CONTINUOUS PARAMETER FREE FILLED FUNCTION METHOD

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1 Jurnal Karya Asl Lorekan Ahl Matematk Vol 7 No (05) Pae Jurnal Karya Asl Lorekan Ahl Matematk CONINUOUS PARAMEER REE ILLED UNCION MEHOD Herlna Naptupulu, Ismal Bn Mohd Rdwan Pya 3,,3 School o Inormatcs Appled Mathematcs, Unverst Malaysa erenanu, erenanu, Malaysa naptupuluherlna@malcom, smal_ayah_rma@yahoocom, rdwanpya@malcom Abstract : In lobal mnmzaton problems there are two dcultes aced by researchers, rstly s how to move rom one local mnmzer to another mnmzer wth less uncton value, secondly s how to decde that the current mnma s the lobal lled uncton method s one o the recent known determnstc methods whch wdely studed by scentsts to overcome these dcultes hs method has capablty n solvn multdmensonal problems o multmodal uncton ecently, s consdered as an easly appled method Varous knds o parameter or parameter ree lled unctons, alorthm methods as well as ts modcatons are proposed or the sake o eectveness ecency n solvn lobal optmzaton problems Unortunately, untl now there s no ecent technque that can be used to compute or to suest any approprate parameter Moreover, many lled uncton methods need to choose more than one startn pont the nvolved unctons have more than one lobal mnmzer In ths paper, we propose a new lled uncton method wthout parameter that ders rom any estn parameter ree lled unctons urthermore, we proposed an alorthm method or solvn unconstraned lobal mnmzaton problems by one startn pont only he perormance o the alorthm has been supported by the numercal results presented n ths paper, whch show that our method s promsn on solvn lobal optmzaton problem Keywords: lled uncton, Newton s method, steepest descent, lobal optmzaton Introducton Consder the unconstraned lobal mnmzaton problem n mn ( ) : () n One o the most worthy o note research areas n mathematcs especally n numercal analyss s to locatn the lobal mnmzer o a uncton o several varables he reason s due to the estence o multple local mnmzers, whch s derent rom the lobal mnmzers here are two man obstacles when we try to obtan the lobal one; how to jump rom one to a lower local mnmum pont how to ve the decson that the current local mnmzers s the lobal soluton Based on these two dcultes, almost all lobal optmzaton problems cannot be solved by classcal nonlnear prorammn technques drectly One o determnstc approach whch hln the lobal optmzaton problems s known as lled uncton method, the method was ntally proposed or smooth optmzaton by Ge [] he Ge s lled uncton o ( ) at solated mnmzer k over a doman D has the orm k P, k, r, ep r ( ) () where r are adjustable parameters Several lled unctons have been proposed or reconsdern the obstacles o Ge s lled uncton Some o the proposed lled uncton are wth parameter(s) (see [4-9], [], [3] others are wthout parameter (see [3], [0], []) However, those estn lled uncton methods are stll have ollown lmtatons () he estn lled unctons do not ve uarantee o the estence o a better local mnmzer 05 Jurnal Karya Asl Lorekan Ahl Matematk Publshed by Pustaka Aman Press Sdn Bhd

