5
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007 777 data, that means, the output data in this case is a vector consisting of three elements whose range is from 0 to 16. Other methods of choosing the output data that can identify the fault elements is also possible and is user dependent. B. Network Training In this work the MLP is trained in the backpropagation mode, be- cause of its suitability for forming mapping implementations. In this training algorithm the weight updation takes place according to (1) where, is the learning rate and is the mean square error at the th instant. The neural network learns from samples of input-output data i.e., and , where is the total number of samples for training. In the present problem radiation pattern was simulated for all 697 number of combination of faults, i.e., . Because the network size is large as compared to the number of available training patterns, so in order to avoid a possible overfitting and to improve net- work generalization, we have adopted the regularization method [9]. In this procedure, the performance function of the feedforward network is modified by adding a term with the mean square error that consists of the mean of the sum of the squares of the network weights and biases. The performance function now becomes (2) where, is the performance ratio, is the target output, is the net- work output, and is the total number of weights and biases. To implement this, the network inputs and outputs were preprocessed so that they fell approximately in the range [0, 1]. C. Training Parameters The efficiency of training depends on the training parameters. The values of the training parameters taken for the training of the present network are as mentioned in Table I. IV. RESULTS AND DISCUSSION The trained network was tested for the results. The network outputs are rounded off to get the fault element numbers and their location. For example, a raw output of 0.297 after rounding off, produces the value 0.3 representing the fault in the 3rd element. Results of some typical fault patterns are given in Fig. 2. In all the cases the network output matches with the simulation results. V. CONCLUSION A MLP neural network trained in the backpropagation mode is used to locate the fault elements in an antenna array from its distorted radia- tion pattern. The results are in excellent agreement with the simulation results. The developed network can be used at the base stations to find out the number and location of the fault elements in the array, from the distorted radiation pattern due to presence of nonradiating elements. Although the methodology was developed for a linear array, the same can be extended to any planar array structure. REFERENCES [1] R. J. Mailloux, Phased Array Antennas Handbook. Norwood, MA: Artech House, 1994. [2] B. K. Yeo and Y. Lu, “Array failure correction with a genetic algo- rithm,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 823–828, 1999. [3] R. J. Mailloux, “Array failure correction with a digitally beamformed array,” IEEE Trans. Antennas Propag., vol. 44, pp. 1542–1550, 1996. [4] H. Steyskal and R. J. Mailloux, “Generalization of an array failure cor- rection method,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 145, no. 4, pp. 332–336, Aug. 1998. [5] J. A. Rodriguez and F. Ares, “Finding defective elements in planar arrays using genetic algorithms,” Progress in Electromagn. Res., PIER, vol. 29, pp. 25–37, 2000. [6] S. Haykins, Neural Networks: A Comprehensive Foundation. New York: IEEE Press/IEEE Computer Society Press, 1994. [7] Incorporated, IE3D Zeland Software. [8] K. C. Lee and T. N. Lin, “Application of neural networks to analyses of nonlinearly loaded antenna arrays including mutual coupling effects,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1126–1132, 2005. [9] H. Demuth and M. Beale, Neural Network Toolbox for Use with Matlab: User’s Guide (Vers. 4). Natick, MA: The Math Works Inc., 2000. A Hybrid Genetic-Algorithm Space-Mapping Tool for the Optimization of Antennas Mario Fernández Pantoja, Peter Meincke, and Amelia Rubio Bretones Abstract—A hybrid global-local optimization technique for the design of antennas is presented. It consists of the subsequent application of a genetic algorithm (GA) that employs coarse models in the simulations and a space mapping (SM) that refines the solution found in the previous stage. The technique is particularly suited to optimization problems for which long computational times are required to achieve accurate solutions. Index Terms—Antenna arrays, genetic algorithms (GAs), optimization methods, space mapping (SM). I. INTRODUCTION The application of genetic algorithms (GAs) as optimization tools for the design of antennas has been an active field of research in the past decade [1]. The main reasons for this interest are related to their robustness, enabling the solution of optimization problems for which local techniques of optimization are not effective, as well as their ver- satility, permitting the successful use of the same schemes to different problems [2]. There are, however, inherent restrictions to the applicability of the GAs. As a consequence of their structure, the problems, for which Manuscript received October 13, 2005; revised February 13, 2006. This work was supported by the Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia” through Programa Nacional de Ayudas para la Movilidad de Profesores de universidad as well as the Danish Technical Re- search Council and Spanish Ministerio de Ciencia y Tecnología, under Projects TEC-2004-06217-C02-01 and TEC-2004-04866-C04-03. M. Fernández Pantoja is with the Facultad de Ciencias, Departamento de Electromagnetismo, University of Granada, 18071 Granada, Spain (e-mail: [email protected]). P. Meincke is with the Technical University of Denmark, Ørsted-DTU, Elec- tromagnetic Systems, DK-2800 Lyngby, Denmark (e-mail: [email protected]. dk). A. Rubio Bretones is with the Facultad de Ciencias, Departamento de Electromagnetismo, University of Granada, 18071 Granada, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2007.891556 0018-926X/$25.00 © 2007 IEEE

