Roaming Behavior of Unconstrained Particles

Transcription

1 23 BRICS Cogress o st Computatioal BRICS Coutries Itelligece Cogress & o th Computatioal Brazilia Cogress Itelligece o Computatioal Itelligece Roamig Behavior of Ucostraied Particles AP Egelbrecht Departmet of Computer Sciece Uiversity of Pretoria Pretoria, South Africa egel@cs.up.ac.za Abstract It has bee show recetly that ucostraied particles that follow the positio ad velocity update rules of a stadard global best particle swarm optimizatio algorithm leave the boudaries of the search space withi the first few iteratios of the search process. Provided that a better solutio does ot exist outside of the search boudaries, these roamig particles are evetually pulled back withi the search boudaries. This article illustrates the cosequece of roamig particles should better solutios exist outside of the search boudaries, amely that particles are pulled outside of the search boudaries ad that such ifeasible solutios are foud. The article also evaluates the hypothesis that it is the roamig behavior of ucostraied particles that improves the ability of particle swarm algorithms to locate feasible solutios outside of the particle iitializatio space. I. INTRODUCTION Recetly, Egelbrecht [] provided empirical evidece that particles that are ot subject to ay boudary costraits leave the boudaries of the search space withi the first iteratios of the optimizatio process. The study i [] assumed that a better solutio does ot exist outside of the boudaries defied for the optimizatio problem. Uder this assumptio, it was show that particles are evetually pulled back ito the search boudaries to coverge o a feasible solutio withi the search boudaries. This behavior of ucostraied particles was referred to as roamig behavior, ad it was poited out that, although particles are pulled back ito feasible space, much computatioal effort is wasted by searchig outside of the search boudaries. While ot formally ivestigated, it was poited out i [] that this roamig behavior may result i ifeasible solutios should a better solutio exsit outside of the search boudaries ad if o boudary costrait mechaism is employed to esure feasible particle positios. This article follows o the work preseted i [] to allude to the dager of ifeasible solutios due to particle roamig behavior. Should a better solutio exist outside of the search boudaries, ucostraied particles are pulled outside of the search boudaries, due to persoal best ad the global best positios beig outside of the search boudaries. Empirical evidece of ifeasible solutios due to particle roamig behavior is provided. Note that the mai objective of this study is to emphasize the cosequeces of particle roamig behavior, so that particle swarm optimizatio (PSO) practisioers are aware of this. The aim is ot to suggest remedies to prevet roamig behavior, as may boudary costrait mechaisms exist to prevet particles from leavig the search space [2], [3], [4]. As a secod objective, this study shows that it is the roamig behavior of particles that allow particles to fid feasible solutios outside of the particle iitializatio space, but still withi the search boudaries. The first behavior is illustrated for a stadard global best (gbest) PSO [5], the guarateed covergece PSO () [6], the (CPSO) [7], [8], ad the bareboes PSO () [9]. Due to space costraits, results for the secod behavior are preseted for oly the ad the. The remaider of this paper is orgaized as follows: Sectio II provides a compact summary of the algorithms cosidered i this study. A overview of the roamig behavior of particles is provided i Sectio III. Sectio IV provides empirical evidece that ifeasible solutios are foud by ucostraied particles should better solutios exist outside of the search boudaries. Sectio V provides empirical evidece that the roamig behavior has the advatage that solutios withi bouds, but outside of the particle iitializatio space ca be foud. II. PARTICLE SWARM OPTIMIZATION ALGORITHMS The stadard PSO as itroduced by [5] implemeted oe of two eighborhood topologies: either a star topology where a particle s eigborhood is the etire swarm, or a rig eighborhood