2 Herlna Naptupulu et al () Some lled unctons n the lteratures requre the assumpton that the objectve uncton o lobal optmzaton problem has only a nte number o local mnmzer () Many lled uncton methods ve an assumpton that every local mnmzer has the derent values, e ( ) ( y ), Almost all lled uncton methods requre parameter(s) to be adjusted y One Dmensonal Parameter ree lled uncton In [3], Goh et al proposed a new class o lled uncton whch does not requre any parameter to be selected n ndn the lobal mnmzer hey used the dea o nteraton to buld the lled uncton ollown s the orm o parameter ree lled uncton proposed n [3] where k P, k s an solated local mnmum pont o Under some condtons on the uncton satsy the denton o lled uncton ([3]) () s k ds k k k ( ) () s k ds k ( ), the uncton P(, k ) s a lled uncton o ( ) () 3 wo Dmensonal Parameter ree lled uncton In ths paper, our proposed parameter ree lled uncton s an approach to nd the lobal mnmzer o a multmodal uncton on, under the ollown assumptons : () s a contnuously derentable uncton () has only a nte number o mnmzers, () ( ) as ( ) ( ) ( ) he assumpton () mples the estence o a closed bounded doman D such that D contans all mnmzers o ( ) the value o ( ) when s on the boundary o D s reater than any values o ( ) when s nsde D where Consder a uncton : D where D s a bo dened by ji Assume that js, I S I S I, S, I, S D (3) ( j,) are the nmum supremum o the nterval j ji, (, ) D s a current local mnmzer o nto our sub domans D, D, D 3, D 4 such that 85 js respectvely he doman D can be dvded by D D D D D D, where D [, ], [, ] S S D [, S], [ I, ] 3 3 D3 [ I, ], [, S] 4 4 D4 [, ], [, ] I I (3)

3 Jurnal KALAM Vol 7 No, Pae Symbol j denotes the nterval o varable llustraton o doman D wth ts sub doman j D n sub doman (,,3,4) D (,,3,4) ure 3 shows the (, ) ure 3 he subdomans D, D, D 3 D 4 separated by (, ) One etenson o parameter ree lled uncton [3] s so-called IHR (wo dmensonal Ismal Herlna Rdwan) uncton,, dened on sub doman, s ven by, D (,,3,4), s,, ds,,, s, ds he denton o IHR uncton s ven n Denton 3 ts valdty s proved by heorem 3- heorem 33 Denton 3 mnmzer () () j (, ) (,;,,3,4 n D ( j ) s called IHR uncton o 4 D 0 D ) s a mamzer o j, j (, ) j (, ) or all D j has no statonary pont n the set H ( ) ( ), D () here est a pont ( ) ( ), 0, \ j where H W D such that s, ds ( j ) W, s ds ( j ) ( ) (33) at an solated local s a statonary pont o heorem 3 I then Proo (, ) s an solated mnmzer o the objectve uncton s a mamzer o lled uncton j (,; j,,3,4 ) 86 on 4 D D, 0

4 Snce hen or all,, 0 or all sub doman D (,,3,4), we have s,, ds 0 or, s, ds 0 or D, j, 0(,;,,3,4 j, 0 j, j ) hereore Herlna Naptupulu et al heorem 3 I (, ) s a local mnmzer o (, ) 87 on 4 D D,,,,, \, H D, then has no statonary pont n the set Proo Suppose that, H,, Snce IHR H satses Consder IHR, s a twce derentable uncton, then n sub doman ( (, ) (, )) 0 hereore,, means that or the radent o, also 0 has no statonary pont n ( s, ) (, ) ds 0 (, s) (, ) ds 0 ( (, ) (, )) 0 hus 0, means that has no statonary pont n hereore, n sub doman there s no statonary pont o IHR n he smlar proo can be appled to, 3 H heorem 33 I H,,,, W (, ) 0,, D \ (, ) where H H s, ds ( j ) W, s ds ( j ) then there s a pont (, ) H such that Proo 4 s a statonary pont o j Consder n sub doman D Suppose that or (, ) H satses hen s, ds 0 D D (,; j,,3,4 ),,, s ds, (, ) (, ), (, ) 0,0 holds

5 Jurnal KALAM Vol 7 No, Pae hereore, there ests a statonary pont o Net consder hen n sub doman D or (, ) H Also or (, ) H, s ds 0,,, holds satses, (, ) s ds, (, ) (, ) 0,0 hereore there est a statonary pont o or (, ) H Smlar proo can be appled to, 3 4 he j (,; j,,3,4 ) n (33) has some specal eatures whch are descrbed n ths secton Consder the rst equaton o (33) wrtten as,,, s ds, he radent o s ven by s,,,,, ds here are two cases to be consdered Case ( ) ( ) Clearly that, (a) 0, 0 0 (b) 0, 0 0 Case ( ) ( ) Clearly that, (a) 0, 0 0 (b) 0, 0 0 or or urthermore, the matr o second dervatves o s ven by,, s ds ts determnant s, s, D Det ds (34) 88

