Convergence boundaries of complex-order particle swarm optimization algorithm with weak stagnation: dynamical analysis

This paper addresses the area of particle swarm optimization (PSO) algorithms and, in particular, investigates the dynamics of the complex-order PSO (COPSO). The core of the COPSO adopts the concepts of complex derivative and conjugate order differential in the position and velocity adaption mechanisms to improve the algorithmic performance. The work focuses on the analytical stability analysis of the COPSO in the case of weak stagnation. The COPSO is formulated in the form of a control structure, and the particle dynamics are represented as a nonlinear feedback element. In a first phase, a state-space representation of the different types of COPSO is constructed as a delayed discrete-time system for describing the historical memory of particles. In a second phase, the existence and the uniqueness of the equilibrium point of the COPSO variants are discussed and the stability analysis is derived analytically to determine the convergence boundaries of the COPSO dynamics with weak stagnation. Simulations illustrate the proposed ideas, such as the area of stability of the COPSO equilibrium point and the performance of the algorithms.