A Hybrid Genetic-Algorithm Space-Mapping Tool for the Optimization of Antennas

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Page 1: A Hybrid Genetic-Algorithm Space-Mapping Tool for the Optimization of Antennas

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007 777

data, that means, the output data in this case is a vector consisting ofthree elements whose range is from 0 to 16. Other methods of choosingthe output data that can identify the fault elements is also possible andis user dependent.

B. Network Training

In this work the MLP is trained in the backpropagation mode, be-cause of its suitability for forming mapping implementations. In thistraining algorithm the weight updation takes place according to

wk+1ih = w

kih � �

@Ek

@wih

(1)

where, � is the learning rate and Ek is the mean square error at the kthinstant.

The neural network learns from samples of input-output data i.e.,ak and bk; k = 1; 2; . . . ; N , where N is the total number of samplesfor training. In the present problem radiation pattern was simulated forall 697 number of combination of faults, i.e., N = 697. Because thenetwork size is large as compared to the number of available trainingpatterns, so in order to avoid a possible overfitting and to improve net-work generalization, we have adopted the regularization method [9]. Inthis procedure, the performance function of the feedforward network ismodified by adding a term with the mean square error that consists ofthe mean of the sum of the squares of the network weights and biases.The performance function now becomes

E = 1

N

N

i=1

(ti � ai)2 + (1� )

1

n

n

j=1

w2j (2)

where, is the performance ratio, ti is the target output, ai is the net-work output, and n is the total number of weights and biases.

To implement this, the network inputs and outputs were preprocessedso that they fell approximately in the range [0, 1].

C. Training Parameters

The efficiency of training depends on the training parameters. Thevalues of the training parameters taken for the training of the presentnetwork are as mentioned in Table I.

IV. RESULTS AND DISCUSSION

The trained network was tested for the results. The network outputsare rounded off to get the fault element numbers and their location. Forexample, a raw output of 0.297 after rounding off, produces the value0.3 representing the fault in the 3rd element. Results of some typicalfault patterns are given in Fig. 2. In all the cases the network outputmatches with the simulation results.

V. CONCLUSION

A MLP neural network trained in the backpropagation mode is usedto locate the fault elements in an antenna array from its distorted radia-tion pattern. The results are in excellent agreement with the simulationresults. The developed network can be used at the base stations to findout the number and location of the fault elements in the array, from thedistorted radiation pattern due to presence of nonradiating elements.Although the methodology was developed for a linear array, the samecan be extended to any planar array structure.

REFERENCES

[1] R. J. Mailloux, Phased Array Antennas Handbook. Norwood, MA:Artech House, 1994.

[2] B. K. Yeo and Y. Lu, “Array failure correction with a genetic algo-rithm,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 823–828,1999.

[3] R. J. Mailloux, “Array failure correction with a digitally beamformedarray,” IEEE Trans. Antennas Propag., vol. 44, pp. 1542–1550, 1996.

