Exploratory Toolkit for Evolutionary And Swarm based Optimization

Transcription

1 Exploratory Toolkt for Evolutonary And Swarm based Optmzaton Namrata Khemka, Chrstan Jacob Unversty of Calgary, Calgary, AB, Canada, T2N1N4 {khemka, Optmzaton of parameters or systems n general plays an ever ncreasng role n mathematcs, economcs, engneerng, and lfe scences. As a result, a wde varety of both tradtonal analytcal, mathematcal and non tradtonal algorthmc approaches have been ntroduced to solve challengng and practcally relevant optmzaton problems. Evolutonary optmzaton methodsnamely, genetc algorthms, genetc programmng, and evoluton strategesrepresent a category of non tradtonal optmzaton algorthms drawng nspratons from the process of natural evoluton. Partcle swarm optmzaton represents another set of more recently developed algorthmc optmzers nspred by socal behavours of organsms such as brds [8] and socal nsects. These new evolutonary approaches n optmzaton are now enterng the stage, and are thus far very successful n solvng real world optmzaton problems [12]. Although these evolutonary approaches share many concepts, each one has ts strengths and weaknesses. The best way to understand these technques s through practcal experence, n partcular on smaller scale problems or on commonly accepted benchmark functons. In [11], we descrbe how evoluton strateges and partcle swarm optmzers compare on benchmarks prepared for a much more complex optmzaton task regardng a knematc model of a soccer kck. The Mathematca notebooks that we created throughout these evaluaton experments and for the fnal desgn of the muscle control algorthms for the soccer kck are now also avalable through a webmathematca nterface. The new Evolutonary & Swarm Optmzaton web ste s ntegrated wth the collecton of notebooks from the EVOLVICA package, whch covers evoluton based optmzers from genetc algorthms and evoluton strateges to evolutonary programmng and genetc programmng. The EVOLVICA database of notebooks, along wth the newly added swarm algorthms, provde a large expermentaton and nqury platform for ntroducng evolutonary and swarm based optmzaton technques to those who ether wsh to further ther knowledge n the evolutonary computaton doman or requre a streamlned platform to buld prototypcal strateges to solve ther optmzaton tasks. Makng these notebooks avalable through a webmathematca ste means that anyone wth an nternet browser avalable wll have nstant access to a wde range of optmzaton algorthms. Avgnon, June th Internatonal Mathematca Symposum

2 2 Namrata Khemka & Chrstan Jacob 1. Introducton Evolutonary optmzaton methodsgenetc algorthms (GA) [7], genetc programmng (GP) [13], and evoluton strateges (ES) [14]are a branch of non tradtonal optmzaton methods drawng nspratons from the processes of natural evoluton. Partcle swarm optmzers (PSO), on the other hand, are nspred by the socal behavour of brd flockng [10]. Over the last two years, we have been nvestgatng the performance of evoluton and swarm based optmzers n the doman of bomechancs, whch we developed together wth the Human Performance Laboratory at the Faculty of Knesology, Unversty of Calgary [1],[2],[3]. In ths partcular bomechancal applcaton, numercal optmzaton algorthms are used to desgn equpment for sports actvtes. The nvolved smulatons of muscle movements are very tme consumng and hgh dmensonal, thus makng ther evaluatons costly and dffcult. Smulatng a soccer kck s an example of such a model whch nvestgates muscle actvaton patterns wthn the leg and foot when kckng a soccer ball towards the goal. The specfc objectve n ths case s to obtan a hgh ball speed; n order to mnmze the goal keeper s chances to catch the ball. In 1998, Cole appled a (1+ Λ) ES to ths model [1],[2]. More recently we presented mproved adaptatons of the model parameters through a partcle swarm optmzer [11][12]. The man focus of ths paper, however, s not on the optmzaton of parameters for the nvestgated soccer kck model. Instead, we present what we have learnt from our comparson studes of the evoluton and swarm based optmzers on a set of selected benchmark functons. These benchmark studes turned out to be extremely useful n understandng the ntrcaces n performance regardng three optmzers: (1) the orgnally used (1+ Λ) ES, (2) a canoncal ( basc ) PSO (bpso), and a (3) a partcle swarm optmzer wth nose nduced ( random ) nerta weght settngs (rpso). We wll descrbe and analyze the performance of each of these optmzers on fve benchmark functons n two, four, and ten dmensons and try to project these fndngs to performance characterstcs we have also found n the real world applcaton of the dscussed soccer kck model, whch poses a 56 dmensonal optmzaton problem. The Mathematca notebooks that we created, provde us wth nsghts regardng the relatons between control parameteres and system performance of optmzers under study. Consequently, we gan a better understandng of the algorthms on mult dmensonal real world problems. Ths paper s organzed as follows. In Secton 2, we gve descrptons of the three optmzaton algorthms used n our comparson. An ntroducton of the benchmark functons and an outlne of the expermental setups follows n Secton 3 and Secton 4, respectvely. We dscuss the expermental results, and summarze our lessons learnt n Secton 5. The accompanyng webmathematca ste s presented n Secton 6. Secton 7 concludes the paper. 2. The Three Contenders: ES, bpso, and rpso The three contenders for our comparatve study of evoluton and swarm based optmzaton algorthms are: (1) a relatvely smple (1+Λ) Evoluton Strategy, (2) a canoncal ( basc ) PSO (bpso), and (3) a partcle swarm optmzer wth nose nduced ( random ) nerta weght settngs (rpso). The followng subsectons present these approaches n more detal. 8th Internatonal Mathematca Symposum Avgnon, June 2006