topology where a particle s eighborhood is defied by its immediate eighbors. This study cosiders PSO algorithms that use the star topology. Uder this assumptio, for the gbest PSO, particle positios x i are updated usig x i (t +)=x i (t)+v i (t +) () ad the velocities are updated usig the iertia weight model [] as follows v ij (t +) = wv ij (t)+c r j (t)[y ij (t) x ij (t)] + c 2 r 2j (t)[ŷ j (t) x ij (t)] (2) where w is the iertia weight, v ij (t) is the velocity of particle i i dimesio j =,..., x, x ij (t) is the positio of particle i i dimesio j, y ij (t) is particle i s persoal best positio i dimesio j, ŷ j (t) is the global best positio i dimesio j, c ad c 2 are positive acceleratio costats used to scale the cotributio of the cogitive ad social compoets respectively, ad r j (t),r 2j (t) U(, ) are radom values i the rage [, ], sampled from a uiform distributio /3 $3. 23 IEEE DOI.9/BRICS-CCI.&.CBIC /BRICS-CCI-CBIC

2 It is importat to ote that this study selects the global best positio as the best persoal best positio, hece usig a memory-based global best update. Particle positios are iitialized withi the give domai ad velocities are iitialized to zero []. No boudary costrait mechaism is employed for the first part of the study, i.e. to illustrate that roamig particles will fid better positios outside of the bouds if they exist. For the secod part of the study, i.e. to show that particles have the ability to fid solutios outside of the iitializatio space (but withi the boudaries), a costrait is imposed o persoal best positios: A persoal best positio is oly updated if the ew positio is better ad withi the search boudaries. Due to the memory-based global best update, this persoal best costrait guaratees that the global best postio will be feasible. The guarateed covergece PSO () developed by [6] forces the global best particle to search withi a cofied regio for a better positio, prevetig early stagatio of particles. Let τ be the idex of the global best particle, so that y τ = ŷ. chages the positio update of the global best particle to x τj (t +)=ŷ j (t)+wv τj (t)+ρ(t)( 2r 2 (t)) (3) which is obtaied usig equatio () if the velocity update of the global best particle chages to v τj (t+) = x τj (t)+ŷ j (t)+wv τj (t)+ρ(t)( 2r 2j (t)) (4) where ρ(t) is a scalig factor that determies the size of the boudig box aroud the global best particle withi which a better positio is search for [6]. Note that oly the global best particle is adjusted accordig to Equatios (3) ad (4); all other particles use Equatios () ad (2). Clerc [7] ad Clerc ad Keedy [8] developed the costrictio PSO (CPSO), where the velocity update of Equatio (2) chages to v ij (t+) = χ[v ij (t)+φ (y ij (t) x ij (t))+φ 2 (ŷ j (t) x ij (t))] (5) with the costrictio coefficiet, 2κ χ = 2 φ (6) φ(φ 4) where φ = φ + φ 2,φ = c r, ad φ 2 = c 2 r 2. Equatio (6) is used uder the costraits that φ 4 ad κ [, ]. I the (), proposed by Keedy [9], particles do ot follow search trajectories, but are sampled from a Gaussia distributio cetered aroud the average of the particle s persoal best positio ad the global best positio. If it is assumed that c = c 2, the velocity update chages to ( yij (t)+ŷ ij (t) v ij (t +) N 2 The positio update chages to ),σ,σ= y ij (t) ŷ ij (t) (7) x ij (t +)=v ij (t +) (8) III. ROAMING BEHAVIOR Helwig ad Waka [] coducted a theoretical aalysis of iitial particle swarm behavior, providig formal proofs that most particles leave the boudaries of the search space withi the first iteratios. Empirical evidece of this roamig behavior of ucostraied particles was provided by Egelbrecht [], ad the followig cosequeces of particle roamig behavior were idetified: Should a better solutio be foud outside of the defied bouds of the optimizatio problem, ad o boudary costrait mechaism employed, persoal best positios are also pulled outside the bouds of the search space. The cosequece of these ifeasible solutios is that global best positios are also pulled outside of the search space, resultig i a ifeasible optimum beig foud. If roamig particles do ot fid better solutios outside of the search space, they are evetually pulled back ito feasible space. However, this