The pigeon-inspired optimization (PIO) algorithm is a newly presented swarm intelligence optimization algorithm inspired by the homing behavior of pigeons. Although PIO has demonstrated effectiveness and superiority in numerous fields, particularly in practical engineering optimization, there have been few results concerning the theoretical foundations of PIO. This paper conducts convergence analysis of basic PIO in a continuous search space in two aspects. First, we analyze the convergence of each pigeon’s expected position using a difference equation and prove that the average position of each pigeon in the swarm will converge to the same value. To further study the stochastic global convergence property of the pigeon swarm, we apply the martingale theory to investigate the basic PIO swarm sequence, and achieve a sufficient condition to guarantee global convergence of the basic PIO. Our theoretical analysis shows that this convergence depends upon the accumulation of the minimum probability with which the pigeon swarm jumps to the global-optimal region at each iteration. The mathematical methods proposed in this study, particularly the martingale technique, also provide a new effective approach for the theoretical analysis of bio-inspired algorithms in continuous optimization.
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This work was supported by Natural Science Foundation of China-Guangdong Joint Fund (Grant No. U1501254), National Natural Science Foundation of China (Grant Nos. 61772225, 61876207), Guangdong Natural Science Funds for Distinguished Young Scholar (Grant No. 2014A030306050), the Ministry of Education — China Mobile Research Funds (Grant No. MCM20160206), Guangdong High-level Personnel of Special Support Program (Grant No. 2014TQ01X664), International Cooperation Project of Guangzhou (Grant No. 201807010047), and Science and Technology Program of Guangzhou (Grant Nos. 201804010276, 201707010227, 201707010228).
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Zhang, Y., Huang, H., Wu, H. et al. Theoretical analysis of the convergence property of a basic pigeon-inspired optimizer in a continuous search space. Sci. China Inf. Sci. 62, 70207 (2019). https://doi.org/10.1007/s11432-018-9753-5
- pigeon-inspired optimization (PIO)
- convergence analysis
- continuous optimization
- swarm intelligence