3 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 3 1. (1 + Λ) Evoluton Strateges: ES Evoluton Strateges (ES) have been a successful evolutonary technque for solvng complex optmzaton problems snce the 1960 s [14]. ES evolves vectors of real numbers and the genetc nformaton s nterchanged between these vectors through recombnaton operators. Slght varatons ( mutatons ) on these vectors are obtaned by evolvng strategy parameters whch determne the amount of mutatons appled to each vector component. As stated above, the orgnal experments on the soccer kck smulaton were conducted by Cole usng a (1 + Λ) ES scheme [1], [2], [3]. As these experments were dffcult to repeat, we also used ths verson for our comparson test. In the (1 + Λ) ES scheme, a sngle parent s mutated Λ tmes. Each of the newly created offsprng are evaluated and the parents and the offsprng are added to the selecton pool. The sngle best ndvdual among the 1 Λ solutons n the pool survves and becomes the parent for the next teraton. Below we descrbe the (Μ/Ρ + Λ) strategy whch s a generalzaton of the (1 + Λ) scheme, where Μ parents generate Λ offsprng wth recombnaton of Ρ ndvduals. (Μ/Ρ + Λ) ES algorthm Step 1: Intalze the populaton of sze n by randomly assgnng locatons P p,, p,, p n and strategy parameters S s 1,, s,, sn. 1 Step 2: Generate Λ Μ offsprng by randomly selectng and recombnng Ρ ndvduals from the pool of Μ parents: p p rec = Χ(p 1 Step 3: Mutatons: p 1,..., p Ρ 1 ) where, p 1 j P rec = (Χ(p1, p k1 ),..., Χ(p d, p kd )). new := p rec + z where z := Σ(N 0 S 1... N 0 S d ) N Α S returns a Gaussan dstrbuted random value around Α wth varance S. Step 4: Evaluate the ftness of all ndvduals. Step 5: Select the Μ best ndvduals to serve as parents for the next generaton. Step 6: If the termnaton crteron s met: STOP, Otherwse, go to Step Basc Partcle Swarm Optmzaton: bpso As the basc partcle swarm optmzer (bpso) we use Eberhart s and Kennedy s orgnal PSO verson [10]. Inspred by both socal behavour and brd flockng patterns, the partcles fly through the soluton space and tend to land on better solutons. The search s performed by a populaton of partcles, each havng a locaton vector P p,, p,, p, and representng a potental soluton n a d dmensonal 1 j d search space. Each partcle also keeps track of ts velocty vector, Ν, whch determnes the drecton and how far a partcle wll move n the next teraton. The ftness of a partcle s determned by an evaluaton functon p. Partcles move through the search space n dscrete tme steps. In order to provde a balance between local (wth a hgher tendency to converge to soluton n close vcnty) and global search (lookng for Avgnon, June th Internatonal Mathematca Symposum