is doe over a large umber of iteratios, wastig effort searchig ifeasible space. Roamig particles have a egative ifluece o swarm diversity calculatios, sice diversity icreases as particles move further outside the bouds of the search space. This is a problem for PSO algorithms that use measures of diversity to cotrol the search trajectories of particles. A additioal cosequece of the roamig behavior of particles, as idetified i this study, is that it eables particles to locate solutios outside of the particle iitializatio space, but withi boudary costraits provided that some form of boudary costrait hadlig mechaism is employed. The focus of this study is to provide empirical evidece to illustrate the disadvatage of fidig ifeasible solutios if ucostraied particles are used ad the advatage of fidig feasible solutios outside of the iitializatio space if costraied particles are used. IV. INFEASIBLE SOLUTIONS OUTSIDE OF BOUNDARIES The purpose of this sectio is to provide empirical evidece that particles that are ot subject to a boudary costrait will fid a better, ifeasible solutio outside of the search space boudaries should such better solutio exsit. This is as a direct result of the roamig behavior of particles. Sectio IV-A summarizes the experimetal procedure, while Sectio IV-B provides ad discusses the obtaied results. A. Experimetal Procedure The cosequece of particle roamig behavior is illustrated for the,, CPSO, ad. For this purpose, bechmark fuctios have bee used as summarized i Table I. Note that the domais of these fuctios were selected to esure that better miima exist outside of the bouds. Each of the fuctios were evaluated i 3 dimesios. Shiftig a fuctio, f l, was implemeted usig f S l (x) =f l (z)+β where z = x γ ad β is a costat. Rotatio was doe usig Salomo s method [2], with z = x γm, where M is 5

3 the rotatio matrix. A ew rotatio matrix was computed for each idepedet ru of the algorithm. The rotated fuctio is referred to as fl R, computed by multiplyig the decisio vector x with the traspose of the rotatio matrix. Details for the shifts ad rotatios are as follows: Shifted rotated Ackley: A liear rotatio matrix was used with a coditio umber of, β = 4, ad γ = 32. Shifted rotated Griewak: A liear rotatio matrix was used with a coditio umber of 3, β = 8 ad γ = 6. Shifted Norwegia: β =ad γ =. Shifted Rosebrock: β = 39 ad γ = 6. Shifted Schwefel.2: β =ad γ = 5. Shifted Spherical: β = 45 ad γ = 2. For all of the algorithms, swarm sizes of 3 particles were used, ad each algorithm executed for 5 idepedet rus of 5 iteratios. The cotrol parameters were set to the same values as follows: The iertia weight was set to , while the values of the acceleratio coefficiets were set to This choice is based o [3], where it was show that such parameter settigs facilitate coverget behaviour. For the CPSO, κ =. ad c = c 2 =2.5. These values are equivalet to the iertia weight ad acceleratio coefficiet values as give above for the other PSO algorithms. B. Results ad Discussio Table II summarizes the fitess of the best solutio foud at the last iteratio ad the swarm diversity as averages ad deviatios over the 5 idepedet rus for each fuctio ad algorithm. Figures ad 2 illustrate fitess, diversity, ad percetage positio violatios for the Ackley ad Eggholder fuctios respectively. These figures are represetative of behaviors observed for the other fuctios. The first major observatio from the table ad these figures is that swarm diversity remaied very high after 5 iteratios despite the fact that the average fitess of the global best particles coverged. This observatio exlcudes the Eggholder ad Schwefel 2.26 fuctios for which a average fitess of -INF was obtaied (this is due to the fact that particles cotiue to move outside of the defied boudaries to better positios, due to the shape of these fuctios), ad a icomputable diversity. For these fuctios, particles cotiued to accelerate to positios of large egative fitess for each iteratio. Figure 2(a) shows that swarm diversity for the Eggholder fuctio exploded to extremely large values withi the first few iteratios. These large swarm diversity values are idicative of large dispersio