Modification of Real-Number and Binary PSO Algorithms for Accelerated Convergence

214 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 1, JANUARY 2011

Modification of Real-Number and Binary PSOAlgorithms for Accelerated Convergence

Arezoo Modiri, Student Member, IEEE, and Kamran Kiasaleh, Senior Member, IEEE

Abstract—Modifications in the velocity calculation of the par-ticle swarm optimization (PSO) algorithm are proposed. The sug-gested modifications aim to arrive at a faster, more straightfor-ward and still robust search procedure as compared to the con-ventional method. Two main factors, i.e., personal best influenceand initial velocity values, are evaluated. It is shown that in prob-lems with wide-range parameters, the effect of personal best loca-tions is intrinsically encompassed by that of global best locations,thereby allowing for further simplification of the PSO algorithmby eliminating the factor which accounts for the personal best so-lutions in the velocity calculation. This simplification expedites theconvergence procedure in real PSO. It is also shown that the ini-tial velocity values can be modified to enhance the performancein terms of achieving better solution when compared with the ex-isting algorithms, particularly in binary PSO. In order to validatethe viability of the proposed procedure, the performances of thereal-number and binary PSO algorithms with different velocitycalculations are assessed in 1000-run sets, and pros and cons arestudied. In particular, the performance of the proposed algorithm,when used to design software defined thinned array antennas, isshown to be superior to those of the existing algorithms.

Index Terms—Evolutionary optimization, particle swarm opti-mization, side lobe level, thinned antenna array.

I. INTRODUCTION

E VOLUTIONARY algorithms (EAs) are based on sto-chastic variation of a number of particles (individuals)

in a solution space. The major difference between various EAalgorithms resides in the variation operators. A comprehensivecomparison of EAs in electromagnetic can be found in [1] wherethree groups of EAs are introduced: evolutionary programming(EP), evolution strategies (ES) and genetic algorithms (GAs).After several assessments, the author concluded that the EP’ssimplicity, ease of implementation, and flexibility offer designengineers with a valuable tool for efficient design of microwavedevices. PSO algorithm falls within the EP group, and hencethe above conclusion also applies to the PSO algorithm.

Motivated by the behavior of groups of organisms duringtheir food-searching activities, Eberhart and Kennedy intro-duced the PSO algorithm in 1995 [2]. The idea was furtherdeveloped in the ensuing years [3]–[7]. PSO has been usedalong with a variety of analytical and numerical tools for

Manuscript received September 22, 2009; revised May 13, 2010; acceptedJuly 09, 2010. Date of publication November 01, 2010; date of current versionJanuary 04, 2011. This work was supported in part by National Institute of Jus-tice (NIJ) under Grant #2006IJCXK043.

The authors are with the Department of Electrical Engineering, Uni-versity of Texas at Dallas, Richardson, TX 75080-3021 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TAP.2010.2090460

different electromagnetic applications, a good list of which canbe found in [8]. Although PSO has already shown considerableadvantages in electromagnetic designs [9]–[14], it should stillbe noted that the EAs’ efficiencies are problem-dependent,and their performances are impacted significantly by the initialsolutions [15]–[17].

In general, there are two approaches one may take for the pur-pose of optimization. The first approach makes an attempt tofind a suitable solution as fast as possible, whereas the secondapproach is concerned with the best possible solution regard-less of the number of iteration. Since the first approach is oftenselected for a large number of electromagnetic problems, thispaper targets the same approach as well.

PSO has some deficiencies, which limit its performance. Toelaborate, it is possible that particles get locked in a local bestsolution, which prevents them from reaching the best solution;or, the optimization may turn to be time consuming due to largenumber of required iterations and/or particles. The latter oc-curs when the fitness evaluations go beyond simple mathemat-ical calculations (as is the case in a majority of electromag-netic problems). Thus, PSO in its current form seems to be in-appropriate for time-sensitive optimization problems, often en-countered in software-defined applications (such as smart beam-forming of thinned array antennas). Hence, this study proposesmethods to enhance the performance (in terms of convergencetime) of PSO algorithm while preserving its robustness.

In the ensuing sections, after a brief description of PSO, twomajor modifications in velocity vector are introduced and theirperformances are compared to that of the existing PSO.