4 P p,, p,, p 1 j d Ν 4 Namrata Khemka & Chrstan Jacob p search space n dscrete tme steps. In order to provde a balance between local (wth a hgher tendency to converge to soluton n close vcnty) and global search (lookng for overall good solutons), an nerta weght Ω (step 5) was suggested by Eberhart [4],[5]. In algorthm bpso, ths term s set to a constant value of Ω = 1. bpso algorthm Step 1: Intalze the partcle populaton by stochastcally assgnng locatons P p,, p,, p n and veloctes V Ν 1,, Ν,, Ν n. 1 Step 2: Evaluate the ftness of all partcles: P p,, p,, p n. 1 Step 3: Keep track of the locatons where each ndvdual had ts hghest ftness so far: P p best 1,, p best,, p best n. Step 4: Keep track of the poston wth the global best ftness: p best max g p P p. Step 5: Modfy the partcle veloctes based on the prevous best and global best postons: Ν new Ω Ν φ 1 p best p φ 2 p best g p Step 6: Update the partcle locatons: p new p Ν new for 1 n. Step 7: If the termnaton crteron s met: STOP, Otherwse, go to Step Random Partcle Swarm Optmzaton: rpso for 1 n. Prevous work on PSO suggests that the so called nerta weght Ω should be annealed (dpso) over tme n order to obtan better results [6]. Ths tme decreasng nerta weght facltates a global search at the begnnng and the later small nerta weght fne tunes the search space. Snce the annealed value s dependent on tme, the number of teratons must be known n advance. However, n most real world scenaros, lke the soccer kck optmzaton, t s extremely dffcult to get to know the number of necessary teraton steps n advance. The random rpso verson tres to allevate ths problem by assgnng a random number to Ω (Step 5) n each teraton as follows: Ω 0.5 Random 2 Ths generates a unformly dstrbuted random number n the nterval [ 1 4, 3 4 ]. 8th Internatonal Mathematca Symposum Avgnon, June 2006

5 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 5 3. Benchmarks In the lght of the No Free Lunch Theorem [16], t s, dffcult to dentfy a clearly superor search or optmzaton algorthm n any comparson. Therefore our purpose s not to show whch one of the three algorthms s outperformng the others n any case, but to fnd out whch of these optmzers s better suted for specfc optmzaton challenges. In partcular, we also want to nvestgate whether an algorthm s performance characterstcs n two dmensons where vsualzaton and manual nspecton s easest and most accesble transfers to hgher dmensons. We evaluate the performance of (1 + Λ) ES scheme and both versons of the partcle swarm algorthms on a small set of numercal benchmark functons. We use the fve benchmark functons llustrated and descrbed n more detal below. We explore each of these benchmark search spaces for dmensons d = 2, d = 4, and d = 10. The frst three functons are unmodal, that s, wth a sngle global optmum. The last two functons are multmodal, where the number of local maxma ncreases exponentally wth the problem sze [15]. f 1 : Sphere d f 1 x 2 j, 5.12 x j 5.12, x 0. j 1 Ths s a smple, symmetrc, smooth, unmodal search space (nverted parabola), and s known to be easly solved by all algorthms. As n our case, t s manly used to calbrate control parameters of the optmzers. f 2 : Edge y x d f 2 j 1 x j d j 1 x j, 10 x j 10, x 0. Functon f 2 has shared edges makng t more dffcult to fnd good solutons around the rdges. Avgnon, June th Internatonal Mathematca Symposum

6 6 Namrata Khemka & Chrstan Jacob f 3 : Step y x d f 3 12 j 1 x j, 5.12 x j 5.12, x 0. We have ncluded the lnear surface functon, f 3,n order to see whether the alogrthms perform a gradent ascent strategy. f 4 : Ackley f y x d d 2 x j 1 j 1 d d j 1 Cos2Πx j 20, 30 x j 30, x 0. The Ackley functon s more dffcult as search algorthms tend to settle n any of the local optma, makng t more challengng to dentfy the global optmum x y f 5 : Grewangk d f j x j 2 d x j Cos j, 100 x j 100, x 0. j 1 8th Internatonal Mathematca Symposum Avgnon, June 2006

7 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 7 Grewangk s functon has hundreds of local optma above the sphere. We ncluded t to compare the algorthms performances on Grewangk and the sphere functon y x Expermental Setup For each of the three optmzers ES, bpso, and rpso we performed 20 experments on each of the fve test functons f 1 to f 5 for dmensons d = 2, d = 4, and d = 10. A dfferent random number seed s used for each of the 20 runs. The ntal ndvduals (ncludng those for ES) are unformly dstrbuted throughout the search space. Each ntal populaton, generated for an ES experment, s also used for the bpso and rpso experments. Ths ensures that all comparable runs start wth the same ntal dstrbuton of ndvduals. The termnaton crteron for all runs was to stop when the maxmum number of teratons t max = 1,500, was reached. By that tme all the optmzers had already reached ther convergence phase (Fgure 2). The parameter settngs for the three algorthms are descrbed n Tables 1, 2, and 3. populaton sze, n 10 locaton range, p j p low, p hgh vares velocty range, v j v low, v hgh 10 % of p j explotaton rate, φ exploraton rate, φ 1 1 Table 1. bpso parameter settngs populaton sze, n 10 locaton range, p j p low, p hgh vares velocty range, v j v low, v hgh 10 % of p j explotaton rate, φ exploraton rate, φ Table 2. rpso parameter settngs populaton sze selecton pool, 1 Λ 1 9 mutaton step sze radus 1 Table 3. ES parameter settngs Avgnon, June th Internatonal Mathematca Symposum