of the particles, ad the fact that swarm diversity values are sigificatly larger tha the extet of the domais poits to the roamig behavior of particles beig pulled further away from the search boudaries. The percetage of particles that violate boudaries, for at least oe dimesio, supports this observatio. The cosequece of particle roamig is clearly illustrated i the average percetage of persoal best positios that violate boudary costraits ad the percetage of simulatios for which the global best positio violates boudary costraits. I additio to the Ackley ad Eggholder profiles, Figure 3 illustrates the global best positio violatios for the other fuctios. Note how the percetage of simulatios for which the global best positio violates boudary costraits reached % withi the first few iteratios. This also implies that all the persoal best positios will violate the boudary costraits. These results provide clear support for the observatio that particles fid better solutios outside of the search boudaries if such better solutios exist. It is iterestig to ote that oly for the Salomo fuctio did the diversity decrease over time, albeit still to a large value. For all the other fuctios, diversity cotiued to icrease, idicatig that particles were movig further away from oe aother. Note that diversity was caluclated as the average distace that particles are from the swarm ceter. V. FEASIBLE SOLUTIONS OUTSIDE OF INITIALIZATION SPACE The aim of this sectio is to illustrate that the roamig behavior of boudary costraied particles allows particles to fid feasible solutios outside of the particle iitializatio space. Sectio V-A provides the experimetal procedure, ad Sectio V-B presets ad discusses the results. A. Experimetal Procedure This sectio empirically aalyzes the ability of PSO algorithms to locate optima outside of the rage i which particle positios have bee iitialized. The hypothesis is that the roamig behavior of particles facilitate exploratio outside of the iitializatio bouds to obtai the best solutio, but still withi the problem boudaries. For the purposes of this study, persoal best positios were costraied to remai withi the problem boudaries, which will esure that global best positios are feasible due to the memory-based global best update strategy used. The algorithms used the same cotrol parameter values as give above. Oly the ad results are preseted. The ad CPSO exhibited similar behavior. The fuctios used for this study iclude the Ackley ad Salomo fuctios defied i Table I, but for domais of [ ,32.768] ad [-,] respectively. I additio to these fuctios, the fuctios defied i Table III have bee used. The rotatios for the Griewak ad Rosebrock fuctios were doe usig a radomly geerated orthoormal matrix with coditio umber of oe. All of the fuctios except the Eggholder ad Schwefel 2.26 fuctios have their global miimum at x =(,,...,), with f(x )=. The global miimum of the Schwefel 2.26 fuctio is at x = ( ,..., ), with f(x )= The global miimum of the Eggholder fuctio is located ear the maximum boudary of the search space. For all of the fuctios, except the Eggholder ad Schwefel 2.26 fuctios, the itializatio rage for particle positios were take as 5%, %, 5%,..., % of the extet of the problem domai, startig from the maximum value. For both the Eggholder ad Schwefel 2.26 fuctios, the global miimum is located close to the maximum boudary of the 6

4 Average Diversity of Best Solutio Average Particle Boudary Violatios (a) Fitess. (b) Diversity. (c) Particle Boudary Violatios Average Pbest Boudary Violatios pbest PSO (d) Persoal Best Boudary Violatios (e) Global Best Boudary Violatios Fig.. Ackley Fitess, Diversity, ad Boudary Violatio Profiles for Ifeasible Solutios Study Average Diversity of Best Solutio 5e+6 4.5e+6 4e+6 3.5e+6 3e+6 2.5e+6 2e+6.5e+6 e Average Particle Boudary Violatios Average Pbest Boudary Violatios pbest PSO (a) Diversity (b) Particle Boudary Violatios (c) Persoal Best Boudary Violatios (d) Global Best Boudary Violatios Fig. 2. Eggholder Fitess, Diversity, ad Boudary Violatio Profiles for Ifeasible Solutios Study 7

