Applied Mathematics and Computation

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1 Aled Mathematcs ad Comutato 04 (008) Cotets lsts avalable at SceceDrect Aled Mathematcs ad Comutato oural homeage: A dervato of the umber of mma of the Grewak fucto Hudae Cho a, Fracsco Olvera a, *, Seth D. Gukema b a Deartmet of Cvl Egeerg, Teas A&M Uversty, 3136 TAMU, College Stato, TX , Uted States b Deartmet of Geograhy ad Evrometal Egeerg, Johs Hoks Uversty, Baltmore, MD, Uted States artcle fo abstract Keywords: Grewak fucto Local mma Otmzato Mult-modal otmzato The Grewak fucto s commoly used to test the ablty of dfferet soluto rocedures to fd local otma. It s mortat to kow the eact umber of mma of the fucto to suort ts use as a test fucto. However, to the best of our kowledge, o attemts have bee made to aalytcally derve the umber of mma. Because of the comle ature of the fucto surface, a umercal method s develoed to restrct doma saces to hyerrectagles satsfyg certa codtos. Wth these doma saces, a aalytcal method to cout the umber of mma s derved ad roosed as a recursve fuctoal form. The umbers of mma for two search saces are rovded as a referece. Ó 008 Elsever Ic. All rghts reserved. 1. Itroducto The Grewak fucto [1] has bee wdely used to test the covergece of otmzato algorthms [ 15] because ts umber of mma grows eoetally as ts umber of dmesos creases [7,14]. The fucto s defed as follows: f ð~þ ¼ 1 X Y cos ff þ 1; 4000 ¼1 ¼1 wth ½ 600; 600Š where s the umber of dmesos of the fucto. The global mmum s located at ~ 0 wth a value of 0. The actual umber of mma may ot be mortat whe global otmzato s erformed, but t eeds to be kow to test techques that search for local otma. Most studes vaguely meto the umber of mma of the Grewak fucto [7 9], ad, to the best of our kowledge, o aalytcal dervato to determe t has bee gve the lterature. Kowg the umber of mma s crtcal f the Grewak fucto serves as the bass for evaluatg algorthms desged to fd local mma as well as global oes (.e., mult-modal otmzato). I some cases [14], the umber of solutos gve s cosstet wth aalytcal results. For eamle, [14] comared the ablty of NcheSO, best SO, lbest SO, sequetal chg, ad determstc crowdg based o the umber of mma foud through umercal searches. However, further work wth aother algorthm has foud a dfferet umber of solutos tha foud by [14]. I order to address ths ssue ad rovde a cosstet bass for comarg algorthms, ths aer aalytcally derves the umber of mma of the Grewak fucto. We develo a aroach three basc stes. Frst, we restrct the search sace to a hyerrectagle. Secod, we show that the hyerrectagle s the mamum ossble hyerrectagle of the Grewak fucto wth whch local mma o the Grewak fucto corresod to taget ots o a smler surface. Thrd, we develo a aalytcal aroach for coutg the umber of the taget ots o the smler surface. Ths aroach yelds a accurate cout of the umber of local mma of the Grewak fucto wth the defed hyerrectagle. * Corresodg author. E-mal addresses: hcho.eg@gmal.com (H. Cho), folvera@cvl.tamu.edu (F. Olvera), sgukema@hu.edu (S.D. Gukema) /$ - see frot matter Ó 008 Elsever Ic. All rghts reserved. do: /.amc