II. PSO ALGORITHM

The principles of PSO algorithm can be found in various ref-erences in literature (e.g., in [17], [18]), thus, this paper does notgo through basic descriptions of the algorithm.

Since it is often possible to merge several fitness (objective)functions into one, single-objective optimization is the focus ofmost optimization problems. Single-objective PSO algorithmshave been classified in two groups, real-number (RPSO) andbinary (BPSO) [8], and both are considered in this study.

The only key operator in PSO is the velocity calculationwhich has been investigated in several studies (e.g., [15], [8],[19]). In this paper, benefiting from the results available inthe literature, we propose and compare the performance of anew velocity vector. To that end, let us examine the velocityfunction of the PSO algorithm.

It is generally assumed that there are particles and param-eters (or dimensions) to be optimized. The particles’ positions

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MODIRI AND KIASALEH: MODIFICATION OF REAL-NUMBER AND BINARY PSO ALGORITHMS 215

and velocities are stored in matrices. More precisely, thevelocity function is given by

(1)

where denotes the matrix of particles’ velocities in theth iteration, is the position matrix in the th iteration, and

or “personal best” is the matrix of the locations inthe solution space, which are remembered by each particle as itsmost successful position discovered during its own exploration.The th row in corresponds to the n-dimensional locationof the th particle. or “global best” is the matrixof the best locations found (in terms of the fitness value) byall particles. Furthermore, and are two random variablesuniformly distributed on [0, 1]; and finally, and are threecoefficients which can be used to modify the algorithm in orderto achieve the desired performance.

In the conventional PSO, both RPSO and BPSO, the initialvelocity values are defined as it is shown in (2).

(2)

Where is the matrix of initial velocities, and,

and are the maximum and minimum allowed velocities inthe th dimension. A mapping function is used for BPSO toquantize real values [8].

In order to limit the search to the solution space, a boundarycondition should be introduced. In [15], several boundary con-ditions were assessed and invisible wall was claimed to show abetter performance. This study uses invisible wall as the mainboundary condition as well, which means that the fitness valuecalculation will be skipped for particles that fly out of the solu-tion space. Upon the return of the particle, the fitness value cal-culation will resume. However, absorbing wall boundary condi-tion is also considered yet as a second option.

III. PROPOSED VELOCITY VECTOR

As mentioned in introduction, although PSO outperforms itscounterparts such as GA, in many ways, it has some deficien-cies. Basic PSO has been modified in different ways to improvethe drawbacks; in [20] various modified methods were intro-duced and their performances were compared. Furthermore, anew method, named tabu-searching PSO (TS-PSO), was intro-duced. In TS-PSO a second swarm and a number of tabu con-ditions are introduced. Equations (5)–(7) in [20] show that thesolution is gained by leading each particle of the second swarmaway from any dimension of the personal best and/or global bestlocation associated to the same particle that enters the tabu con-dition. This method is claimed to prevent the algorithm fromgetting locked in local best solutions but optimization lengthincreases.

Since a few rows of are qualified to become rows in, the idea of ignoring unqualified locations in , as an al-

ternate to the move-away procedure in TS-PSO, comes to mind.The goal of optimization is to move toward best solution, thus,

moving towards locations identified by which will neverbecome one of the locations identified by, seems to be a mod-ifiable step. Hence, in this study, we propose the following ve-locity function:

(3)

The values assigned to velocity vector coefficients ( and) in the literature have been, in large part, obtained using ex-

perimentation [18]. Thus, they can be modified according to op-timization goal.

We modify the manner by which the initial velocity vector isselected as well. The motivation behind this modification is sup-ported by nature. For instance, bees start their flower-searchingmovement with a speed that is higher than that of the subse-quent steps. This characteristic has already been added to RPSOby decreasing in each step of the iteration. However, such anapproach has shown no impact on BPSO [8]. In the proposedalgorithm, the velocity vector is initialized using the followingstatement:

(4)

The novelty of the equation is in its two-state discrete na-ture in compare to conventional velocity of (2). In this paper, asin conventional PSO, represents a velocity value equal to

but in the opposite direction, and is equal to the pa-rameter range for RPSO. In BPSO, is selected with regardsto the mapping function [8]. Since each parameter in BPSO caneither be 1 or 0, we propose to be 0. is assumed to be6 as in literature [8].