6 Now suppose that (34) (a) (b) (c) s a local mnmum o s a local mamum o s a saddle pont o s a statonary (crtcal) pont o D D D urthermore, the radent the matr o second dervatve o Now suppose that (a) (b) (c) o j Herlna Naptupulu et al Accordn to the second dervatve test (,,3,4), s ds,,,,, s ds, s a statonary pont o D 0 D, s s a local mnmzer o, s s a local mamzer o D 0 s a saddle pont o D 0 By consdern the rst second partal dervatves o can be eplored Suppose that pont o j, (,,3,4; j,) led between Consder these two cases Case ( ) ( ) Clearly that a) s a mnmzer o b) j are ven by s the determnant o (35) hen ds 0 ds 0 j are two nearest mnmzer o (o one dmensonal) respect to j j j 0 j when (35) (,,3,4; j,), the specal eatures, -as (see ure 3), snce j s a mamzer o j (o one dmensonal) respect to j -as (see ure 3b), snce j j j 0 when j We do not search the pont (mamzer o j wth respect to (b)) snce ths pont s not the nearest zero o current mnmzer 0 0 j s a statonary -as as eplaned n case Consder only the pont (a 89

7 Jurnal KALAM Vol 7 No, Pae mnmzer o j wth respect to n other words the net mnmzer, j -as as eplaned n case (a)) whch s attaned whle, does not passed by yet decrease, Case ( ) ( ) Snce or such a ure 33b) j then ure 3 Illustraton or specal character o j j j s an nlecton pont o j j 0 (o one dmensonal) wth respect to j -as (see ure 33 Illustraton or specal character o j 4 Alorthm Method Numercal Eperment In ths secton we propose the alorthm method or solvn lobal optmzaton problem usn parameter ree lled uncton, combne wth radus o curvature, Newton s method steepest descent method ollown are the step o the alorthm Data (ntalzaton) specy ntal pont 0, doman D, real number d >0 set,, j, m step Specy ntal step sze o steepest descent0 0, mnmze ( ) startn at 0 to obtan mnmzer m step Construct IHR uncton m (, ) at step 3 Compute m ma, m m j m 90 or m : d ma m, m s too small, where 3/ 3/ ( ) / ( ) /, ( ) / ( ) /

8 9 Herlna Naptupulu et al step 4 I then choose vector drectons, where ( e (,), e (, ), e3 (,), e 4 (, ) ) o to step 5 else stop step 5 Set step 6 Set ntal pont c e 4 c : 0 D j n 0 j: j m e m I then o to step 7 else then o to step 5 else j ; : ; o to step 4 step 7 I ( 0 ) ( m ) then set 0 : 0, o to step else one o () or () s true () ( ) j j m cme m c me () j j c e ( c) e m m m m then o to step step 8 else o to step 6 step 8 Solve (, ) 0, (,,,4; j,) usn Newton s method wth ntal pont c: c j m step 9 I () () hold then set 0 :, m: m o to step else c: c o to step 6 () j / 0 or j j / 0 or j () ( ) ( m ) 0 / 0 4 or j / 0 4 or j 0 obtan Some benchmark test unctons are used or observn the capablty o the Parameter ree lled uncton alorthm, the Vsual C++ proram are used or the alorthm Problem S Hump Back Camel uncton 4 6 4, ; D, ([-5,5],[-5,5]) 3 here are lobal mnmum o ths uncton e, (008984,0765) ( , 0765) wth uncton value ( ) 0363 Problem hree Hump Back Camel uncton 4 6, 05 ; D, ([-3,3],[-3,3]) 6 he lobal mnmum o ths uncton s (0,0) wth ( ) 0 Problem 3 Shubert uncton I 5 5, cos ( ) cos ( ) ; D, ([-0,0],[-0,0]) hs uncton has 760 mnma here are 8 lobal mnmum wth uncton value ( ) 8673