2 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) Secto elaborates o the characterstcs of the fucto surface ad redefes the roblem of coutg the umber of mma to make t aalytcally tractable. Because of the comle ature of the fucto surface, the doma sace eeds to be restrcted to hyerrectagles foud by the umercal method troduced Secto. Although the aalytcal method to determe the umber of mma derved Secto 3 caot be aled to arbtrary doma saces, t should be oted that the method does ot mss ay mma wth hyerrectagles satsfyg certa codtos. As most otmzato algorthms are tested wth fed hyerrectagles, t remas ractcal to use hyerrectagles as doma saces for testg may otmzato algorthms.. Redefto of the roblem The artal dervatve of the Grewak fucto wth resect to s of ð~þ ¼ s ff o 000 þ ff Y cos ff : ¼1; It s dffcult, f ot mossble, to aalytcally solve ths o-lear system volvg varables. Global ad local mma have to satsfy the followg codtos: f 0 ; ð~þ ¼ s ff 000 þ ff Y cos ffff ¼ 0 for ¼ 1;...; ; ð1þ ¼1; f 00 ; ð~þ ¼ þ 1 Y cos ff > 0 for ¼ 1;...; ;; ðþ ¼1 ¼1 where f 0 00 ;ð~þ ad f; ð~þ are the frst ad secod dervatves of f ð~þ, resectvely. Note that s a de for dmesos. Iequalty () s requred to esure that mama are ot take to accout. By rearragg (), we obta Q ¼1 cos ff >. Because the rego of o-ostve values of Q 000 ¼1 cos ff satsfyg (1) ad () (.e., f ð~þ ¼1 þ 1 at local mma) s outsde of the rego of ts ostve values (.e., f ð~þ < ¼1 þ 1 at local mma), roblem domas ths aer are restrcted such that Y cos ff > 0: ð3þ Sce a value of s small for low dmesos, ot much orto of the fucto sace s lost. Eq. (1) ca be rewrtte as 000 follows: s ff ¼ ff " # 1 Y cos ff ; ð4þ 000 ¼1; where Q ¼1; cos 0 because Q ¼1 cos > 0. ff ff Because f ð~þ meets the surface ¼1 at the global mmum ad ear local mma, we wll fd the mma of f ð~þ by fdg the taget ots of f ð~þ o the smler surface ¼1 ad dervg the relatosh betwee these two sets of ots. I the followg, taget ots refer to the taget ots of the Grewak fucto o the surface ¼1 uless otherwse oted. Sce we oly wat to kow the umber of mma, ther eact coordates are ot of drect terest. I ths aer, the umber of mma s drectly derved by coutg the umber of taget ots assocated wth them. Because the taget ot assocated wth the global mmum s the global mmum tself, ths method also takes to accout the global mmum. Therefore, roblem domas have to be carefully defed so that there ests oe mmum for each taget ot. As or creases, f 0 ; ð~þ also teds to crease alog the le ad, evetually, o ots satsfyg 000 (1) are foud, whch makes global otmzato easer [7]. Because there are hgh correlatos betwee dmesos hghdmesoal roblems, t s hard to determe whether or ot there are local mma satsfyg 0 < Q ¼1 cos < 1by sectg f 0 ; ð~þ surfaces searately. It s ecessary to kow the mamum etet of each beyod whch there are o local mma assocated wth taget ots as show Fg. 1. For ¼ 1, t s trval to check the mamum etet of 1 because all the ots le o f1;1 0 ð 1Þ. For, a umercal aalyss s requred to estmate the corers of the hyerrectagle beyod whch there est taget ots ot assocated wth ay local mma. Whle taget ots are evely dstrbuted at every ff, local mma