As it has been mentioned in [1], [8], [18] and [19], the sug-gested values for , and are either 1.49 for both, or 2 for both.

is also suggested to decrease during iterations linearly from0.9 to 0.4 in RPSO, and have constant value of 1 in BPSO. Pre-vious studies recommend for better convergence[8], and therefore, this paper focuses on as well.

In Subsection A, it is illustrated that the proposed velocityfunction of (3) reinforces a scenario in which the particles havecollectively made a correct movement toward the global bestsolution on the average. However, the performance of this ap-proach needs to be examined in the event where particles havenot taken the correct step toward the global best solution. Hence,in the ensuing sections, the performance of the proposed algo-rithm will be compared to the existing algorithm under realisticscenarios.

In Sections IV and V, the results of simplified velocity func-tion and the proposed initial values are studied in RPSO andBPSO, respectively. It is shown that modifying PSO velocityvector as suggested, mainly increases the convergence speed butmay yield (depending on the problem) a better solution than thatobtained using the conventional method too.

A. Statistical Analysis

The viability of (3) can even be proved mathematically byevaluating the average position of particles in each iterationcycle. Let and denote the position and velocity of


the th particle in the th dimension during the th iteration re-spectively. In PSO algorithm, this position is achieved by

(5)

Using the conventional velocity function (1), the average par-ticle velocities are given by

(6)

where and denote the average personal best posi-tion vector of all existing particles and the global best positionvector in the th iteration respectively. and are the positionand velocity vectors averaged over i.

Substituting in (6) and using (5), the averageposition vector is simplified to

(7)

If the algorithm moves the particles in the correct direction,should approach . Therefore, the average position

of the particles in a successful iteration cycle in which the ma-jority of particles have moved to positions with better fitnessvalues (compared to previous iteration) can be approximated as

(8)

One can observe that the average position of particles in (8)may also be achieved by using the proposed velocity function,given by (3) in above procedure.

IV. EVALUATION OF THE PROPOSED VELOCITY

FUNCTION IN RPSO

In order to have comparable results, the benchmark fitnessfunctions used in [1] and [20], shown in (9)–(11), are used. Toelaborate, the functions shown in (9)–(11) all possess a globalminimum that occurs at the origin of the n-dimensional solutionspace. So the goal of optimization is to approach this zero valueby moving particles in an -dimensional solution space.

Rastigrin function:

(9)

Griewank Function:

(10)

Rosenbrock function:

(11)

Where . To assess the impact of pro-posed modifications separately, the results of the following fourmethods are investigated (the initial solutions are identical forall methods during each run throughout this paper).

Fig. 1. AIN behavior of the four methods over gradually increasing sample runsin a 10D problem with 5, 10 and 20 particles for (a) Rastigrin, (b) Griewank,and (c) Rosenbrock functions and parameter range of �� .

• Method1: , and the initial velocity is given by(2) (conventional method).

• Method2: , and the initial velocity is given by(4).

• Method3: , and the initial velocity is givenby (2).

• Method4: , and the initial velocity is givenby (4).

In order to have comprehensive statistical study, the conver-gence behaviors in different cases are observed in 500 iterationsduring 1000 runs, and the average results associated with eachbenchmark function are depicted in terms of the achieved av-erage final value (AFV) and the average iteration number (AIN)required to attain a given AFV. The results are averaged over aset of runs. The number of sample runs in the set increases by20, i.e., the results of sample runs are averagedand plotted. In this manner, in addition to achieving consistentstatistical results, we are also able to measure the number ofsamples required for attaining reliable results.

A 10D scenario is assumed with particle num-bers of 5, 10, and 20. The parameters can vary in the ranges of

, and . Although a highernumber of particles are generally considered in RPSO, two ob-jectives are obtained here by these selections. First, the (m,n)values are chosen so that they match the parameters for theensuing binary PSO study, allowing for a comparative study.Second, since speed of convergence is a priority in this study,key solutions for accelerating the convergence while preserving


Fig. 2. AFV behavior of the four methods over gradually increasing sampleruns in a 10D problem with 5, 10 and 20 particles for (a) Rastigrin, (b) Griewank,and (c) Rosenbrock functions and parameter range of �� .

