A smoothing method for a class of generalized Nash equilibrium problems

Jun-Feng Lai,Zong-Chuan Wen,Jian Hou

doi:10.1186/s13660-015-0607-6

Abstract

The generalized Nash equilibrium problem is an extension of the standard Nash equilibrium problem where both the utility function and the strategy space of each player depend on the strategies chosen by all other players. Recently, the generalized Nash equilibrium problem has emerged as an effective and powerful tool for modeling a wide class of problems arising in many fields and yet solution algorithms are extremely scarce. In this paper, using a regularized Nikaido-Isoda function, we reformulate the generalized Nash equilibrium problem as a mathematical program with complementarity constraints (MPCC). We then propose a suitable method for this MPCC and under some conditions, we establish the convergence of the proposed method by showing that any accumulation point of the generated sequence is a M-stationary point of the MPCC. Numerical results on some generalized Nash equilibrium problems are reported to illustrate the behavior of our approach.

Highlights

This paper considers the generalized Nash equilibrium problem with jointly convex constraints (GNEP)
The idea of using an exact penalty approach to the GNEP was proposed by Facchinei and Pang [ ] and Facchinei and Kanzow [ ], but the disadvantage of this method is that a nondifferentiable NEP has to be solved to obtain a generalized Nash equilibrium
Another approach for solving the GNEP is based on the Nikaido-Isoda function

Summary

Introduction

This paper considers the generalized Nash equilibrium problem with jointly convex constraints (GNEP). Be the Cartesian product of the strategy sets of all players, a vector x∗ ∈ X is called a Nash equilibrium, or a solution of the NEP, if each block component x∗,ν is a solution of. The idea of using an exact penalty approach to the GNEP was proposed by Facchinei and Pang [ ] and Facchinei and Kanzow [ ], but the disadvantage of this method is that a nondifferentiable NEP has to be solved to obtain a generalized Nash equilibrium Another approach for solving the GNEP is based on the Nikaido-Isoda function. A regularized version of the Nikaido-Isoda function was first introduced in [ ] for NEPs further investigated by Heusinger and Kanzow [ ], they reformulated the GNEP as a constrained optimization problem with continuously differentiable objective function. The regularized NI function is given by α(x, y) :=

We now define the corresponding value function by

Let rik

Gl zk dk

Nash equilibrium at x

We set

The constraints are