Theoretical analysis of the convergence property of a basic pigeon-inspired optimizer in a continuous search space

Abstract

The pigeon-inspired optimization (PIO) algorithm is a newly presented swarm intelligence optimization algorithm inspired by the homing behavior of pigeons. AlthoughPIO has demonstrated effectiveness and superiority in numerous fields, particularly in practicalengineering optimization, there have been few results concerning the theoretical foundations of PIO. This paper conductsconvergence analysis of basic PIO in a continuous search space in two aspects. First, we analyze the convergence of each pigeonsexpected position using a difference equation and prove that the average position of each pigeon inthe swarm will converge to the same value. To further study the stochastic globalconvergence property of the pigeon swarm, we apply the martingale theory to investigatethe basic PIO swarm sequence, and achieve a sufficient condition to guarantee global convergenceof 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.