9 Jurnal KALAM Vol 7 No, Pae Problem 4 Shubert uncton II 5 5, cos ( ) cos ( ) ( +453) ( 08003) D, ([-0,0],[-0,0]) hs uncton has 760 mnma he lobal mnmum s ( 453, 08003) wth uncton value ( ) 8673 Problem 5 Shubert uncton III 5 5, cos ( ) cos ( ) ( +453) ( 08003) D, ([ 0,0],[ 0,0]) hs uncton has 760 mnma he lobal mnmum s ( 453, 08003) wth uncton value ( ) 8673 he computatonal results are summarzed n tables or each eample he symbol used n the tables are as ollows Symbols k,, c, d, denote number o teraton, evaluated uncton n j (33), an nteer to be multpled wth, postve real number whch replaces s too small, respectvely he symbols 0,, k represents an ntal pont or steepest descent method, a pont whch satses ( 0 ) ( k ) or a pont or soluton o (, k ) 0 whch satses ( ) ( k ), k-th mnmzer obtaned rom steepest descent method respectvely he symbols s the uncton value k o All the lobal mnmzer o each testn unctons are wrtten n bold he numercal results ven n able 4-45, shows that the IHR alorthm method s succeed n solvn two varable unconstraned lobal optmzaton problems o ven unctons able 4 Results or S Hump Back Camel uncton k 0, j, c, Results 0-0 = 0 3, c =4, = =(,-), 0 = =(607, ), 0 =(0753, ), 0 =( ,-06658), 0 = 00 =( ,-07657),, c =8, = =(09366,033347), 0 = 00 3 =( , ), =( ,07656), = 045 = = = = = = k j able 4 Results or hree Hump Back Camel uncton k 0, j, c, Results 0-0 =(-,-), 0 0 = 0 =(-74755, ),, c =3, =05 0 = =(-04755,066), 0 = =( , ), =

10 Herlna Naptupulu et al 0 = 05 =(( , ), = k , j, c, 0 = 000, c =, =000063, d=0 able 43 Results or Shubert I uncton Results 0 =(-5, 5), =(-453,3), 0 =(-053,7), 93 0 = =( ,80566), 0 = 000 =(-08003,80566),, c =5, = , d=0 0 = 000, c =, = , d=0 0 = 000, c =0, = , d=0 0 = 000, c =3, = , d=0 0 = 000 3, c =3, = , d=0 0 = 000, c =3, = , d=0 0 = 000, c =0, = , d=0 0 =(9968,480566), =(04784,485806), 3 =(033444,485806), 0 =(073444,55806), =(08704,539569), 4 =(08784,54886), 0 =(878,74886), =(5559,60878), 5 =(54886,60878), 0 =(60886,54878), =(608753,54886), 6 =(60878,54886), 0 =(34878,80886), =( ,60878), 7 =(-70835,60878), 0 =(-64835,54878), = = -507 = = =( ,54886), 8 =( ,54886), 0 =(-47857,4886), = = = = -07 = = = -566 = = = = = = = = =( , ), 0 = =( ,-08003), 9, c =8, = , d=0 0 =(54046,479968), = = = = = = =(54046,479968), 0 = = =(54886,485806), 0 3, c =3, = , d=0 0 =(485843,54886), = =(48886,545806), 0 = = 000 =(485806,54886),, c =8, 0 =(845806,8886), 0 =