PSO’s intrinsic robustness are sought. In real cases, it is almostimpossible to reduce the number of dimensions in order to speedup the solution. However, decreasing the number of particlescan drastically shorten the optimization time. Thus, we do notstep beyond particles in the first stage of our study.AIN/AFV results are shown in Figs. 1–4.

From Figs. 1–4, two interesting trends are discovered. Thefirst one is the high similarity between the results of the threemodified methods, particularly in terms of convergence speed.The second common trend is that the modified methods mostlyexpedite the convergence by almost 18% at the expense ofhigher errors for lower parameter ranges. However, as theparameter range increases, the modified methods begin toslow down, while getting much lower errors with respect toMethod1. For Rastigrin and Griewank, this transition happenssomewhere between and , whereas for Rosen-brock it occurs somewhere between and ; i.e., inaddition to exhibiting encouraging convergence behavior, themodified methods outperform the conventional one, both inspeed and error rate, provided that the proper parameter rangesare selected. For Griewank, is such a range.

In order to cope with the realistic computation limitations ofa standard 2.83 GHz and 3.23 GB RAM PC running MATLAB,the procedure is performed for each function separately and the

Fig. 3. AIN behavior of the four methods over gradually increasing sampleruns in a 10D problem with 10 particles for (a) Rastigrin, (b) Griewank,and (c) Rosenbrock functions and parameter range of ��, and��.

resulting figures are grouped together as in Figs. 1–4. For thesake of clarity, the figure in the right column of Fig. 4(c) as wellas some other figures which will appear later in this section areshown in logarithmic scale.

To summarize the results, the modified methods outperformthe conventional one in terms of convergence speed for Rasti-grin function better than the two others, and in terms of the bestfinal value for Rosenbrock function. However, the cost of finalvalue error is only acceptable for high parameter ranges.

Figs. 1–4 reveal the AFV and the required AIN, but they donot explain the gradual convergence behavior, which is of greatimportance in this study. Fig. 5 demonstrates the fitness valuevariation during optimization process, averaged over 1000 runsfor 10D problem with 10 particles. It is obvious that modifiedmethods are superior for the typical range with termi-nation criterion of 500 iterations. For the case of Griewank, allmethods perform almost similarly.

Since average fitness trends may not be adequate for pre-senting the success rate, in Fig. 6 the probability density func-tion of the final fitness value in 1000 runs for 500 iteration sce-nario are plotted for Rastigrin function. In this figure, MFFVdenotes the maximum final fitness value. To elaborate, the fourbars at 0.1MFFV show the probability of the final fitness value’sbecoming less than or equal to 0.1MFFV, for the four methods.Similarly, the bars at 0.2MFFV show the probability of the final


Fig. 4. AFV behavior of the four methods over gradually increasing sampleruns in a 10D problem with 10 particles for (a) Rastigrin, (b) Griewank,and (c) Rosenbrock functions and parameter range of �� , and��.

fitness value’s becoming less than or equal to 0.2MFFV andmore than 0.1MFFV. Note that the probability of ending withsmaller fitness values increases in modified methods as the pa-rameter range increases. Similar trend is observed by studyingother functions. For the sake of conciseness, we have omittedthose results.

In addition to convergence performance, the behavior of par-ticles has also been studied in [20] as yet another performancecriterion. Fig. 7 depicts how the average position of a sampleparticle in a sample dimension varies. The results are averagedover 1000 runs for Rastigrin function assuming

, and parameter range of . In order to have a better view,two areas in Fig. 7(a) are zoomed out in figures (b) and (c). It isnoteworthy that the modifications offer the particles a slightlygreater degree of freedom in the initial phase of the exploration,but faster convergence in later iterations. Similar trend is ob-served by studying other sample particles, other dimensions,and other functions. Again for the sake of brevity, those resultsare not included.

As it has been declared earlier, this study emphasizes on time-sensitive optimizations, where rather than achieving optimumsolution in long optimization procedures, expeditious conver-gence to an acceptable solution is targeted ([21], [22]). For in-stance, when an antenna is designed to resonate at a desired fre-

Fig. 5. Fitness value of the four methods averaged over 1000 runs in a 10Dproblem with 10 particles for (a) Rastigrin, (b) Griewank, and (c) Rosenbrockfunctions and parameter range of ��.