[4] H. Steyskal and R. J. Mailloux, “Generalization of an array failure cor-rection method,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag.,vol. 145, no. 4, pp. 332–336, Aug. 1998.

[5] J. A. Rodriguez and F. Ares, “Finding defective elements in planararrays using genetic algorithms,” Progress in Electromagn. Res., PIER,vol. 29, pp. 25–37, 2000.

[6] S. Haykins, Neural Networks: A Comprehensive Foundation. NewYork: IEEE Press/IEEE Computer Society Press, 1994.

[7] Incorporated, IE3D Zeland Software.[8] K. C. Lee and T. N. Lin, “Application of neural networks to analyses of

nonlinearly loaded antenna arrays including mutual coupling effects,”IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1126–1132, 2005.

[9] H. Demuth and M. Beale, Neural Network Toolbox for Use withMatlab: User’s Guide (Vers. 4). Natick, MA: The Math Works Inc.,2000.

A Hybrid Genetic-Algorithm Space-Mapping Tool for theOptimization of Antennas

Mario Fernández Pantoja, Peter Meincke, andAmelia Rubio Bretones

Abstract—A hybrid global-local optimization technique for the design ofantennas is presented. It consists of the subsequent application of a geneticalgorithm (GA) that employs coarse models in the simulations and a spacemapping (SM) that refines the solution found in the previous stage. Thetechnique is particularly suited to optimization problems for which longcomputational times are required to achieve accurate solutions.

Index Terms—Antenna arrays, genetic algorithms (GAs), optimizationmethods, space mapping (SM).

I. INTRODUCTION

The application of genetic algorithms (GAs) as optimization toolsfor the design of antennas has been an active field of research in thepast decade [1]. The main reasons for this interest are related to theirrobustness, enabling the solution of optimization problems for whichlocal techniques of optimization are not effective, as well as their ver-satility, permitting the successful use of the same schemes to differentproblems [2].

There are, however, inherent restrictions to the applicability of theGAs. As a consequence of their structure, the problems, for which

Manuscript received October 13, 2005; revised February 13, 2006. This workwas supported by the Secretaría de Estado de Universidades e Investigación delMinisterio de Educación y Ciencia” through Programa Nacional de Ayudas parala Movilidad de Profesores de universidad as well as the Danish Technical Re-search Council and Spanish Ministerio de Ciencia y Tecnología, under ProjectsTEC-2004-06217-C02-01 and TEC-2004-04866-C04-03.

M. Fernández Pantoja is with the Facultad de Ciencias, Departamento deElectromagnetismo, University of Granada, 18071 Granada, Spain (e-mail:[email protected]).

P. Meincke is with the Technical University of Denmark, Ørsted-DTU, Elec-tromagnetic Systems, DK-2800 Lyngby, Denmark (e-mail: [email protected]).

A. Rubio Bretones is with the Facultad de Ciencias, Departamento deElectromagnetismo, University of Granada, 18071 Granada, Spain (e-mail:[email protected]).

Digital Object Identifier 10.1109/TAP.2007.891556

0018-926X/$25.00 © 2007 IEEE

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high computational times are needed to accurately simulate eachpossible solution, remain yet excessively costly. To overcome thisproblem, several efforts have been devoted to find more efficientoptimization schemes, resulting not only in improved versions of thegenetic algorithms, e.g., micro-genetic algorithms (�GA) [3] and hy-brid taguchi genetic algorithms [4], but also in new global techniquesof optimization derived from different philosophies, e.g., particleswarm optimizations [5] and ant colony optimizations [6]. Theseimprovements, along with the increasing capability of computers andthe development of parallel codes [7], have led to satisfactory solutionsfor more complex problems. Moreover, in problems for which theaccuracy of the optimized solution is not critical, a usual procedure fordecreasing the total computational time is to reduce the computationalburden of the models by, for example, using a coarse meshing of thecomputational grid in simulators based on finite element methods[8] or decreasing the number of basis functions in codes applyingthe method of moments [9]. Unfortunately, an estimation of the errorintroduced by these approaches is often difficult to make.