8 8 Namrata Khemka & Chrstan Jacob 5. Dscusson of the Results The results of the evoluton based optmzer and the two partcle swarm optmzaton technques are brefly dscussed n ths secton. 1. Phenotype Plots Fgure 1 gves an example of the populaton dynamcs resultng from each of the three algorthms (ES column 1, bpso column 2, and rpso column 3) appled over a certan number of teratons. The ndvduals are represented as dots and n each plot three teratons (red, blue, green) are depcted. In order to acheve a far comparson, all three algorthms start from the same ntal populatons. The behavour of the ndvduals s seen at dfferent teratons makng t easy to compare and contrast the movement of the ndvduals and study ther convergence behavour. For example, f 1 s plotted at teratons 2 (red), 7 (blue), and 40 (green). In comparson to the (1+Λ) ES scheme, we observe that the partcle swarm ndvduals (both bpso and rpso) search the soluton space more thoroughly and n multple drectons n comparson to the (1+Λ) ES scheme and have hgher exploraton capabltes. The ES ndvduals stay close to each other wthn a certan mutaton radus. Ths s a typcal effect of usng ths partcular scheme of ES. The ndvduals of algorthm ES converge to a local optmum soluton for functons f 4 and f 5. Ths also llustrates the fact that ES ndvduals exhbt strong local search behavours, whch n ths case s manly due to a relatvely small mutaton step sze. f 1 ES, bpso, rpso : Iteratons 2, 7, 40 f 2 ES, bpso, rpso : Iteratons 2, 6, 20 f 3 ES, bpso, rpso : Iteratons 2, 7, 60 f 4 ES, bpso, rpso : Iteratons 2, 20, 60 8th Internatonal Mathematca Symposum Avgnon, June 2006

9 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 9 f 5 ES, bpso, rpso : Iteratons 2, 20, 60 Fgure 1. Phenotypcal plots for the 2 dmensons versons of the benchmark functons f 1 to f 5 from top to bottom. From left to rght the columns show snapshots of typcal optmzaton runs for ES, bpso, and rpso. The populatons of the frst, second, and thrd snapshot are represented as red, blue, and green spheres respectvely. 2. Convergence Plots The convergence plots represent mean ftness values computed over all twenty runs for all 1,500 teratons. Ths graph helps n demonstratng the convergent behavour of the ndvduals of a partcular algorthm and also llustrates whch of the three algorthms has the fastest ftness convergence speed. Fgure 2 summarzes the results are shown for d = 2 (column 1), d = 4 (column 2), and d = 10 (column 3). In almost all cases, algorthm ES s comparable to rpso. Algorthm bpso turns out to be the slowest n terms of convergence speed for d = 2. However, as the number of dmensons ncreases (d = 4), the convergence rate of ES decreases, and n ten dmensons, ES converges more slowlyand towards a lower ftness values. All the three algorthms show relatvely steep ascents durng the frst 100 or 200 teratons. After whch rpso seemed to rapdly level off wthout makng any further progress (for most of the test cases) durng the rest of the smulaton.e., the swarm stagnates (no changes observed n terms of fndng a better ftness value). For nstance, on f 1, partcle swarms stagnate and flatten out wthout any further mprovements. However, the convergence rate of ES on functon f 3, gradually slowed down but dd not completely level off, whch ndcates that f t were allowed to run longer t may have dscovered better solutons. Another observaton made for functon f 5 n two dmensons s that rpso has the slowest convergence rate. Ths s n lne wth the results of the phenotype plot (Fgure 1) where the partcles do not converge to one locaton. f 1 ES, bpso, rpso : dmensons 2, 4, 10 Avgnon, June th Internatonal Mathematca Symposum