are ot. If the boudary of a doma sace s located betwee a taget ot ad ts corresodg local mmum, the umber of taget ots s ot the same as the umber of local mma. For ths reaso, a roblem doma U s defed as U ¼ð0; ff k Þ where k N. The mamum value of k, k ;ma, s defed such that the largest local mmum assocated wth a taget ot s located ð ff ðk;ma 1Þ; ff k;ma Þ. Usg the erodcty of the se curve, the k th local mmum, ~ k ¼ð k 1 ;...; k Þ, s obtaed by solvg the followg shfted verso of (4): ff

3 696 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) Fg. 1. Out-most rego of oe dmeso of the Grewak fucto beyod whch there est o more mma. Note that oly the taget ot o the lefthad sde of the gray rego s assocated wth a local mmum. roblem domas should be smaller tha the hyerrectagle defed by the gray rego for the method reseted ths aer to be vald. s 0 k ff ¼ 0 k ff " # þ ðk 1Þ Y k 1 cos ff ; ð5þ 000 ¼1; where 01 ¼ 1 ad 0 k ¼ k ff ðk 1Þ. For oe-dmesoal roblems, (5) s further smlfed by settg ¼ 1 ad Q ¼1; cos ff ¼ 1. If the both sdes of (5) meet at 0 ¼ 3 ff Q, there are o local mma at ths ot because the value of ¼1 cos does ot satsfy (3). By solvg 3 þ ða 1Þ 000 " # 1 Y cos ffff ¼ 1; ¼1; where a R, we obta $ % k ;ma ¼ bac ¼ 1000 Y cos ff þ 1 4 ; ¼1; where bc s the mamum teger less tha or equal to a gve umber (.e., the floorg fucto). However, sce the Grewak fucto s defed wth ½ 600; 600Š, ff k;ma must be less tha or equal to ma ¼ 600. Therefore, k ;ma s $ % ma k ;ma ¼ m ff 1000 ; Y cos ff þ 1 ð6þ ¼1; 4 ad, gve a oe-dmesoal doma sace ð0; ff k Þ where 1 6 k 6 k ;ma, k s the umber of local mma. I roblems of more tha oe dmeso, because the osto of a local mmum oe as s hghly correlated wth those the other aes, t s ot trval to aalytcally solve (5) for all dmesos. The values of cosð ff Þ ad k ;ma for ¼ 1;...; ca be umercally estmated wth the seudo code reseted Fg.. The subroute defed Fg. 3 s used to solve (5) for each dmeso at a tme. 0 k foud ths way may ot be the correct oe because the correlato betwee dmesos s ot take to accout whe solvg (5). A estmated value of 0k s used to evaluate Q ¼1; cos, whch s teratvely lugged to (5) to estmate the et value of 0 k. Oce k ;ma s estmated, a roblem doma eeds to be defed. Defe a roblem doma by U ¼ð0; ;ma Þ, where 0 < ;ma 6 ff k;ma, such that ;ma does ot have to be ff k where 1 6 k 6 k ;ma. Whe Q ¼1; cos ff s greater tha 0, a local mmum s foud ð ff k 1 ff ff ; k Þ because cosð ff Þ s greater tha 0 satsfyg (3), ad (4) ca hold true oly ths rage. Lkewse, whe Q ¼1; cos s less tha 0, a local mmum s foud ð ff k 3 ff ff ; k ff Þ. ff Thus, ;ma eeds to avod these rages because, otherwse, t s ossble to fd local mma ot assocated wth taget ots at ¼ ff k ff, whch meas that the aalytcal method troduced ths aer caot be aled. Therefore, the allowable rage of ;ma s ether ff ff

4 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) Fg.. seudo code to estmate cos ff for ¼ 1;...;. ff k ff ff ; k 3 ff Fg. 3. seudo code for the get subroute. or ff k ff ff ; k 1 ff : The above codtos for ;ma ca be terreted as ad 0 < ;ma 6 ff k;ma ( $ % ) ff _ s a eve teger ;ma X ¼ s a multle of ff ð7þ : ð8þ