Fig. 6. Probability density function of the final fitness value for four methods,with 1000 runs in a 10D problem for Rastigrin function with parameter rangeof (a) ��, (b) ��, and (c) ��.

quency, refining the solution to improve a reflection coefficientof dB to, let’s say dB, although of some mathemat-ical interest, is of little practical interest. In fact, the minimiza-tion criterion in these types of problems is met far in advanceof the actual minimum [23]. Thus, convergence speed to meeta design criterion is a more meaningful approach than an ap-proach that strives to achieve the global optimum at all cost. Toillustrate this point, in Appendix A, a microstrip antenna designscenario is discussed.

To further elaborate on the tradeoff between the optimizationspeed and the error rate, let us introduce error as

(12)

In (12), regardless of the parameter range, error is definedas the difference between the final fitness value achieved bythe optimization with termination criterion of 500 iterations


Fig. 7. (a) Average sample particle position in an arbitrary dimension for thefour methods over 1000 runs with 10 particles in a 10D problem for Rastigrinfunction with parameter range of �� , (b) zoom of 1–100 iterations,and (c) zoom of 200–300 iterations.

Fig. 8. The success rate trend of the four methods in 1000 runs in a10D problem for Rastigrin function with parameter range of (a) ��,(b) ��, and (c) �� with respect to the error value.

, and the global minimum , which iszero for the selected benchmark functions. Figs. 8–10 depictthe percentage of the 1000 runs for which error falls belowcertain prescribed value. To clarify the parameter range effect,10-dimensional Rastigrin function, for instance, has maximumvalue of 250 when the three parameters are in the range of

, whereas the maximum value increases to 25e9 for theparameter range of . Therefore, the difficultyof approaching the global minimum of zero in a wider solutionspace should be kept in mind while inspecting the curves.

As expected, a gradual movement toward higher success rateis observed in Figs. 8–10 for the curves related to the modifiedmethods versus those of the conventional one as one increasesthe parameter range. However, as it is also observed in Fig. 6,the above trend is more evident in Rosenbrock and Rastigrinfunctions than in Griewank’s.

Furthermore, absorbing wall boundary condition for a 10Dproblem with 5, 10 and 20 particles is examined. Fig. 11 shows

Fig. 9. The success rate trend of the four methods in 1000 runs in a10D problem for Griewank function with parameter range of (a) �� ,(b) ��, and (c) �� with respect to the error value.

Fig. 10. The success rate trend of the four methods in 1000 runs in a10D problem for Rosenbrock function with parameter range of (a) �� ,(b) ��, and (c) �� with respect to the error value.

Fig. 11. Fitness value of the four methods averaged over 500 runs in a 10Dproblem with 5 particles for Rosenbrock function with parameter range of�� using absorbing boundary condition.

the convergence behavior for Rosenbrock function in the same10D problem with 5 particles. It is apparent that the results aresimilar to those of the invisible wall. Figures related to the othercases and functions are eliminated since they also show analo-gous outcomes. Since it was concluded from previous study that


Fig. 12. Fitness value of the four methods averaged over 500 runs with 30 par-ticles for (a) Rastigrin, (b) Griewank, and (c) Rosenbrock functions with param-eter range of �� .

500 runs are enough for collecting reliable average results, thesample number is reduced to 500 runs.

Finally, to provide a comparable analysis with studiesavailable in the literature, the parameters of the search arechosen similar to those presented in [1] and [20]. It is assumedthat PSO has a population size of particles, and thefitness functions have three parameters ranging in

. The termination criterion is defined by amaximum iteration number of 500. Fig. 12 depicts the conver-gence behavior averaged over 500 runs.

It is observed in Fig. 12 that the modified methods yield muchbetter performance in terms of the convergence speed and finalfitness value for Rastigrin and Rosenbrock, but for Griewank,the results of all methods are similar. In comparison, TS-PSOalgorithm requires more than 5000 iterations to achieve similarresults for parameters with shorter range [20].

It was shown in this section that the two modifications pro-posed for PSO yield similar improvements in the RPSO algo-rithm behavior. The modifications generally speed up the algo-rithm. As the parameter range increases, the superiority of theproposed modifications improves greatly both in convergencespeed and final value, particularly for Rastigrin and Rosenbrockfunctions. It is also shown that run number of less than hundredsis not sufficient for making statistical decision.