10 10 Namrata Khemka & Chrstan Jacob f 2 ES, bpso, rpso : dmensons 2, 4, 10 f 3 ES, bpso, rpso : dmensons 2, 4, 10 f 4 ES, bpso, rpso : dmensons 2, 4, 10 f 5 ES, bpso, rpso : dmensons 2, 4, 10 Fgure 2. Ftness plots of the benchmark functons f 1 to f 5 from top to bottom. From left to rght the columns llustrate the convergence behavours of the three algorthms (ES: blue, rpso: orange, bpso: green) n 2, 4, and 10 dmensons. 3. Success Plots The best ftness value obtaned at the end of each run s llustrated n Fgure 3. Here, for each functon, the ftness values of all 20 runs are plotted n ascendng order from left to rght. Therefore, each graph dsplays the success rate of each algorthm on a partcular functon. The best (rght most pont), worst (left most pont), and mean ftnesses can also be easly derved from these graphs. In all 20 runs, algorthm ES fnds worse solutons than both PSO algorthms for d = 2, 4, and 10. Ths s observed by the left most blue pont n Fgure 3. Ths s n lne wth the results of the phenotype plots (Fgure 1), especally for the multmodal functons f 4 and f 5, where the ES ndvduals are unsuccessful n fndng the global optmum. The hgher exploraton capabltes of the PSO algorthms seem to facltate the dscovery of better solutons n comparson to the local ES scheme. In the phenotype plots (Fgure 1) one can also observe that the bpso partcles do not converge to one soluton only. However, there s always at least one partcle that fnds the global optmum. Therefore, both PSO algorthms (bpso and rpso) are comparable n terms of fndng the best soluton, as llustrated by the rght most green and orange 8th Internatonal Mathematca Symposum Avgnon, June 2006

11 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 11 the global optmum. Therefore, both PSO algorthms (bpso and rpso) are comparable n terms of fndng the best soluton, as llustrated by the rght most green and orange ponts n Fgure 3. f 1 ES, bpso, rpso : dmensons 2, 4, 10 f 2 ES, bpso, rpso : dmensons 2, 4, 10 f 3 ES, bpso, rpso : dmensons 2, 4, 10 f 4 ES, bpso, rpso : dmensons 2, 4, 10 f 5 ES, bpso, rpso : dmensons 2, 4, 10 Fgure 3. Success rato plots of the benchmark functons f 1 to f 5 from top to bottom. From left to rght the columns show the best ftness value obtaned at the end of each run of the three algorthms (ES: blue, rpso: orange, bpso: green) n 2, 4, and 10 dmensons. Avgnon, June th Internatonal Mathematca Symposum

12 12 Namrata Khemka & Chrstan Jacob 4. Parameter Range Plots For the followng analyses we only look at algorthm performance on the 10 dmensonal benchmarks. In Fgure 4 we vsualze how for each varable range changes durng the course of the evolutonary search. For example, n the frst row of Fgure 4 the vertcal bars represent the range for each of the ten varables, over all teratons, lmted to all those solutons that have a ftness of at least Τ 5 (the hghest ftness s zero). Knowng how the value ranges change s mportant especally when explorng real world optmzaton problems, snce t can provde nsghts on whether the ftness functon s senstve wth respect to a partcular varable or not. It also clearly vsualzes the stage of convergence n each dmenson. Fgure 4 shows that algorthm ES mantans a hgh parameter range n comparson to the partcle swarm algorthms. Ths s n lne wth the results of Fgure 2, where ES has the slowest convergence speed for d = 10. Evoluton strateges seem to consstently keep wder parameter ranges. Both partcle swarm algorthms show comparable ranges. 8th Internatonal Mathematca Symposum Avgnon, June 2006

13 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 13 f 1 ES, bpso, rpso : Τ 5 f 2 ES, bpso, rpso : Τ 5 f 3 ES, bpso, rpso : Τ 1 f 4 ES, bpso, rpso : Τ 5 f 5 ES, bpso, rpso : Τ 5 Fgure 4. Parameter range plots for the 10 dmensonal versons of the benchmark functons f 1 to f 5 from top to bottom. From left to rght the columns show the parameter range for each of the ten dmensons over a certan threshold value (Τ) for ES, bpso, and rpso. Avgnon, June th Internatonal Mathematca Symposum