5 698 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) I case ;ma does ot satsfy (8) because teger values for ;ma are referred, we eed to make sure that there are o local mma $ % ;ma ff ff ; ;ma ; ð9þ where 0 < ;ma 6 ff k;ma ad ;ma R X. Ths test ca be doe drectly by checkg whether or ot the dstace the th as betwee ;ma ad the closest taget ot whose coordate s greater tha ;ma s greater tha the ossbly largest dstace betwee them. The closest taget ot whose coordate s greater tha s & t ð Þ¼ ff ff : Lkewse, the largest dstace betwee a local mmum ad ts corresodg taget ot s obtaed by calculatg t ð k ;ma Þ k ;ma because t ð k Þ s the taget ot assocated wth k, ad the dstace betwee them also creases as creases. If t ð ;ma Þ ;ma s greater tha t ð k ;ma Þ k ;ma, there must be oe local mmum ð ;ma ; t ð ;ma ÞÞ alog the th as, whch meas that there are o local mma the rage defed by (9). Whe ;ma satsfes all the requremets descrbed above, a doma sace ca be eteded to U ¼½ ;ma ; ;ma Š 8 f1;...; g because the egatve doma sace ð ;ma ; 0Þ s symmetrcal to ð0; ;ma Þ, ad the aalytcal method derved the followg secto takes to accout both regos mlctly. 3. Dervato of the umber of mma I the revous secto, the roblem was redefed so that the umber of taget ots s the same as the umber of mma. The cose fucto s defed ½ 1; 1Š ad, thus, the rage of the fucto Q ¼1 cos ff s also restrcted to ½ 1; 1Š. Cosequetly, 1 Q h ¼1 cos ff 1 has a value ½0; Š ad f ð~þ 4000 ¼1 ; ¼1 þ. Therefore, taget ots of f ð~þ le o the surface ¼1 whe Q ¼1 cos ff s 1. The absolute value of cos ff s 1 whe s a multle of ff. The tmes cosð ff Þ equals to 1 deeds o the rage of or ½ ;m ; ;ma Š. The umber of ff k, where k Z, wth ths rage s calculated as N ¼ ;ma ff ;m ff þ 1: The umber of s satsfyg cos ff ¼ 1s þ 1 N þ ¼ ;ma ff ;m ff ad the umber of s satsfyg cos ff ¼ 1s þ 1 1 : N ¼ N N þ ¼ ;ma ff ;m ff Now, the umber of mama ad mma ca be eressed as M ¼ Q ¼1 N. Coutg the umber of -tules the set ( ) 1 A ¼ cos ffffff ;...; cos ffffff ½ 1; 1Š Y cos ¼ 1 1 ¼1 ff s a combatoral roblem where combatos take lace wthout reettos. Ay elemet, cos ff,of-tules belogg to the set A must have a value of 1 or 1 because, otherwse, the absolute value of Q ¼1 cos ff caot be 1. Because Q ¼1 cos should be 1, a eve umber of elemets a -tule have a value of 1, ad the other elemets have a value ff of 1. Therefore, the umber of -tules the set A ca be eressed as Xbc bc ¼ X ð ÞðÞ ; ¼0 ¼0 where s the bomal coeffcet. Ecode -tules A as ða 1 ; a ;...; a Þ where a s ether 1 or 1. If 1 ad 1 are substtuted wth þ ad symbols, resectvely, -tules A ca be rereseted as (+, +,, +), (,, +,, +), (, +,, +,, +) (.e., oe -tule of ad two eamles of, resectvely), ad so o. Note that there are a eve umber 0