V. EVALUATION OF THE PROPOSED VELOCITY

FUNCTION IN BPSO

In this section, we assess the performance of the proposed al-gorithm in BPSO scenario, which has been focus of some recentstudies [19]. BPSO is highly useful in problems with two-stateparameters, such as irregular shaped planar antenna design andarray antenna design [8], but to extend its appliance to smarttype of applications requiring fast resulting, it is not appropriateyet. This paper studies thinned antenna arrays which are peri-odic arrays with a number of non-illuminating elements. In addi-tion to general array applications, these types of antennas can beused where software-defined beam-forming or pattern-shapingis required by using the currents of elements as tuning factors.

Fig. 13. Symmetric half-wavelength thinned linear array.

In this section, we first study 2n-element linear half-wave-length thinned antenna array design, using BPSO, assuming

. It has been proven for BPSO that the gradual de-crease of has no effect on performance, thus, as recommendedin literature, we assume to be constantly equal to “1” [8].

The optimization goal is to minimize the side lobe level (SLL)in the antenna’s broadside. The lobes are assessed in de-grees angular distance from main lobe centre. The fitness func-tion, shown in (13), is minimized

(13)

The array factor formula can be found in many referencessuch as (26) in [8]. Assuming the antenna in xy plane, in thispaper, the SLL is assessed in the plane including the array itselfand broadside for linear array, but for planar arrays anumber of -planes are considered.

In order to have consistent study, first, the convergence be-haviors in cases similar to those encountered in RPSO are ob-served in 1000 runs, and the average results associated with eachcase are gathered in two figures depicting AIN and AFV values.Then, planar array antenna cases are studied.

As the complexity of fitness function evaluation increases,the necessity of achieving better results in a fewer number ofiterations becomes more obvious. In addition, as the numberof particles reduces, the calculation time reduces significantly,although the chance of convergence to appropriate results alsodrops. Thus, a problem-dependent trade-off should be consid-ered. Since this study considers large number of runs, the di-mensionality of the problems are selected according to addressmemory and time limitations.

A. 2n-Element Symmetric Linear Array

The linear arrays are assumed to be symmetric arrays with 2nelements, as shown in Fig. 13.

A 10D scenario with particle numbers of 5, 10and 20 are considered and AIN and AFV curves are shownin Figs. 14–15. The performance of Method2 and Method4in terms of AIN and AFV for BPSO (similar to the RPSOcase) is superior to Method1. However, although Methods2 andMethod3 for RPSO yielded superior performance as comparedto the conventional method, only Method2 yielded superiorperformance consistently for BPSO case, while Method3 of-fered inconsistent performance advantage over the conventionalmethod in our studies. The major reason for this distinctionbetween RPSO and BPSO is due to insertion of the two-level(binary) quantization [7]. In fact, =0 (Method3) may evendegrade the convergence performance unless it is used togetherwith proposed initial velocity (as in Method4). The figuresclearly show that the convergence speed (AIN) is significantlyimproved by Method2 and Method4, whereas the AFV gain isjust slightly enhanced.


Fig. 14. AIN behavior of the four methods over gradually increasing sampleruns in a 10D problem with 5, 10 and 20 particles.

Fig. 15. AFV behavior of the four methods over gradually increasing sampleruns in a 10D problem with 5, 10 and 20 particles.

Fig. 16. Non-symmetric thinned planar antenna array.

Also, it is important to mention that since in BPSO each pa-rameter can either be 1 or 0, parameter range will not be a con-siderable factor.

B. Non-Symmetric Planar Array

Similar study is repeated for the more complicated case ofplanar half-wavelength, non-symmetric thinned array antennas(Fig. 16). A problem is first considered. In orderto shorten the total run time, 100 iterations are performed in 700runs, and the swarm size is limited to two particles. The numberof runs is reduced to 700 from 1000 according to the study re-sults in Section A, showing reliable run number. The pattern isassessed in 6, 8 and 10 -planes (

, and , respectively). The superiorityof Method2 and Method4 both in terms of convergence speedand final result is evident in Fig. 17.

