VIP), optimization problems, fixed point problems, complementarity problems, Nash equilibrium of noncooperative games, saddle point, vector minimization problem and the Kirszbraun problem (see e.g., [1,2,3,4]). This paper proposes two modifications of Algorithm 1 (see [47]) for a class of pseudomonotone equilibrium problems motivated from some recent results (see [28, 48, 49]) These resulting algorithms combine the explicit iterative extragradient method with the subgradient method and the inertial term that is used to speed-up the iterative sequence towrads the solution. Assumption 2.1 We have the following assumptions on the bifunction f : H × H → R which are useful to prove the weak convergence of the iterative sequence {xn} generated by our proposed algorithms. Combining (1) and (2) we obtain λnf (yn, y) – λnf (yn, xn+1) ≥ xn – xn+1, y – xn+1 , ∀y ∈ Hn. Lemma 3.2 Let {xn} and {yn} be generated from the Algorithm 1, the following relation holds: λn f (xn, xn+1) – f (xn, yn) ≥ xn – yn, xn+1 – yn. Remark 4.1 The knowledge of the Lipschitz-type constants is not mandatory to build up the sequence {xn} in Algorithm 2 and to get the convergence result in Theorem 4.1

Nash–Cournot oligopolistic equilibrium model

Conclusion