A New Algorithm to Solve the Generalized Nash Equilibrium Problem

Luping Liu,Akemi Gálvez,Wensheng Jia

doi:10.1155/2020/1073412

Abstract

We try a new algorithm to solve the generalized Nash equilibrium problem (GNEP) in the paper. First, the GNEP is turned into the nonlinear complementarity problem by using the Karush–Kuhn–Tucker (KKT) condition. Then, the nonlinear complementarity problem is converted into the nonlinear equation problem by using the complementarity function method. For the nonlinear equation equilibrium problem, we design a coevolutionary immune quantum particle swarm optimization algorithm (CIQPSO) by involving the immune memory function and the antibody density inhibition mechanism into the quantum particle swarm optimization algorithm. Therefore, this algorithm has not only the properties of the immune particle swarm optimization algorithm, but also improves the abilities of iterative optimization and convergence speed. With the probability density selection and quantum uncertainty principle, the convergence of the CIQPSO algorithm is analyzed. Finally, some numerical experiment results indicate that the CIQPSO algorithm is superior to the immune particle swarm algorithm, the Newton method for normalized equilibrium, or the quasivariational inequalities penalty method. Furthermore, this algorithm also has faster convergence and better off-line performance.

Highlights

In 1952, Debreu [1] came up with the generalized Nash equilibrium problem (GNEP), in which each player’s feasible strategy set depends on the strategies of other players
Some numerical examples illustrate that the algorithm is effective. e GNEP is transformed into the nonlinear equation problem in Section 3. e coevolutionary immune quantum particle swarm optimization algorithm (CIQPSO) algorithm is designed and the algorithm is applied in solving the GNEP
Some numerical examples are given in this paper, Examples 1 and 2 prove that the CIQPSO algorithm is better than the immune PSO (IPSO) algorithm, and its convergence speed is faster and not relies on the selection of the initial point

Summary

Introduction

In 1952, Debreu [1] came up with the generalized Nash equilibrium problem (GNEP), in which each player’s feasible strategy set depends on the strategies of other players. Using the regularized indicator Nikaido–Isoda-type function, Lalitha and Dhingra [9] presented two constraint optimization formulas for the GNEP; Dreves [10] considered a linear GNEP and a best-response approach for equilibrium selection in two-player GNEP; Izmailov and Solodov [11] analyzed error bounds and Newton-type methods for the GNEP. We use the CIQPSO algorithm to solve the GNEP by constructing an appropriate fitness function and analyze and compare some numerical examples to show that the algorithm is effective. Inspired by research works mentioned above, our paper mainly studies swarm intelligence algorithms to solve the GNEP.

Model and Assumptions

The Design of the CIQPSO Algorithm

Numerical Experiments

Conclusions