11 Jurnal KALAM Vol 7 No, Pae =000066, d=0 0 = 000, c =3, =000066, d=0 0 = 000 3, c =8, =000066, d=0 0 = 000 3, c =3, =000066, d=0 0 = 000 4, c =5, =000066, d=0 0 = 000, c =5, =000066, d=0 0 = 000, c =3, =000066, d=0 0 = 000 3, c =34, =000066, d=0 0 = 000, c =, =000066, d=0 0 = 000 0, c =5, =000066, d=0 0 = 000, c =3, =000066, d=0 =(485799, ), =(485806,-08003), 0 =(545806,-4003), =(54847,-453), 3 =(54886,-453), 0 =(-0736,47487), =( ,485799), 4 =(-08003,485806), 0 =(-4003,545806), =(-4476,54886), 5 =(-453,54886), 0 =(-4453,4886), =(-7083,-453), 6 =(-70835,-453), 0 =(-0835,-6453), =(-4549,-70835), 7 =(-453,-70835), 0 =(-0859,-76835), =( ,-77083), 94 0 = = = =(-08003,-77083), 0 =(-76003, ), =( ,-08003), 9 =(-77083,-08003), 0 =(-73083,-04003), =(-70839,485806), 0 =(-70835,485806), = = = = = =(-0835,-04943), 0 = -456 =(-4549,-08003), =(-453,-08003), 0 =(-0859,-4003), 0 = =( ,-453), 0 = 000 =(-08003,-453),, c =5, =000066, d=0 0 =(49968,-6453), 0 = -456 =(48577,-70835), 0 = =(485806,-70835), 3, c =3, 3 0 =(545806,-76835), 0 =

12 Herlna Naptupulu et al =000066, d=0 0 = 000 3, c =65, =000066, d=0 0 = 000, c =3, =000066, d=0 =(54847,-77083), 4 =(54886,-77083), 0 =(-7574,5969), =( ,54886), 5 =(-77083,54886), 0 =(-3083,088864), =( ,-77083), 0 = =(-70835,-77083), 3, c =3, =000066, d=0 0 = =(-76835,-7083), =( ,-70835), 7 =(-77083,-70835), 4 0 = = = k 0-3 0, j, c, 0 = 000, c =5, =000869, d=0 0 = 000 4, c =, = , d=0 0 = 000, c =3, = , d=0 0 = 000 able 44 Results or Shubert II uncton Results 0 =(05,09), 0 =-4969 =(03366, ), 0 =(33366,38883), =(565,63), =(54805,608599), 0 =(50805,568599), 0 0 =(-7075,-64595), 3 =(-7083,-4499), 0 =(-483,3750), = = 074 = = = =(-503,-08003), 4 =(-453,-08003), able 45 Results or Shubert III uncton k 0, j, c, Results 0-0 = 000, c =9, = , d=0 0 = 000 4, c =3, =000068, d=0 0 =(-,), 0 =-034 =(08965,-08003), 0 =(6696,499968), =(477877,-05857), =(485536,-08003), 0 =(-4506, ), 0 0 = = = = = = 664 = = =(-34464,-70003), 0 = = =(-453, ), 3, c =47, = , d=0 0 = = -477 =(797487,3934), 0 = 959 =(-08308,-9877), =

