Implementation of Augmented Lagrangian Methods for Equilibrium Problems

Actual implementation of augmented Lagrangian algorithms requires a solution of the subproblem generated at each iterate, which is the most challenging task. In this paper, we propose two approaches to make the augmented Lagrangian algorithms, introduced in Iusem and Nasri (RAIRO Oper Res 44:5-26, 2010) for equilibrium problems, computer amenable. The first algorithm that we suggest here incorporates the Newton method and the other one benefits from the Shor subgradient method to solve the subproblems that are produced when the augmented Lagrangian algorithms are applied to the equilibrium problem. We also illustrate our findings by numerical results which are obtained when our algorithms are implemented to solve quadratic equilibrium problems and certain generalized Nash equilibrium problem, including the river basin pollution problem, a particular case of the equilibrium problem. Moreover, we compare our numerical results with those presented in Matioli et al. (Comput Optim Appl 52:281-292, 2012) and Tran et al. (Optimization 57:749-776, 2008) for the same test problems.