In this communication, an additional stage is introduced in the opti-mization procedure to assure the accuracy of the final result. This ad-ditional step, based on the space-mapping (SM) technique [10], allowsGA operators to employ coarse models in the simulations to find anapproximate solution of the problem. Once attained, SM provides anaccurate solution of the problem with a relatively low computationalcost. SM techniques, in conjunction with different local optimizationproblems, have previously proven effective for solving different opti-mization problems in electromagnetics [11], [12]. To the knowledge ofthe authors, the application of this technique to improve the results ofgenetically designed antennas has not previously been published.

As an example of optimization the hybrid algorithm GA-SM is usedto select the lengths and feeding points of an array of 3� 3 patch an-tennas on a finite ground plane. The results show the advantages ofapplying this scheme.

II. HYBRID GA-SM OPTIMIZATION ALGORITHM

A flowchart of the algorithm, presented in Fig. 1, consists basicallyof two different strategies applied consecutively. First, a GA optimizeroffers, by means of repetitive, fast computer simulations of possiblesolutions, a low-accuracy optimal resolution of the problem. This in-termediate result is referred to as a coarse optimal solution. Second,an accurate simulation of the coarse optimal solution is performed toverify whether the coarse solution is acceptable to provide a final so-lution to the problem. If this accurate analysis shows unsatisfactorycharacteristics, i.e., displacements of the resonance frequencies or in-creased levels of input reflection coefficients, a subsequent stage basedon SM is initiated. In this stage, a local optimizer using both coarseand fine models is employed to produce an accurate solution of theproblem, denoted by the fine optimal solution. This solution resemblesthe coarse solution offered by GA in those parameters chosen to bethe objectives of the optimization. Consequently, the designer must de-fine the parameters that vary in the optimization process, denoted by�xc and �xf , and the characteristics (basis functions, precision of inte-grals, etc.) of both coarse and fine models. The correct choices at thispoint will be essential for the success of the optimization. The coarsemodel should be as fast as possible, but keeping a certain similarity be-tween its responseRc(�xc) and the responseRf (�xf ) offered by the finemodel. Otherwise, the SM stage will not work properly. Once selected,the GA optimizer provides the optimal coarse solution, denoted by �x�c ,by means of applying genetic operators only over coarse simulations.Once determined that the deviation of the response calculated by an ac-curate simulation Rf(�x

c) of the optimal coarse solution is higher thanaccepted, the SM algorithm seeks for a mapping ~P between the fine andcoarse models �xc = ~P (�xf ), so that Rf (�xf ) � Rc(�xc). To determine

Fig. 1. Flowchart of the hybrid GA-SM algorithm.

~P , an iterative local optimization is performed. The key steps in theSM algorithm are the parameter-extraction phase, which ascertains thecoarse model that better fits a certain fine model, the update-mappinglevel, which alters the estimate of ~P using a Broyden equation [12],and the invert-mapping level, which determines the fine model for thenext iteration. If the coarse and fine models are properly chosen, thisiterative process converges at the fine optimal solution �x�f when the re-sponses Rf(�xf ) and Rc(�xc) are similar up to a previously fixed levelof precision. More details on the SM can be found in [10].

It is worth pointing out several considerations to clarify the method.First, the SM stage acts only to align the coarse and fine optimal solu-tions. It does not look for better solutions that could appear if the GAoptimization had been made with accurate simulations. As long as theresponses of the coarse and fine solutions have sufficient similarity, thefine solution satisfies the initial objectives in the same degree as thecoarse optimal solution does. Second, general statements related to thecomputational time added by the SM stage are hard to establish. Theirvalidity will strongly depend on various factors such as the number ofparameters to be optimized or the accuracy in the resemblance of theoptimal fine and coarse solutions. In any case, as a rule, the maximumreduction in computational time of this method, compared to a GA op-timization based on accurate models, is determined by the reduction incomputational time between fine and course simulations for a singlecase. Third, we remark that the flowchart described offers different al-ternatives of implementation, depending on the specific GA schemesand/or SM techniques to be arranged in the scheme of Fig. 1.

