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This paper proposes an estimation of particle swarm distribution algorithm (EPSDA) to solve the nonlinear bilevel programming problem (NBLP) by embedding the estimation distribution algorithm (EDA) into the particle swarm optimization (PSO). One Gaussian function is selected to construct the probability distribution for the superior candidate from the present population before executing the velocity and position update rule at each iteration in PSO. Thus, some new individuals viewed as an offspring population are generated from the probability distribution to replace some inferior particles in the current population for making up a new population. Therefore, this proposed algorithm combines PSO (the local search method) and EDA (the global search method) through updating the population to enhance the efficiency of solving NBLP. In experiments, we select four representative examples for linear, quadratic, nonlinear, high-dimensional nonlinear cases to carry out sensitivity analysis on parameters of the proposed algorithm. The results reveal that linearly decreasing inertia weight and adaptive acceleration coefficients are better than constant parameters. Furthermore, the multivariate Gaussian distribution can achieve better performance compared with the normal distribution. These four examples are also used to compare the performance of EPSDA with ones of separate PSO and EDA by setting constant parameters. The experiments show that EPSDA is better than separate PSO and EDA from both the quality of solution and the computational efficiency. The results on four high-dimensional non-convex nonlinear examples demonstrate feasibility and efficiency of EPSDA to solve the high-dimensional nonlinear bilevel programming from the iteration number and computational time.

The Estimation of Particle Swarm Distribution Algorithm With Sensitivity Analysis for Solving Nonlinear Bilevel Programming Problems