Similarly, 3 3, 4 4 and 5 5 planar arrays are studiedwith 3, 4 and 5 particles, respectively. This time, the pattern isassessed in no more than 6 -planes , 200iterations are performed in each of 700 runs. In order to alterthe fitness function, a success boundary of dB is added to

Fig. 17. Fitness value trend of the four methods averaged over 700 runs in a16D problem with 2 particles. Optimization is performed in (a) 6�-planes, (b) 8�-planes, and (c) 10 �-planes.

Fig. 18. AIN and AFV behavior of the four methods averaged over graduallyincreasing sample runs in a 9D problem with 3 particles.

the fitness function introduced in (13), resulting in the followingfitness function:

(14)

Figs. 18–20 show the AIN/AFV, density, and average fitnesstrend for the 3 3 array, respectively. The results confirm theprevious conclusions. Method2 and Method4 are superior tothe conventional method and Method3, both in terms of con-vergence speed and final value. Fig. 19 shows that the prob-ability of achieving a smaller final fitness values is higher forMethod2 and Method4 as compared to other methods. An im-portant point underscored by Fig. 20 is that wherever the opti-mization procedure is terminated, Method2 and Method4 out-perform the other two techniques. Similar trend is observed bystudying other cases, but, for the sake of conciseness, the resultsare not included here. It is also notable in Fig. 18 that 500 runsare sufficient for concluding statistical results.

As in RPSO, absorbing-wall scenario is investigated, andsimilar outcomes are achieved. In Fig. 21, the optimizationresults associated with the 3 3 array with three particles


Fig. 19. Probability density function of the final fitness value for four methods,with 700 runs in a 9D problem with 3 particles in 6 �-planes.

Fig. 20. Fitness value trend of the four methods averaged over 700 runs in a9D problem with 3 particles in 6 �-planes.

Fig. 21. Fitness value trend of the four methods averaged over 500 runs in a 9Dproblem with 3 particles in 6�-planes using absorbing wall boundary condition.

averaged over 500 runs, using absorbing boundary condition,are depicted. It is apparent that the conclusions of invisibleboundary condition can be extended to absorbing wall case. Theoverall results confirm that the proposed methods (Method2and Method4) yield superior performance for BPSO.

Finally, to underscore the time-consuming nature of the opti-mization process, 2.8, 3.4, and 5.5 hours were required for the700 runs of simple 4 4 array with only 2 particles for patternassessment in 6, 8 and 10 -planes, respectively, and angularresolution of degrees. To underscore the com-puting requirement, it is notable that adding two more particlesto 6 -plane case converts 2.8 hours of computing on a stan-dard 2.83 GHz and 3.23GB RAM PC to 5.25 hours. Assuming

a realistic software-defined thinned array scenario, a substan-tially smaller number of runs will be required. Thus, signifi-cance of a method or methods with achieved satisfactory resultsin a smaller number of iteration becomes quite evident.

VI. CONCLUSION

This paper has studied modified PSO algorithms used for op-timizing various problems. As the first modification to the con-ventional PSO, the effect of omitting personal best coefficient invelocity function, both in RPSO and BPSO, was studied. As thesecond suggested modification, the impact of modifying the ini-tial velocity on performance of PSO algorithm was investigated.

The modifications were shown to give better results forbenchmark mathematical optimizations with wide-range pa-rameters. In particular, superior performances were observedfor Rastigrin and Rosenbrock functions, as well as thinnedantenna array design. More specifically, we have shown thatthe use of the proposed algorithm will result in the reduction ofthe number of iterations by almost 18% to achieve considerablybetter final result in RPSO, while for BPSO, the proposedalgorithm resulted in a reduction in the optimized SLL valueof the antenna by 1–2 dB for non-symmetric thinned planararray. Although the SLL gain remained small for linear thinnedarrays, the convergence speed was higher when comparedwith the conventional method in both linear and planar arrays,making the proposed algorithm suitable for implementation insoftware-define radios.

APPENDIX A

In this Appendix, fitness function for designing a simple rect-angular patch antenna is examined in order to demonstrate thesimilarity between the studied benchmark functions and thoseof some practical problems. In fact, due to computation com-plexity, numerical tools are applied to execute the electromag-netic simulations of the class of problems discussed below.

