Solving Bilevel Multiobjective Programming Problem by Elite Quantum Behaved Particle Swarm Optimization

and Applied Analysis 3 min y f ( x, y ) s.t. g ( x, y ) ≥ 0, 2.1 where F x, y and f x, y are the upper level and the lower level objective functions, respectively. G x, y and g x, y denote the upper level and the lower level constraints, respectively. Let S { x, y | G x, y ≥ 0, g x, y ≥ 0}, X {x | ∃y, G x, y ≥ 0, g x, y ≥ 0}, S x {y | g x, y ≥ 0}, and for the fixed x ∈ X, let S X denote the weak efficiency set of solutions to the lower level problem, the feasible solution set of problem 2.1 is denoted as: IR { x, y | x, y ∈ S, y ∈ S X }. Definition 2.1. For a fixed x ∈ X, if y is a Pareto optimal solution to the lower level problem, then x, y is a feasible solution to the problem 2.1 . Definition 2.2. If x∗, y∗ is a feasible solution to the problem 2.1 , and there are no x, y ∈ IR, such that F x, y ≺ F x∗, y∗ , then x∗, y∗ is a Pareto optimal solution to the problem 2.1 , where “≺” denotes Pareto preference. For problem 2.1 , it is noted that a solution x∗, y∗ is feasible for the upper level problem if and only if y∗ is an optimal solution for the lower level problem with x x∗. In practice, we often make the approximate Pareto optimal solutions of the lower level problem as the optimal response feed back to the upper level problem, and this point of view is accepted usually. Based on this fact, the EQPSO algorithm may have a great potential for solving BLMPP. On the other hand, unlike the traditional point-by-point approachmentioned in Section 1, the EQPSO algorithm uses a group of points in its operation, thus the EQPSO can be developed as a new way for solving BLMPP. We next present the algorithm based on the EQPSO is presented for 2.1 . 3. The Algorithm 3.1. The EQPSO The quantum behaved particle swarm optimization QPSO is the integration of PSO and quantum computing theory developed by 35–38 . Compared with PSO, it needs no velocity vectors for particles and has fewer parameters to adjust. Moreover, its global convergence can be guaranteed 39 . Due to its global convergence and relative simplicity, it has been found to be quite successful in a wide variety of optimization tasks. For example, a wide range of continuous optimization problems 40–45 are solved by QPSO and the experiment results show that the QPSO works better than standard PSO. Some improved QPSO algorithms can refer to 46–48 . In this paper, the EQPSO algorithm is proposed, in which an elite strategy is exerted for global best particle to prevent premature convergence of the swarm, and it makes the proposed algorithm has good performance for solving the high dimension BLMPPS. The EQPSO has the same design principle with the QPSO except for the global optimal particle selection criterion, so the global convergence proof of the 4 Abstract and Applied Analysis EQPSO can refer to 39 . In the EQPSO, the particles move according to the following iterative equation: z 1 p − αtmBestt − zt ∗ ln ( 1 u ) if k ≥ 0.5, z 1 p α ( mBest − zt ∗ ln ( 1 u ) if k < 0.5, 3.1