5 (a) Shifted Rotated Ackley (b) Shifted Rotated Griewak (c) Shifted Norwegia (d) Shifted Rosebrock (e) Shifted Schwefel (f) Schwefel 2.26 (g) Salomo (h) Shifted Spherical Fig. 3. Global Best Boudary Violatio Profiles for Ifeasible Solutios Study search space. That is, for the 5% iitializatio rage, for the Griewak fuctio, particles were uiformly iitialized i the rage [54, 6] 3, for the % iitializatio rage particle were uiformly iitialized i the rage [48, 6] 3, ad so o. B. Results ad Discussio Figure 4 illustrates the average best fitess obtaied over the 5 idepedet rus for the ad. Also idicated i the figures is the deviatio i best fitess over the 5 rus. For the rotated Griewak, rotated Rosebrock, Salomo, ad Schwefel.2 fuctios, both algorithms showed o sigificat differece i average best fitess over the differet iitializatio rages. For all of the iitializatio rages, the average best fitess values were very similar, with small deviatios except for the 5% iitializatio rage. For iitializatio rages less tha 5%, which exclude the global miimum, the algorithms were able to locate better solutios outside of the iitializatio rage. For the Ackley fuctio, iitializatio rages 5% to 35% showed o sigificat differece i average best fitess, at a high average value of just below 2. From iitializatio rage of 4% ad higher, solutios with a average fitess of aroud 2.5 were obtaied. The Eggholder fuctio showed a similar tred, but with performace improvig oly for larger iitializatio rages. Note that the global miimum is located close to the maximum boudary of the search space, ad that particles for small iitializatio rages are iitialized very far from the global miimum. Despite this, particles were able to locate positios of better quality outside of the iitializatio rage. The Schwefel 2.26 fuctio also has its global miimum 8

6 located close to the maximum boudary. For this fuctio all iitializatio rages resulted i good solutios, with o sigificat differece i performace. Note that iitializatio i the etire domai resulted i the worst performace, clearly idicatig success i locatig better solutios outside of iitializatio rages. The Schaffer 6 fuctio exhibited a iterestig behavior, with performace deterioratig with icrease i the iitializatio rage. The best performace was obtaied for a small iitializatio rage of 5% with iitial particle positios far from the global miimum. VI. CONCLUSIONS The mai objective of this paper was to poit out a disadvatage of the roamig behavior of particles i stadard particle swarm optimizatios. If o boudary costrait mechaism is employed, ad if a solutio of better quality exists outside of the boudaries, the roamig particles pull their persoal best positios, ad cosequetly the global best positio, outside of the boudaries ito ifeasible space, ever to retur to feasible space. The importat outcome of this observatio is that a boudary costrait mechaism has to be used to esure that a solutio withi the boudaries defied for the optimizatio problem is foud. I additio to this observatio, empirical results were preseted to show a advatage of the roamig behavior of particles, i that better positios outside of the iitializatio rage ca be obtaied. Should a boudary costrait mechaism be used, the such solutios will be feasible. This fidig shows that PSO ca obtai good solutios eve if the optimum does ot lie withi the iitializatio rage. The ability of particles to locate solutios outside of the iitializatio rage is importat for optimizatio problems that do ot have boudary costraits, such as eural etwork traiig. The roamig behavior of particles may also play a positive role i costraied optimizatio problems where pockets of feasible space exist withi ifeasible space, as roamig may aid i particles crossig ifeasible space ito feasible space. These advatages to roamig behavior will be ivestigated i future studies. Additioally, it will be iterestig to aalyze roamig behavior for evolutioary algorithms such as differetial evolutio ad evolutioary programmig. REFERENCES [] A. Egelbrecht, Particle Swarm Optimizatio: Velocity Iitializatio, i Proceedigs of the IEEE Cogress o Evolutioary Computatio. IEEE Press, 22. [2] S. Cheg, Y. Shi, ad Q. Qi, Experimetal Study o Boudary Costraits Hadlig i Particle Swarm Optimizatio: From Populatio Diversity Perspective, Iteratioal Joural of Swarm Itelligece Research, vol. 2, o. 3, pp , 2. [3] W. Chu, X. Gao, ad S. Sorooshia, Hadlig Boudary Costraits for Particle Swarm Optimizatio i High-Dimesioal Search Space, Iformatio Scieces, vol. 8, o. 2, p , 2. [4] X.