6 of symbols, ad the others are all þ s. The umbers of values satsfyg a ¼þad a ¼ are N þ ad N, resectvely. Coutg all the ossble ~ vectors geeratg -tules the set A ca be doe recursvely terms of. Let S be the umber of mma for -dmesoal roblems. The smlest form s S 1 ¼ N þ 1 for ¼ 1. For ¼, there are S 1 mma whe a s fed to þ because S 1 umber of 1 s satsfyg Q 1 ¼1 a ¼þalso satsfy Q 1 ¼1 a a ¼ Q ¼1 a ¼þ.Ifa s fed to, Q 1 ¼1 a must be, ad the umber of a 1 satsfyg ths codto s M 1 S 1 (.e., the umber of mama for ¼ 1). Therefore, S ¼ S 1 N þ þðm 1 S 1 ÞN. Geeralzg ths recursve form, the followg equatos are obtaed: S 1 ¼ N þ 1 f ¼ 1; ð10þ S ¼ S 1 N þ þðm 1 S 1 ÞN f > 1 ð11þ for ½ ;m ; ;ma Š8 f1;...; g. Now, (10) ad (11) ca be eaded as follows: S 1 ¼ k l 1;ma m 1;m þ 1 f ¼ 1; ð1þ ;ma S ¼ S 1 ffffff ; m ffffff þ 1 " $ % & # Y 1 ;ma ;m þ ¼1 ff ff þ 1 S 1 ;ma ffffff þ 1 ;m ffffff 1 f > 1 for ½ ;m ; ;ma Š8 f1;...; g. 4. Results ad dscusso H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) Fg. 4 ad Table 1 reset the mamum estmated umber of local mma, k ;ma, ad the largest local mmum, k ;ma, o the th as. They defe hyerrectagles wth whch (1) ad (13) ca be aled. Outsde these regos, the aalytcal method reseted ths aer caot be used to cout the umber of mma. Fg. 4 shows k ; ma for dfferet dmesos. For 43, the umercal algorthm Fg. eereced dffcultes fdg k ;ma, ad o lots were draw. Ths result mght be caused by reducg the search sace by all drectos. However, because the umber of mma wth oly a small fracto of hyerrectagles defed by k ;ma h s so hgh eve for ¼ 3 (e.g., 115 mma ½ 8; 8Š 3, a subsace of k ;ma ; k ;ma 8 f1; ; 3g), t would be ractcally eough to defe doma saces for u to ¼ 40. Table 1 shows k ;ma ad k ;ma estmated for u to three-dmesoal roblems. Note that k ;ma for the same vares wth because of the correlato betwee dmesos. Whe defg a doma sace by U¼½ ;ma ; ;ma Š8 f1;...; g, we eed to make sure 0 < ;ma 6 t k ;ma. Ths codto satsfes (7) because t k ; ma ¼ ff k;ma for all the cases Table 1. Also, ;ma has to satsfy (8) or (9). As a set of eamles, doma saces U ¼½ ma ; ma Š were evaluated for where ma f14; 8g. Note that, for the sake of smlcty, doma saces were chose such that all ;ma ¼ ma. For ma ¼ 14, (8) holds true whe = 1 or. The ð13þ k.ma =1 = =3 =4 =5 =10 =0 =30 = Fg. 4. k ;ma versus for dfferet roblem dmesos.

7 700 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) Table 1 Mamum estmated umber of local mma ad the largest local mmum each dmeso k ;ma k ;ma t k ;ma k ;ma s the mamum estmated umber of local mma o the th as wth ð0; 600Þ; k ;ma s the largest local mmum o the th as; t k ;ma ts corresodg taget ot; ad t k ;ma k ;ma s the largest dstace the th as betwee them. t k ;ma k ;ma s Table Numbers of mma for ½ 14; 14Š ad ½ 8; 8Š ½ 14; 14Š ½ 8; 8Š closest taget ot whose coordate s greater tha 3;ma ¼ 14 s t 3 ð14þ, ad the dstace the 3rd as betwee 3;ma ad t 3 ð14þ s t 3 ð14þ 14 ¼ :3. Ths dstace s greater tha t 3 k 3;ma 3 k 3; ma 3 ¼ 1:4 for ¼ 3 as show Table 1. Ths meas that the local mmum assocated wth t 3 ð 3;ma Þ ests ð 3;ma ; t 3 ð 3;ma ÞÞ, ot the rage defed by (9) for 3;ma ¼ 14. For ma ¼ 8, (8) holds true whe ¼ or 3. A vsual secto of the 1 as ad a umercal aalyss show that there are o local mma the rage defed by (9) for 1;ma ¼ 8. Because ma f14; 8g satsfes the boudary codtos secfed by (8) ad (9), we ca safely use (1) ad (13) to calculate the umber of mma of the Grewak fucto. Table shows the umbers of mma for the two search saces for u to three dmesos. 