An elite quantum behaved particle swarm optimization (EQPSO) algorithm is proposed, in which an elite strategy is exerted for the global best particle to prevent premature convergence of the swarm. The EQPSO algorithm is employed for solving bilevel multiobjective programming problem (BLMPP) in this study, which has never been reported in other literatures. Finally, we use eight different test problems to measure and evaluate the proposed algorithm, including low dimension and high dimension BLMPPs, as well as attempt to solve the BLMPPs whose theoretical Pareto optimal front is not known. The experimental results show that the proposed algorithm is a feasible and efficient method for solving BLMPPs.
Bilevel programming problem (BLPP) arises in a wide variety of scientific and engineering applications including optimal control, process optimization, gameplaying strategy development, transportation problem, and so on. Thus the BLPP has been developed and researched by many scholars. The reviews, monographs, and surveys on the BLPP can refer to [
For the multiobjective characteristics widely existing in the BLPP, the bilevel multiobjective programming problem (BLMPP) has attracted many researchers to study it. For example, Shi and Xia [
Particle swarm optimization (PSO) is a relatively novel heuristic algorithm inspired by the choreography of a bird flock, which has been found to be quite successful in a wide variety of optimization tasks [
In this paper, a global convergence guaranteed method called as EQPSO is proposed, in which an elite strategy is exerted for global best particle to prevent premature convergence of the swarm. The EQPSO is employed for solving the BLMPP in this study, which has not been reported in other literatures. For such problems, the proposed algorithm directly simulates the decision process of the bilevel programming, which is different from most traditional algorithms designed for specific versions or based on specific assumptions. The BLMPP is transformed to solve multiobjective optimization problems in the upper level and the lower level interactively by the EQPSO. And a set of approximate Pareto optimal solutions for BLMPP are obtained using the elite strategy. This interactive procedure is repeated until the accurate Pareto optimal solutions of the original problem are found. The rest of the paper is organized as follows. In Section
Let
Let
For a fixed
If
For problem (
The quantum behaved particle swarm optimization (QPSO) is the integration of PSO and quantum computing theory developed by [
The process of the proposed algorithm for solving BLMPP is an interactive coevolutionary process. We first initialize population and then solve multiobjective optimization problems in the upper level and the lower level interactively using the EQPSO. For one time of iteration, a set of approximate Pareto optimal solutions for problem 1 is obtained by the elite strategy which was adopted in Deb et al. [
In Steps 4 and 8, the global best position is chosen at random from the elite set
The notations of the algorithm.

The 

The 

The 

The population size of the upper level problem. 

The subswarm size of the lower level problem. 

Current iteration number for the overall problem. 

The predefined max iteration number for 

Current iteration number for the upper level problem. 

Current iteration number for the lower level problem. 

The predefined max iteration number for 

The predefined max iteration number for 

Nondomination sorting rank of the upper level problem. 

Crowding distance value of the upper level problem. 

Nondomination sorting rank of the lower level problem. 

Crowding distance value of the lower level problem. 

The 

The offspring of 

Intermediate population. 
In this section, three examples will be considered to illustrate the feasibility of the proposed algorithm for problem (
(a) Generational Distance (
(b) Spacing (
All results presented in this paper have been obtained on a personal computer (CPU: AMD Phenon II X6 1055T 2.80 GHz; RAM: 3.25 GB) using a c# implementation of the proposed algorithm.
Example
Figure
Results of the Generation Distance (GD) and Spacing (SP) metrics for the above six examples.
Example  GD  SP 

Example 
0.00024  0.00442 
Example 
0.00003  0.00169 
Example 
0.00027  0.00127 
Example 
0.00036  0.00235 
Example 
0.00058  0.00364 
Example 
0.00039  0.00168 
The obtained Pareto optimal front of Example
The obtained solutions of Example
Example
Figure
The obtained Pareto optimal front of Example
The obtained solutions Example
Example
Example
This problem is more difficult compared to the previous problems (Examples
The obtained Pareto front of Example
Figure
The obtained Pareto front of Example
Example
Example
Figure
Figure
The obtained Pareto front of Example
The obtained Pareto front of Example
Example
Example
Figure
The obtained front of Example
The obtained front of Example
In this paper, an EQPSO is presented, in which an elite strategy is exerted for global best particle to prevent the swarm from clustering, enabling the particle to escape the local optima. The EQPSO algorithm is employed for solving bilevel multiobjective programming problem (BLMPP) for the first time. In this study, some numerical examples are used to explore the feasibility and efficiency of the proposed algorithm. The experimental results indicate that the obtained Pareto front by the proposed algorithm is very close to the theoretical Pareto optimal front, and the solutions are also distributed uniformly on entire range of the theoretical Pareto optimal front. The proposed algorithm is simple and easy to implement, which provides another appealing method for further study on BLMPP.
The authors are indebted to the referees and the associate editor for their insightful and pertinent comments. This work is supported by the National Science Foundation of China (71171150, 71171151, 50979073, 61273179, 11201039, 20101304), the Academic Award for Excellent Ph.D. Candidates Funded by Wuhan University and the Fundamental Research Fund for the Central Universities (no. 201120102020004), and the Ph.D. shorttime mobility program by Wuhan University.