1286 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 4, APRIL 2011 Self-Adaptive Differential Evolution Applied to Real-Valued Antenna and Microwave Design Problems Sotirios K. Goudos, Member, IEEE, Katherine Siakavara, Member, IEEE, Theodoros Samaras, Member, IEEE, Elias E. Vafiadis, Member, IEEE, and John N. Sahalos, Life Fellow, IEEE Abstract—Particle swarm optimization (PSO) is an evolutionary algorithm based on the bird fly. Differential evolution (DE) is a vector population based stochastic optimization method. The fact that both algorithms can handle efficiently arbitrary optimization problems has made them popular for solving problems in electro- magnetics. In this paper, we apply a design technique based on a self-adaptive DE (SADE) algorithm to real-valued antenna and microwave design problems. These include linear-array synthesis, patch-antenna design and microstrip filter design. The number of unknowns for the design problems varies from 6 to 60. We compare the self-adaptive DE strategy with popular PSO and DE variants. We evaluate the algorithms’ performance regarding statistical results and convergence speed. The results obtained for different problems show that the DE algorithms outperform the PSO variants in terms of finding best optima. Thus, our results show the advantages of the SADE strategy and the DE in general. However, these results are considered to be indicative and do not generally apply to all optimization problems in electromagnetics. Index Terms—Differential evolution (DE), evolutionary algo- rithms (EAs), linear array synthesis, microwave filter design, optimization methods, particle swarm optimization (PSO), patch antenna design. I. INTRODUCTION S EVERAL evolutionary algorithms (EAs) have emerged in the past decade that mimic biological entities behavior and evolution In this paper we consider particle swarm optimiza- tion (PSO) [1] and Differential evolution (DE) [2], [3]. PSO [1] is an evolutionary algorithm based on the bird fly. It is an easy-to-implement algorithm. PSO has been used successfully in constrained and unconstrained electromagnetic design prob- lems [4]–[24]. Differential evolution (DE) [2], [3] is a population-based sto- chastic global optimization algorithm. Several DE variants or strategies exist. An overview of both PSO and DE algorithms and the hybridizations of these algorithms with other soft com- puting tools can be found in [25]. The classical DE strategy has Manuscript received December 13, 2009; revised May 08, 2010; accepted August 28, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: sgoudo@physics.auth.gr; skv@auth.gr; theosama@auth.gr; vafiadis@auth.gr; sahalos@auth.gr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2109678 been applied to microwave structures [26]–[28], antenna design [29]–[35], signal optimization [36] and microwave imaging ap- plications [37]–[44]. DE produced better results than PSO on numerical bench- mark problems with low and medium dimensionality (30 and 100 dimensions) [45]. However, on noisy test problems, DE was outperformed by PSO. In [46] a comparative study between DE and PSO variants is presented for the design of radar absorbing materials (RAM). The number of problem dimensions was 10 and DE outperformed the PSO variants in terms of convergence speed and best values found. The shape reconstruction of a per- fectly conducting 2-D scatterer using DE and PSO is presented in [40], [44]. Also both algorithms have been applied to 1-D small-scale inverse scattering problems [43]. In these cases, DE outperformed PSO. In [47] a comparison between DE, PSO and Genetic algorithms (GAs) for circular array design is presented. DE and PSO showed similar performances and both of them had better performance compared to GAs. One of the DE advantages is that very few control param- eters have to be adjusted in each algorithm run. However, the control parameters involved in DE are highly dependent on the optimization problem. Therefore, it is not always an easy task to tune these parameters. Recently a novel DE strategy has been applied to numerical benchmark problems that self-adapts the control parameters (SADE) [48]. SADE has been applied suc- cessfully to a microwave absorber design problem [49]. In this paper, SADE is compared with other algorithms. The comparison is performed on common real-valued antenna and microwave design problems. These problems include linear-array synthesis with sidelobe level suppression and null control in specified directions. In order to evaluate the algorithms’ performance combined with a numerical solver, we apply the algorithms to the design of a dual-band E-shaped patch-antenna and of a microstrip bandpass filter. As numerical solver we employ FEKO [50], a commercially available EM solver. We compare the SADE with two PSO variants and the classical DE/rand/1/bin strategy. The numerical results show the advantages of the SADE approach and the DE in general. However, these results cannot lead to the general conclusion that DE outperforms PSO in all optimization problems in electromagnetics. This paper is organized as follows: In Section II we describe the PSO and DE algorithms. We present the numerical results in Section III. Finally, we give the conclusion in Section IV. 0018-926X/$26.00 © 2011 IEEE