IEEE ANTENNAS AND WIRELESSPROPAGATION LETTERS, VOL. 5, 2006 235 Some Insight Over New Variations of the Particle Swarm Optimization Method Stefano Selleri, Senior Member, IEEE, Marco Mussetta, Student Member, IEEE, Paola Pirinoli, Member, IEEE, Riccardo E. Zich, Member, IEEE, and Ladislau Matekovits, Member, IEEE Abstract—The Particle Swarm Optimization (PSO) method re- cently gained high popularity in electromagnetics. Here few vari- ations over the standard algorithm, referred to as Meta PSO, are proposed and the results of their application to the optimization of a microwave microstrip-line filter are presented. Index Terms—Microwave filters, optimization, particle swarm. I. INTRODUCTION E VOLUTIONARY algorithms and in particular the genetic algorithms (GA), as well as the simulated annealing (SA) have been widely used in electromagnetic applications in the last years (see, among many others, [1]–[6]). More recently, a new stochastic optimization technique has rapidly gained pop- ularity in the Electromagnetic Community: the Particle Swarm Optimization (PSO). The PSO has been introduced in the middle of 1990s [7]–[9] and it is based on a “social interaction” metaphor in which the parameter space is searched by controlling the trajectories of a set of particles according to a swarm- or flock-like set of rules. The position of each particle is used to compute the value of the function to be optimized. Individual particles are then attracted, with a stochastic-varying strength, by both the position of their best past performance and the position of the global best past performance of the whole swarm. PSO is akin to the other stochastic methods performing a global search in the parameter space without getting trapped in local minima. In the recent years the interest for its appli- cation to electromagnetic problems has been rapidly increasing [10]–[12], and several papers have been published comparing it with other optimization techniques, mainly with GA [13]–[15]. The use of such optimization techniques, requiring the eval- uation of the cost functions thousands of times, needs partic- ular care for electromagnetic problems, in which generally the cost function is computationally expensive. For this reason, the development of new versions of the PSO, GA, or SA algo- rithms with enhanced properties is a challenging issue. In this Manuscript received February 21, 2006; revised March 2, 2006. S. Selleri is with the Department of Electronics and Telecommunications, University of Florence, I-50134 Florence, Italy (e-mail: stefano.selleri@ unifi.it). M. Mussetta and R. E. Zich are with the Department of Electrical Engi- neering, Politecnico di Milano, I-20133 Milano, Italy (e-mail: marco.mussetta@ polimi.it, riccardo.zich@polimi.it). P. Pirinoli and L. Matekovits are with the Department of Electronics, Politecnico di Torino, I-10129 Torino, Italy (e-mail: paola.pirinoli@polito.it, ladislau.matekovits@polito.it). Digital Object Identifier 10.1109/LAWP.2006.874071 contribution, some new PSO-based techniques, aimed to im- prove the performances of the standard PSO and based on mul- tiple swarm interactions, are presented. The implementation is similar but simpler than those of Cooperative Particle Swarm (CPSO) methods [16]–[18]. These latter splits the domain in subspaces, each searched by a swarm, hence requiring addi- tional functions to reconstruct the point where the cost function is to be evaluated and a more complex way, with respect to con- ventional PSO, to handle and store personal and global bests. The most performant CPSO- (H stands for Hybrid, is the subspace dimension [16]) algorithms relies on the co-evolution of a CPSO and a PSO with exchange of information between the two, leading to an even more complex algorithm. On the other hand, the algorithms here presented exhibit just one or two terms to be summed to the velocity update function, and no other additional complexity. Despite of their simplicity, all the proposed techniques work better than the standard PSO, as shown in Section IV, in which the results of their application to the optimization of a test function and to the design of a mi- crostrip filter are shown. II. BASIC PSO ALGORITHM The standard PSO algorithm is an iterative procedure in which a set of particles, or agents, are charac- terized by their position and the velocity with which they move in the -dimensional space domain of a cost function . A full treatment of the method can be found in [9] but for sake of clarity and uniformity of notations it is briefly summarized in the following. At the beginning positions and velocities have completely random values and , then they are updated iteratively according to the rules (1) (2) being the best position ever attained by particle itself (per- sonal knowledge) and the best position ever attained by the particle swarm (social knowledge); is a friction factor slowing down particles, and are parameters tuning the pulls toward the personal and global best positions and is a random number of uniform distribution in the range. Please note that, if appears more that once in a given formula it is assumed to have different values each time. 1536-1225/$20.00 © 2006 IEEE