5. Coclusos It s dffcult to aalytcally solve the dervatve of the Grewak fucto ad drectly cout the umber of mma because of the comle ature of the fucto surface. The roblem of coutg the umber of mma was redefed as coutg the umber of taget ots lyg o a arabolc surface. A umercal method was develoed to fd hyerrectagles wth whch ths aroach ca be aled, ad the umber of mma of the fucto was aalytcally derved wth these doma saces based o a recursve fuctoal form. The mamum etets of hyerrectagles for u to three dmesos were estmated, ad the umbers of mma for two search saces were rovded as a referece. The umercal ad aalytcal methods troduced ths aer ca be used to determe the eact umber of mma wth the doma sace defed by a hyerrectagle satsfyg certa codtos. The umber of mma derved ths aer ca serve as a soud bass for evaluatg mult-modal otmzato algorthms. Ackowledgemet Ths work was suorted art by the Korea govermet (the Mstry of Scece ad Techology) through the Korea Scece ad Egeerg Foudato Grat (No. M ). Refereces [1] A.O. Grewak, Geeralzed descet for global otmzato, Joural of Otmzato Theory ad Alcatos 34 (1) (1981) [] J. Keedy, Stereotyg: Imrovg artcle swarm erformace wth cluster aalyss, : roceedgs of the Cogress o Evolutoary Comutato, IEEE Servce Ceter, scataway, New Jersey, 000, [3] T. Krk, J.S. Vesterstrøm, J. Rget, artcle swarm otmsato wth satal artcle eteso, : roceedgs of the Cogress o Evolutoary Comutato, vol., Hoolulu, Hawa, 00, [4] J. Rget, J.S. Vesterstrøm, A dversty-guded artcle swarm otmzer the ARSO, Techcal Reort 00-0, Deartmet of Comuter Scece, Aarhus Uverstet, Bg. 540, Ny Mukegade DK-8000 Aarhus C, Demark, 00. [5] X.-F. Xe, W.-J. Zhag, Z.-L. Yag, A dssatve artcle swarm otmzato, : roceedgs of the Cogress o Evolutoary Comutato, Hoolulu, Hawa, 00, [6] R. Brts, A.. Egelbrecht, F. va de Bergh, Scalablty of Nche SO, : roceedgs of the 003 IEEE Swarm Itellgece Symosum, 003, <htt://eeelore.eee.org/ls/abs_all.s?arumber=1073>. [7] M. Locatell, A ote o the Grewak test fucto, Joural of Global Otmzato 5 (003) [8] N. Khemka, C. Jacob, Eloratory toolkt for evolutoary ad swarm-based otmzato, : roceedgs of the 6th Iteratoal Mathematca Symosum, Baff, Alberta, Caada, 004. [9] S. He, Q.H. Wu, J.Y. We, J.R. Sauders, R.C. ato, A artcle swarm otmzer wth assve cogregato, BoSystems 78 (004) [10] A. Aca, A. Guay, Ehaced artcle swarm otmzato through eteral memory suort, : roceedgs of the Cogress o Evolutoary Comutato (005)

8 H. Cho et al. / Aled Mathematcs ad Comutato 04 (008) [11] C.K. Moso, K.D. Se, Bayesa otmzato models for artcle swarms, : roceedgs of the Geetc ad Evolutoary Comutato Coferece, ACM ress, NY, New York, 005, [1] M. Messer, M. Schmuker, G. Scheder, Otmzed artcle swarm otmzato (OSO) ad ts alcato to artfcal eural etwork trag, BMC Boformatcs 7 (006) [13] C.K. Moso, K.D. Se, Adatve dversty SO, : roceedgs of the Geetc ad Evolutoary Comutato Coferece, ACM ress, NY, New York, 006, [14] R. Brts, A.. Egelbrecht, F. va de Bergh, Locatg multle otma usg artcle swarm otmzato, Aled Mathematcs ad Comutato 189 (007) [15] X. Wag, X.Z. Gao, S.J. Ovaska, A hybrd otmzato algorthm based o at coloy ad mmue rcles, Iteratoal Joural of Comuter Scece & Alcatos 4 (3) (007)