III. EXAMPLE OF OPTIMIZATION

To test the adequacy of the method, the determination of properlengths and feeding points for an array of 3� 3 patch antennas ona finite square ground plane to operate at a frequency of 4.5 GHz isproposed as an example of optimization. Given the symmetry of theproblem, as indicated in Fig. 2(a), there are a total of 12 optimization

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Fig. 2. Array of patch antennas on finite ground plane. (a) Top view. (b) Sideview.

parameters, related both to the lengths (L1; . . . ; L6) and to the dis-tances of the feeding points from the center of the patch (d1; . . . ; d6).Fixed quantities of this example are the widths of the patches (W =3 cm), the side length of the ground plane (Lg = 12 cm), and the dis-tance between the antennas and ground (h = 0:15 cm). The substrateused is air.

The use of a global optimizer to solve this problem requires a re-liable code to simulate randomly generated designs. All the resultsshown in this paper are derived from the method-of-moments solutionof the mixed-potential electric-field integral equation with higher-orderLegendre basis functions [13]. Given the specific set of lengths andfeeding points described above, an accurate solution of this problemrequires 6000 basis functions with an analysis time of 4 minutes perfrequency on a 2.2 GHz AMD Opteron processor. As the search spaceconsists of 1012 possible solutions to be considered in the optimizationprocess, the optimization results are reached with a �GA algorithmafter approximately 3000 simulations. If no parallelization is applied,the total optimization time for this simple case could be about ninedays. The application of a coarse model of the individuals, consistingof a decrease in both the number of basis functions and precision of theintegrals which is described in the following section, leads to a fasterresult with the cost of a displacement in the frequency spectrum of thesimulated response of some 100 MHz.

A. Results Applying the Hybrid GA-SM Technique

As a preliminary step for the application of the proposed hybridGA-SM technique the single patch shown in Fig. 3 was analyzed firstwith a fine model and next a coarse model. Fig. 4 illustrates the processof establishing a coarse model and validating the code. One criterionapplied to accelerate the simulations was to reduce the highest polyno-mial order degree by two (sixth order for the fine model versus fourthorder for the coarse model), resulting in 1100 basis functions per squaremeter in the fine simulations in contrast to the 500 basis functions persquare meter in the coarse simulations. In Fig. 4, a comparison withresults published in [14] is included, reflecting not only the accuracyof the fine model but also the resemblance of the coarse response. The

Fig. 3. Patch over finite ground plane. (a) Top view. (b) Side view.

Fig. 4. Input impedance of the patch antenna in Fig. 3 using fine and coarsemodels. (a) Real part. (b) Imaginary part.

parameters of this example were Wp = Lp = 3 cm, Lg = 4 cm,h = 0:15 cm, and df = 0:5 cm.

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TABLE IDIMENSIONS OF COARSE AND FINE OPTIMAL SOLUTIONS

Fig. 5. Input reflection coefficients based on the coarse optimal solution.

Next, the GA-SM optimization was performed. The GA stage, usingonly coarse models of the array, was conducted by means of an elitist�GA algorithm [3], with a population of 5 individuals and a 80% con-vergence for a replacement of the population. The GA operators em-ployed were a tournament selection and a double-point crossover [15].Possible solutions were generated using fixed-point codification, witha total of 12 integer digits ranging from 3 to 3.25 cm for the lengths andfrom 0.33 to 0.60 cm for the distance of the feeding points to the centerof the patch. The values allowed for the patch lengths in the GA processwere established using approximate equations for the resonance fre-quency of rectangular microstrip patch antennas over infinite groundplane , ranging from 4.4 to 4.8 GHz, these corresponding, respectively,to 3.25 and 3 cm. The fitness function F was designed to minimizeat 4.5 GHz the maximum value of the magnitude of the input reflec-tion coefficient for any antenna of the array (F = maxfjS11jig; i =1; . . . ; 6). The resulting dimensions of the optimized antenna are pre-sented in Table I. Fig. 5 shows the magnitude of the input reflectioncoefficient of each antenna element in this coarse solution, indicatingdifferent resonance frequencies for each patch antenna, but all of themnear the desired 4.5 GHz of operation.