13 Jurnal KALAM Vol 7 No, Pae = 000 3, c =3, = , d=0 0 = =( ,-4486), 0 =(-4006,-08486), =(-40687,-08003), 5 =(-453,-08003), 0 4 = = = Concluson rom the numercal results n prevous secton, t s clear that IHR alorthm whch ncorporate IHR uncton, steepest descent method, radus o curvature Newton s method succeeds n solvn unconstraned lobal optmzaton o ven problems by usn one startn pont only Summary o advantaes o our proposed alorthm method are as ollows: () IHR has no parameter to be adjusted () IHR contnuous everywhere () IHR has no eponental nor loarthmc term, (v) IHR alorthm method has a smple stoppn crtera (v) IHR method succeeds n obtann/eplorn lobal mnmzers o objectve uncton whch has more than one lobal mnmzer wth only one startn pont We conclude that our proposed method s a promsn method that eectve ecent compared wth another estn lled uncton method n solvn lobal optmzaton problems Acknowledment hs research was supported by undamental Research Grant Scheme o Malaysa Vot No 5955 Reerences [] Ge, R P: A lled uncton Method or ndn A Global Mnmzer o A uncton o Several Varables Journal Mathematcal Prorammn, 46, 9-04 (990) [] Ma, S, Yan, Y, Lu, H: A Parameter ree lled uncton or Unconstraned Global Optmzaton Journal Appled Mathematcs Computaton, 5, (00) [3] Wen, G K, Mamat, M B, Mohd, I B, Dasrl, Y B: Global Optmzaton wth Nonparametrc lled uncton ar East Journal o Mathematcal Scences (JMS), 6(), 5-64 (0) [4] Xan, L: ndn Global Mnma wth a Computable lled uncton Journal o Global Optmzaton, 9, 5-6 (00) [5] Xan, L: Several lled uncton wth Mtators Journal o Appled Mathematcs Computaton, 33, (00) [6] Zhan, L S, N, C K, L, D, an, W W: A New lled uncton Method or Global Optmzaton Journal o Global Optmzaton, 8, no, pp 7-43 (004) [7] Wu, Z Y, Zhan, L S, eo, K L, Ba, S: New Moded uncton Method or Global Optmzaton Journal o Optmzaton heory Applcaton, 5, 8-03 (005) [8] Wu, Z Y, Lee, H W J, Zhan, L S, Yan, X M: A Novel lled uncton Method Quas - lled uncton Method or Global Optmzaton Journal Computatonal Optmzaton Applcaton, 34, 49-7 (005) [9] Yan, Y, Shan, Y: A New lled uncton Method or Unconstraned Global Optmzaton Journal Appled Mathematcs Computaton, 73, 50-5 (006) [0] Wan, X, Zhou, G: A New lled uncton or Unconstraned Global Optmzaton Appled Mathematcs Computaton, 74, (006) [] Wan, C, Yan, Y, L, J: A New lled uncton Method or Unconstraned Global Optmzaton Journal o Computatonal Appled Mathematcs, 5, (009) [] Ln, Y, Yan, Y, Zhan, L: A Novel lled uncton Method or Global Optmzaton Journal Korean Math Soc, 47, no 6, (00) [3] Xu, Z, Huan, H X, Pardalos, P M, Xu, C X: lled unctons or Unconstraned Global Optmzaton Journal o 96

14 Herlna Naptupulu et al Global Optmzaton, 0, (00) [4] Lan, Y M, Zhan, L S, L, M M, Han, B S: A lled uncton Method or Global Optmzaton Journal o Computatonal Appled Mathematcs, 05, 6-3 (007) [5] Shan, Y L, Pu, D G, Jan, A P: ndn Global Mnmzer wth One-Parameter lled uncton on Unconstraned Global Optmzaton Appled Mathematcs Computaton, 9, 76 8 (007) [6] Gao, C, Yan, Y, Han, B: A New Class o lled unctons wth One Parameter or Global Optmzaton Journal Computers Mathematcs wth Applcatons, 6, (0) [7] Ln, H, Wan, Y, an, L: A lled uncton Method wth One Parameter or Unconstraned Global Optmzaton Appled Mathematcs Computaton, 8, (0) [8] We,, Wan, Y: A New lled uncton Method wth One Parameter or Global Optmzaton Mathematcal Problems n Enneern, 03, Artcle ID 5335, paes (03), do:055/03/5335 [9] Ln, H, Gao, Y, Wan, Y: A Contnuously Derentable lled uncton Method or Global Optmzaton Numercal Alorthm, do: 0007/s (03) [0] An, L, Zhan, L S, Chen, M L: A Parameter ree lled uncton or Unconstraned Global Optmzaton Journal o Shanha Unversty (Enlsh Edton), 8, no, 7-3 (004) [] Shan, Y, L, P, Xe, H: A New Parameter - ree lled uncton or Unconstraned Global Optmzaton Computatonal Scence Optmzaton (CSO), 00 hrd Internatonal Jont Conerence on Computatonal Scence Optmzaton,, (00), do: 009/CSO0059 [] Wen, G K: Global Optmzaton Usn Nonparametrc lled uncton Method (Master hess) Unverst Malaysa erenanu, Malaysa (009) [3] Wan, W, Shan, Y Zhan, L: A lled uncton Method wth one Parameter or Bo Constraned Global Optmzaton Appled Mathematcs Computaton, 94, (007) 97