14 14 Namrata Khemka & Chrstan Jacob 6. "Evolutonary Swarms" webmathematca webste The comparsons conducted provde knowledge and nsghts about the algorthms, whle llustratng the movement of the ndvduals n the soluton space (Fgure 1). They pont out varous characterstcs lke the convergence speed and the success an algorthm has n fndng good solutons. The strengths and weaknesses of the algorthms are exploted, for example, PSO algorthms have a hgher exploraton rate whereas the (1 + Λ) scheme of ES depcts a local search scheme. As stated earler, we have created a set of notebooks to compare (1 + Λ) scheme of ES and both the partcle swarm optmzers on the benchmark functons ( f 1 f 5 ). We provde the nterface to the users for creatng the above graphs (Fgure 1, Fgure 2, Fgure 3, and Fgure 4). We have converted these notebooks to a webmathematca nterface. Ths nteractve ste provdes the hands on experence and an expermental envronment through a selecton of ten benchmark functons along wth the vsualzaton tools (for generatng the mages). Ths ste currently conssts of partcle swarms (PSO) and ts three varantsbasc partcle swarms (bpso), random partcle swarms (rpso), and partcle swarms wth decreasng nerta weght (dpso). We also mplemented both the smple schemes of ES, the (1 + Λ) scheme of ES and the (1, Λ) scheme of ES. The generalzed evoluton strateges, (Μ + Λ) and (Μ, Λ), as descrbed n Secton 2. As t s n general dffcult to know the settngs of varous parameters, we provde suggestons for dfferent settngs. Ths wll help the users to gan further knowledge regardng these optmzers. The webste can be accessed at: 7. Concluson The best way of understandng evolutonary and swarm based algorthm heurstcs s through practcal experence, n partcular on smaller scale problems. The Evolutonary Swarms ste that we have developed wll be ntegrated wth the collecton of notebooks from the EVOLVICA package. Ths database of notebooks, along wth the swarm algorthms, provde a large expermental and nqury platform for ntroducng evolutonary and swarm based optmzaton technques to those who wsh to further ther knowledge n the evolutonary computaton doman. Makng these notebooks avalable through a webmathematca ste means that anyone wth access to the newly bult web pages wll have nstant access to a wde range of optmzaton algorthms. References [1] Cole G., Gerrsten K., Influence of mass dstrbuton n the shoe and plate stffness on ball velocty durng a soccer kck, Addas Salomon AG, [2] Cole G., Gerrsten K., Optmal mass dstrbuton and plate stffness of football shoes, Addas Salomon AG, [3] Cole G., Gerrsten K., Influence of medo lateral mass dstrbuton n a soccer shoe on the deflecton of the ankle and subtabular jonts durng off centre kcks, Addas Salomon AG, th Internatonal Mathematca Symposum Avgnon, June 2006

15 Exploratory Toolkt for Evolutonary and Swarm based Optmzaton 15 [4] Eberhart R.C. andsh Y., Parameter selecton n partcle swarm optmzaton. Evolutonary programmng VII: Proceedngs of the seventh annual conference on evolutonary programmng, [5] Eberhart R.C. andsh Y., A modfed partcle swarm optmzer. Proceedngs of IEEE congress on evolutonary computaton (CEC 1998), [6] Eberhart R.C. and Sh Y., Comparon between genetc algorthms and partcle swarm optmzaton. Evolutonary programmng VII: Proceedngs of the seventh annual conference on evolutonary programmng, [7] Goldberg D., Genetc Algorthms n search, optmzaton, and machne learnng,. Addson Wesley [8] Jacob C., Khemka N., Partcle Swarm Optmzaton n Mathematca An Exploraton Kt for Evolutonary Optmzaton, IMS 04, Proc. Sxth Internatonal Mathematca Symposum, Banff, Canada (2004). [9] Jacob C., Illustratng Evolutonary Computaton wth Mathematca, Morgan Kaufmann Publshers, [10] Kennedy J. and Eberhart R.C., Partcle swarm optmzaton, Proceedngs of IEEE Internatonal Conference on Neural Networks, [11] Khemka N., Comparng PSO And ES: benchmarks and applcaton, M.Sc. Thess, UofCalgary, December [12] Khemka N., Jacob C., Cole.G., Makng soccer kcks better: A study n PSO and ES, Proceedngs of IEEE congress on evolutonary computaton (CEC 2005), [13] Koza J., Genetc Programmng On the programmng of computers by means of natural selecton, MIT press, [14] Rechenberg I., Evoluton Strateges: Nature s way of optmzaton, Optmzaton methods and applcatons, possbltes and lmtatons, 47, Lecture Notes n Computer Scence,1989. [15] Schwfel H.P., Evoluton and optmum seekng, John Wley and Sons, [16] Wolpert D.H. and Macready W.G., No free lunch theorems for search, Santa Fe Insttute, Avgnon, June th Internatonal Mathematca Symposum