-F. Xie ad D.-C. Bi, Hadlig Boudary Costraits for Numerical Optimizatio by Particle Swarm Flyig i Periodic Search Space, i Proceedigs of the IEEE Cogress o Evolutioary Computatio, vol. 2, 24, pp [5] J. Keedy ad R. Eberhart, Particle Swarm Optimizatio, i Proceedigs of the IEEE Iteratioal Joit Coferece o Neural Networks. IEEE Press, 995, pp [6] F. va de Bergh ad A. Egelbrecht, A New Locally Coverget Particle Swarm Optimizer, i Proceedigs of the IEEE Iteratioal Coferece o Systems, Ma, ad Cyberetics. IEEE Press, 22, pp. 96. [7] M. Clerc, The Swarm ad the Quee: Towards a Determiistic ad Adaptive Particle Swarm Optimizatio, i Proceedigs of the IEEE Cogress o Evolutioary Computatio, vol. 3, July 999, pp [8] M. Clerc ad J. Keedy, The Particle Swarm-Explosio, Stability, ad Covergece i a Multidimesioal Complex Space, IEEE Trasactios o Evolutioary Computatio, vol. 6, o., pp , 22. [9] J. Keedy, Bare Boes Particle Swarms, i Proceedigs of the IEEE Swarm Itelligece Symposium. IEEE Press, April 23, pp [] Y. Shi ad R. Eberhart, A modified particle swarm optimizer, i Proceedigs of the IEEE Cogress o Evolutioary Computatio. IEEE, 998, pp [] S. Helwig ad R. Waka, Theoretical Aalysis of Iitial Particle Swarm Behavior, i Proceedigs of the Teth Iteratioal Coferece o Parallel Problem Solvig from Nature, 28, pp [2] R. Salomo, Reevaluatig geetic algorithm performace uder coordiate rotatio of bechmark fuctios, BioSystems, vol. 39, pp , 996. [3] R. Eberhart ad Y. Shi, Comparig Iertia Weights ad Costrictio Factors i Particle Swarm Optimizatio, i Proceedigs of the IEEE Cogress o Evolutioary Computatio, vol., July 2, pp

7 (a) Ackley (b) Eggholder (c) Rotated Griewak (d) Rotated Rosebrock 3 25 (e) Salomo (f) Schaffer (g) Schwefel (h) Schwefel 2.26 Fig. 4. Global Best Boudary Violatio Profiles for Ifeasible Solutios Study Fuctio Domai Fuctio Defiitio.2 Ackley [,32.768] f(x) = 2e.2 Shifted rotated Ackley [ ,] f(x) = 2e Eggholder [-52,52] f(x) = x j= Shifted rotated Griewak [,6] f(x) =+ 4 ( Shifted Norwegia [-.,.] f(x) = x j= Shifted Rosebrock [-3,3] f(x) = x j= Shifted Schwefel.2 [,] f(x) = x Schwefel 2.26 [-5,5] f(x) = x j= Shifted Spherical [,] f(x) = x j= z2 i TABLE I BENCHMARK FUNCTIONS USED FOR INFEASIBLE SOLUTIONS STUDY x j= x2 j e x j= cos(2πx j ) +2+e x j= z2 j x e j= cos(2πz j ) ( +2+e ) (xj+ + 47) si( x j+ + x j /2+47 ) + si( x j (x j+ + 47) )( x j ) ( ) x j= z2 j x j= cos zj j ( )) cos(πzj 3) 99+zj ( (zj+ zj 2)2 +(z j ) ( 2) j ) 2 j= k= z k ( ( )) xj si x j Salomo [-,5] f 4 (x) = cos(2π x j= x2 j )+. x j= x2 j +

8 TABLE II FITNESS AND DIVERSITY RESULTS FOR INFEASIBLE SOLUTIONS STUDY Fitess Diversity Fuctio Algorithm Average Deviatio Average Deviatio Ackley.5E+ 6.93E+ 2.9E+3.3E+3.54E+ 6.68E+ 2.E+3.23E+3 CPSO.55E+ 7.E+ 2.2E+3.24E E+ 9.9E+ 6.5E E+4 Shifted -.9E+2 6.7E-2 5.8E E+3 Rotated -.9E E-2 4.5E+3 2.E+3 Ackley CPSO -.9E E E E+3 -.9E+2 4.8E E E+4 Eggholder -INF INF CPSO -INF INF Shifted -.8E+2.53E E+3 2.7E+ Rotated -.8E+2.37E E+3 3.6E+ Griewak CPSO -.8E+2.32E E E+ -.8E+2.2E E E+ Shifted -4.43E E+34.4E E+39 Norwegia -2.27E+.6E+.55E+6.9E+7 CPSO -4.72E E E E+23 -INF -.5E E+24 Shifted 4.44E+2.6E E+3 4.3E+2 Rosebrock 4.83E+2.77E E+3 3.9E+2 CPSO 4.7E E+2 9.6E+3.23E E+2 5.7E+2.25E+4 9.6E+3 Salomo 6.32E- 3.8E-.82E+2.E E- 2.73E-.9E E+ CPSO 5.8E- 2.57E-.67E E+ 3.64E-.38E-.E+2 4.8E+ Shifted 5.78E-8.3E E+3.75E- Schwefel E E E E- CPSO 7.82E-8.79E E+3 4.8E-.39E-.57E- 7.99E+3.7E+2 Schwefel 2.6 -INF INF CPSO -INF INF Shifted -4.5E+2.37E-3 3.8E+3.7E-6 Spherical -4.5E+2 7.3E-4 3.8E+3.83E-6 CPSO -4.5E+2 4.8E-4 3.8E+3.78E-6-4.5E E-4 3.8E+3.44E-6 TABLE III BENCHMARK FUNCTIONS USED FOR FEASIBLE SOLUTIONS STUDY Fuctio Domai Fuctio Defiitio ( ) Rotated Griewak [-6,6] f(x) =+ x 4 j= z2 j x j= cos zj j Rotated Rosebrock [-,] f(x) = x ( (zj+ z 2 j= j )2 +(z j ) 2) ( ) Schaffer 6 [-,] f(x) = x.5+ si2 (x 2 j +x2 j+ ).5 j= (+.(x 2 j +x2 j+ ))2 Schwefel.2 [-,] f(x) = x ( j ) 2 j= k= x k