Nevertheless, a simulation with a fine model of the same antenna re-veals a shift in the spectrum of approximately 130 MHz (see Fig. 6).For a correction of this effect, the SM stage is introduced. The finespace was defined with only two parameters �xf , each one to scale re-spectively the lengths and the distances from the feeding points, re-

Fig. 6. Input reflection coefficients based on an accurate simulation of thecoarse optimal solution.

Fig. 7. Input. reflection coefficients based on the fine optimal solution.

spectively, of the optimal coarse solution. As pointed out in [11], theconvergence of the model is better achieved when several frequencypoints of analysis are employed. In this case, 11 frequency points weredistributed between 4.25 to 4.75 GHz. The parameter-extraction phasewas accomplished following the aggressive space-mapping approach[12], and taking, as a measure of the similarity between the fine andcoarse models, the mean square error of the difference between theirrespective real parts of the input impedance, for all the patches alongthe frequency points of the analysis. Other relevant selections in thisSM stage were the stopping criteria, set to 10�4, and the numerical es-timate of the analytical Jacobian by using a forward-difference approx-imation. A key point to achieve a quick convergence is to evaluate thesimilarity between fine and coarse-model responses using as measuringfunction the real part of the input impedance rather than the magnitudeof input reflection coefficient, due to the fact that the greater smooth-ness of the former leads to better forward-difference approximations.After a total of three fine simulations and 45 coarse simulations, thealgorithm converged to the values listed in Table I. Fig. 7 shows theeffective correction of the operational point to the 4.5 GHz. If the SMstage is applied over the imaginary part of the input impedance, or overan average calculated on considering the similarity in both the real andthe imaginary parts of the input impedance, then the algorithm con-verges to values identical to those shown in Table I. The computational

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times are of the same order as when only the real part is taken intoaccount.

Finally, an estimation is made of the time saved by using the GA-SMmethod, in contrast to the GA method using only fine simulations.Given that the time consumption for the fine model in this case is 6.5times slower per frequency than the coarse model, and that each fineor coarse simulation made in the SM stage solved 11 frequency points,the optimization was performed roughly 5.25 times faster applying thishybrid technique. Hence, as long as the limit on the time reduction de-pends on the difference between the analysis time of fine and coarsemodels, the designer has to look for faster coarse models to achievegreater reductions. In any event, this process should be carried out care-fully since the SM stage works only effectively if the response of thecoarse model is similar to that of the fine model.

IV. CONCLUSION

In this communication, an efficient scheme has been proposed forthe optimization of antennas. This scheme consists of applying an addi-tional space-mapping technique after a genetic-algorithm optimization,the latter employing a coarse model in the simulation of the antenna re-sponse. The SM stage of the process improves the accuracy of the op-timized results, and the total approach has been demonstrated to offeradvantages in terms of computational cost over the single applicationof GA with a fine-model simulator. Further studies will be conductedto compare the performance of the GA-SM method with other hybridmethods combining local optimization techniques and SM.

REFERENCES

[1] Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimiza-tion by Genetic Algorithms. New York: Wiley, 1999.

[2] R. L. Haupt, “An introduction to genetic algorithms for electromag-netics,” IEEE Antennas Propag. Mag., vol. 37, pp. 7–15, 1995.

[3] K. Krishnakumar, “Micro-genetic algorithms for stationary andnon-stationary function optimization,” in SPIE: Intelligent Controland Adaptive Systems, 1989.

[4] J. T. Tsai, T. K. Liu, and J. H. Chou, “Hybrid Taguchi-genetic algo-rithm for global numerical optimization,” IEEE Trans. Evol. Comput.,vol. 8, no. 4, pp. 365–377, Aug. 2004.

[5] J. Kennedy and R. C. Eberhart, Swarm Intelligence. San Francisco,CA: Morgan Kauffman, 2001.

