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.

In 1952, Debreu [

In recent years, swarm intelligence algorithm has shown the certain potential possibility for solving NP-hard optimization problems [

In the section, we first introduce the model and some assumptions of the GNEP.

Let

If

From now on, we assume that the following conditions are satisfied.

(smoothness assumption and convexity assumption)

The

For each player

First the GNEP is transformed to nonlinear complementarity problem by using the Karush–Kuhn–Tucker (KKT) condition. Therefore, if

Under the above assumptions, system (

For

System (

The key point in this paper is to transform system (

Then, system (

Accordingly, system (

In the CIQPSO algorithm, the fitness function is designed as follows:

Obviously, we can get

Kennedy and Eberhart [

The particle’s trajectory analyses indicated that each particle

In the quantum time-space model, the quantum state of a particle is represented by a wave function. From the view of classical dynamics, when particles move in the attractive potential field whose center is point

The CIQPSO algorithm is proposed by combining the QPSO algorithm and the immune PSO (IPSO) algorithm. The CIQPSO algorithm is to introduce immune memory function and the antibody density inhibition mechanism into the QPSO algorithm. The IPSO has the properties of antigen recognition, immune memory function, and the antibody density inhibition mechanism. Immune memory cells are obtained by preserving

In [

Therefore, we can obtain the probability density selection function from

The design of the CIQPSO algorithm implementation steps is as follows:

Step 1: parameter initialization.

Step 2: compute the fitness value of each particle, if

Step 3: the average best position

Step 4: use formula (

Step 5: compute the fitness value of the updated particle, if

Step 6: randomly generate particles

Step 7: we choose particles

Step 8: we select particles from the memory library to replace particles of poor fitness in the population and the immune system generates a new generation

Step 9: use formula (

Step 10: stopping condition. Whether the maximum iteration number or precision meets termination condition? If yes, stop evolution; otherwise return to Step 2.

The CIQPSO algorithm implementation process is as shown in Figure

The flow diagram of the CIQPSO algorithm.

The CIQPSO algorithm is a new type of bioevolving swarm intelligence algorithm, which is relevant to the evolution rules of genetic algorithm (GA), for instance, natural selection and survival of the fittest. Therefore, we use the quantitative analysis method [

Off-line performance

In

Necessity: since

When

When

We take the limit on both sides of the above equation altogether:

We obtain

Sufficiency: let

If

For both sides of the above equation, take the limit simultaneously:

We obtain

From Theorem

For the sake of testing the performance of the CIQPSO algorithm, we set the parameters as follows: population size is

Considering the GNEP from [

The off-line performance and numerical results are shown in Table

We can derive the number of average iterations of five calculations is 94 from Table

The calculation results of Example

Times | Iteration numbers | Generalized Nash equilibrium solution | Fitness function value |
---|---|---|---|

1 | 122 | 5.6 | |

2 | 110 | 4.2 | |

3 | 43 | 1.2 | |

4 | 88 | 2.9 | |

5 | 108 | 6.8 |

Off-line performance of the CIQPSO algorithm.

Considering the GNEP from [

The off-line performance and numerical results are shown in Table

We can derive the number of average iterations of five calculations is 35 from Table

The calculation results of Example

Times | Iteration numbers | Generalized Nash equilibrium solution | Fitness function value |
---|---|---|---|

1 | 16 | 7.5324 | |

2 | 30 | 3.1302 | |

3 | 65 | 1.7588 | |

4 | 42 | 2.7906 | |

5 | 22 | 1.0214 |

Off-line performance of the CIQPSO algorithm.

Considering the GNEP from [

We use the CIQPSO algorithm to solve this problem, and the numerical results are shown in Table

We can know that the number of average iterations is 2 from Table

The calculation results of Example

Times | Iteration numbers | Generalized Nash equilibrium solution | Fitness function value |
---|---|---|---|

1 | 1 | 0.1456 | |

2 | 2 | 1.1103 | |

3 | 2 | 0.0429 | |

4 | 2 | 1.7806 | |

5 | 3 | 1.0214 |

By some transformations of GNEP and the construction of appropriate fitness functions, we try to solve approximation solutions of the GNEP by using the CIQPSO algorithm. Some numerical examples illustrate that the algorithm is effective. The GNEP is transformed into the nonlinear equation problem in Section

All the data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant no. 11561013) and Guizhou Science Foundation (Grant no. [2020]1Y284).