Int. J. Electron. Commun. (AEÜ) 66 (2012) 107–114 Contents lists available at ScienceDirect International Journal of Electronics and Communications (AEÜ) jou rn al h omepage: www.elsevier.de/aeue Particle Swarm Optimization with Parameter Dependency Walls and its sample application to the microstrip-like interconnect line design O. Tolga Altinoz a , Asim Egemen Yilmaz b,∗ a Electrical and Electronics Engineering, Hacettepe University Bala Vocational School of Industrial Electronics, Bala, Ankara, Turkey b Ankara University, Department of Electronics Engineering, 06100 Tandogan, Ankara, Turkey a r t i c l e i n f o Article history: Received 5 August 2010 Accepted 21 May 2011 Keywords: Particle Swarm Optimization Multidimensional optimization Parameter dependency Reflecting boundary conditions Microstrip-like interconnect line a b s t r a c t In this paper, we first propose and formulate a novel approach in Particle Swarm Optimization (which we call “Particle Swarm Optimization with Parameter Dependency Walls”) for handling the dependencies between the design parameters. After revisiting the definition of Particle Swarm Optimization; we try to visualize the physical meaning of the concepts lying beneath our approach, and demonstrate the existence of analytical solution needed throughout the implementation. In order to illustrate a practical application of our approach, we first revisit the empirical closed-form characteristic impedance expression for the microstrip-like interconnect line with a ground plane aperture. Then, we apply our approach in order to calculate the optimized parameters of microstrip-like interconnect lines in the synthesis problem. The proposed procedure can be useful for the rapid solution of the optimization problems in which there exist dependencies between/among the input variables. © 2011 Elsevier GmbH. All rights reserved. 1. Introduction Daily life frequently forces us to try to find solutions for inverse problems, for which straightforward solution approaches are usu- ally inapplicable. Once the problem is expressed as an optimization problem, namely a minimization/maximization problem repre- sented via a cost (or penalty)/fitness function together with its constraints, the next step would be selection and application of an appropriate method for the solution. Most of the conventional opti- mization methods make several assumptions about the cost/fitness function, such as continuity or differentiability. Some methods also might require a priori information about the behavior of the function, such as the tendency of its gradient. In practice, such information quite often does not exist; moreover, in case such information exists, unfortunately it is already known that the cost/fitness function has discontinuities and/or non-differentiable points. The class of the optimization methods, which is called as “Meta- heuristics”, is advantageous for such cases. Being nothing but systematical trial-and-error procedures, these algorithms neither require a priori information about the cost/fitness functions, nor make assumptions on them. Being able to compute the value of the cost/fitness function is sufficient for these algorithms. Moreover, it ∗ Corresponding author. Tel.: +90 312 203 35 00; fax: +90 312 212 54 80. E-mail addresses: taltinoz@hacettepe.edu.tr (O.T. Altinoz), aeyilmaz@eng.ankara.edu.tr, asimegemenyilmaz@yahoo.com (A.E. Yilmaz). is possible and in most cases very easy to handle the constraints of the problem by means of these algorithms. There are a considerable number of algorithms of this sort (such as Simulated Annealing, Tabu Search, Genetic Algorithm, Particle Swarm Optimization, Ant Colony Optimization, Differential Evolution, etc.) with numerous variants handling multi-dimensional, single- or multi-objective, continuous or combinatorial optimization problems. Due to their nature, most of the algorithms in this class allow parallel imple- mentations, which make the solution of very large scale problems possible. Certainly, due to the random essence and flavor in their very def- inition, Metaheuristics always carry the risk of getting stuck at local optima, or not being able to converge to a reasonable solution. On the other hand, prevention of such situations has sufficiently been studied for all these algorithms. In other words, when carefully implemented (i.e. by surveying the literature and considering the recommendations in the relevant publications), it is quite possible to achieve almost-excellent results for these algorithms. Among the Metaheuristics, our particular interest is on Particle Swarm Optimization in this study. Being a simple but power- ful method, the algorithm has so far been applied in numerous problems in various disciplines. Regardless of its general advan- tages, as will be seen in Section 2, there is another point, which makes the Particle Swarm Optimization method quite attractive for us: handling the parameter dependencies in a quite sim- ple but effective manner. We try to get benefit of this feature and come up with a more concrete formulation throughout this study. 1434-8411/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2011.05.009