[6] C. Coleman, E. Rothwell, and J. Ross, “Investigation of simulated an-nealing, ant-colony optimization, and genetic algorithms for self-struc-turing antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp.1007–1014, Apr. 2004.

[7] D. Levine, “Users Guide to the PGAPACK Parallel Genetic AlgorithmLibrary” Transl.:Argonne National Laboratory Tech. Rep. ANL-95/18,1995.

[8] O. A. Mohammed, “Optimal design of magnetostatic devices: The ge-netic algorithm approach and system optimization strategies,” in Elec-tromagnetic Optimization by Genetic Algorithms, Y. Rahmat-Samiiand E. Michielssen, Eds. New York: Wiley, 1999, ch. 14.

[9] M. Fernández-Pantoja, A. Monorchio, A. Rubio-Bretones, and R.Gómez-Martín, “Direct GA-based optimization of resistively-loadedwire antennas in the time domain,” Electron. Lett., vol. 36, no. 24, pp.1988–1990, Nov. 2000.

[10] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. Bakr,K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,”IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 337–361, Jan.2004.

[11] M. H. Bakr, “Advances in space mapping optimization of microwavecircuits,” Ph.D. dissertation, Univ. Hamilton, Hamilton, ON, Canada,2000.

[12] J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, andK. Madsen, “Electromagnetic optimization exploiting aggressive spacemapping,” IEEE Trans. Microwave Theory Tech., vol. 43, no. 12, pp.2874–2882, Dec. 1995.

[13] E. Jørgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higherorder hierarchical Legendre basis functions for electromagnetic mod-eling,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2985–2995,Nov. 2004.

[14] C. L. Mak, Y. L. Chow, and K. M. Luk, “Finite ground plane effect ofa microstrip patch antenna: A CAD formula of impedance perturbationby synthetic asymptote and GTD,” Proc. Inst. Elect. Eng. Microw. An-tennas Propag., vol. 150, no. 1, pp. 11–17, Feb. 2003.

[15] T. Back, D. Fogel, and Z. Michalewicz, Eds., Handbook of Evolu-tionary Computation. Bristol, U.K.: IOP Publishing, 1997.

Genetical Swarm Optimization: Self-Adaptive HybridEvolutionary Algorithm for Electromagnetics

Francesco Grimaccia, Marco Mussetta, and Riccardo E. Zich

Abstract—A new effective optimization algorithm suitably developed forelectromagnetic applications called genetical swarm optimization (GSO) ispresented. This is a hybrid algorithm developed in order to combine in themost effective way the properties of two of the most popular evolutionaryoptimization approaches now in use for the optimization of electromagneticstructures, the particle swarm optimization (PSO) and genetic algorithms(GAs). The algorithm effectiveness has been tested here with respect to bothits “ancestors,” GA and PSO, dealing with an electromagnetic application,the optimization of a linear array. The here proposed method shows itselfas a general purpose tool able to effectively adapt itself to different electro-magnetic optimization problems.

Index Terms—Array synthesis, evolutionary algorithms, hybridizationstrategies, optimization techniques.

I. INTRODUCTION

The general aim of optimization algorithms is to find a solution thatrepresents a global maximum or minimum in a suitably defined solutiondomain, that means to find the best solution to a considered problemamong all the possible ones [1]. Global search methods present twocompeting goals, exploration and exploitation: exploration is impor-tant to ensure that every part of the solution domain is searched enoughto provide a reliable estimate of the global optimum; exploitation, in-stead, is important to concentrate the search effort around the best so-lutions found so far by searching their neighborhoods to reach bettersolutions. Many search algorithms achieve these two goals using localsearch methods, or global search approaches, or a dedicated combina-tion of both the global and local strategies: these algorithms are com-monly known as hybrid methods.

Manuscript received March 1, 2006; revised October 7, 2006.The authors are with the Department of Electrical Engineering, Politecnico

di Milano, I-20133 Milano, Italy (e-mail: [email protected]).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.2007.891561

0018-926X/$25.00 © 2007 IEEE