Considering cavity model, which is introduced in various ref-erences, such as in [24], the fitness function shown in (15) can beused for a rectangular patch antenna design on a substratewith thickness of . The design objective is to minimize the re-flection at the feeding point by making the input impedanceas close as possible to a defined line impedance of . Let

(15)

be the impedance mismatch function. In this equation

(16)

where

and


In the above, is the feed position. and denotethe feeding area dimensions in and , respectively. m and ncorrespond to mode resonating inside the cavity with thepropagation constant . Furthermore, and denote theangular frequency, permeability and permittivity of the medium,respectively. The ground plane size is assumed to be infinite.

To have a better understanding, let’s assume the simplifiedcase of a square patch , with very small feeding area

, for which one can rewrite (15) as

(17)

where

and

Considering as an optimization parameter, the simi-larity of to benchmark functions becomes apparent. Generally,it is enough to consider the three first modes, i.e.,and . Therefore, the problem transforms to a 3-dimen-sional problem where three parameters are going to be discov-ered by the optimization. Having these three parameters, the realunknowns of the problem, , and , are calculated in a set ofthree equations.

In this problem, the parameters can normally vary in a widerange, especially for the higher order modes (any mode otherthan ) where increases in value because of the pos-sibility of dominator approaching zero. In addition, it is evidentthat the input impedance of the patch increases greatly whenbeing fed close to the edges, which further confirms high pa-rameter range in (17). In other words, this problem is an exampleof a practical design scenario in which an expeditious search to-wards minimum in a wide solution area is more desirable thana search for global optimum minimum.

REFERENCES

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Arezoo Modiri (S’09) received the B.S. degree fromUniversity of Tehran, Tehran, Iran and the M.S. de-gree from the Iran University of Science and Tech-nology, both in electrical engineering, in 2001 and2005, respectively. She is currently working towardthe Ph.D. degree at the University of Texas at Dallas.

Her research interests include antennas, radiationanalysis, optimization methods, body area networks,reflectometry, and computational electromagnetics.


Kamran Kiasaleh (SM’95) received the B.S. (cumlaude), M.S., and Ph.D. degrees from the Universityof Southern California (USC), Los Angeles, in1981, 1982, and 1986, respectively, all in electricalengineering.

In 1986, he joined Teknekron CommunicationSystem, Berkeley, CA, where he was responsible forconducting research on the advanced telecommuni-cations IC products, ASIC design (DS-1 Controller),and was involved in the development of networkcontrol management systems. He was also a Lecturer

in the Department of Electrical Engineering, San Francisco State University,San Francisco, CA, in 1986. Since 1987, he has been with the Erik JonssonSchool of Engineering and Computer Science, University of Texas at Dallas(UTD), Richardson, where he is currently a Full Professor. From Dec. 1996to Dec. 1997, He was on a special assignment with the DSP research anddevelopment center at Texas Instruments Inc., Dallas, where he conductedresearch on various aspects of the 3rd generation wireless communicationsystems. From Sep. 2000 to Jan. 2003, he was with Optical Crossing Inc. (OCI)where he was responsible for the development of the state-of-the-art free-spaceand millimeter-wave communications systems. His research interests include

SDR, synchronization, novel detection schemes for phase noise and fadingimpaired digital communication systems, optical communications, and wirelessas well as optical CDMA systems.

Dr. Kiasaleh was the recipient of the Research Initiation Award (RIA) fromthe National Science Foundation (NSF) and the NASA/ASEE faculty fellowshipaward at the Jet Propulsion Laboratory (JPL) in 1992, where he participated inthe Galileo Optical Experiment (GOPEX) demonstration, the first successfuldemonstration of an optical communications link involving a deep-space ve-hicle. For his participation in this project, he received the NASA Group Achieve-ment Award. In 1993, he was the recipient of NASA/ASEE faculty fellowshipaward at JPL where he participated in Compensated Earth-Moon-Earth LaserLink (CEMERLL) demonstration. Since 2006, he has led a research and de-velopment effort focused on various issues related to software-defined-radios(SDR) for first responders. This effort has been funded by the National Instituteof Justice (NIJ). He was an Associate Editor for IEEE Communications Letters(1999–2006). He is also a member of Eta Kappa Nu and Sigma Xi. He was afaculty advisor to the student chapter of the IEEE Society at the UTD and servedas the Vice President and the President of the Optical Society of North Texas(OSNT) in 1990–1991 and 1991–1992, respectively.

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Modification of Real-Number and Binary PSO Algorithms for Accelerated Convergence

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