Boundary Conditions in Particle Swarm Optimization Revisited

In order to enforce particles to search inside the solution space of interest during the optimization procedure, various boundary conditions are currently used in particle swarm optimization (PSO) algorithms. The performances, however, vary considerably with the dimensionality of the problem and the location of the global optimum in the solution space. In this paper, different boundary conditions are categorized into two groups, namely, restricted and unrestricted, according to whether the errant particles are relocated inside the allowable solution space or not. Moreover, efforts are made to explore different hybrid unrestricted boundary conditions by introducing the favorable characteristics of the reflecting and damping boundary conditions into the existing invisible boundary condition. The performances of the boundary conditions are tested based on both mathematical benchmark functions and a real-world electromagnetic problem: the optimization of a 2-D 16-element array antenna. Simulation results are examined from both the effectiveness and efficiency of the algorithm. Comparisons show that the unrestricted boundary conditions are more efficient when the global optimum is inside the boundary of the solution space, and the damping boundary condition is more robust and consistent when the global optimum is close to the boundary

I. Introduction

The particle swarm optimization (PSO) is a population-based stochastic evolutionary algorithm, noted for its capability of searching for the global optimum of complex problems. Since its inception in 1995 [1], PSO has attracted a lot of attention from a variety of engineering fields, and is recently introduced to the antenna community by Robinson and Rahmat-Samii [2]