An Approach to Solve Linear Equations Using Time- Variant Adaptation Based Hybrid Evolutionary Algorithm

Transcription

1 A Approach to Solve Lear Equatos Usg Te- Varat Adaptato Based Hbrd Evolutoar Algorth 1 A. R. M. Jalal Udd Jaal, 2 M. M. A. Hashe, ad 1 Md. Bazlar Raha. 1 Departet of Matheatcs Khula Uverst of Egeerg ad Techolog Khula-9203, Bagladesh Ph: Et-531, 523 E-al: jaal@ath.kuet.ac.bd ad 2 Departet of Coputer Scece ad Egeerg Khula Uverst of Egeerg ad Techolog Khula-9203, Bagladesh Ph: , Fa E-al: hashe@cse.kuet.ac.bd Abstract For sall uber of equatos, sstes of lear (ad soetes olear) equatos ca be solved b sple classcal techques. However, for large uber of sstes of lear (or olear) equatos, solutos usg classcal ethod becoe arduous. O the other had evolutoar algorths have ostl bee used to solve varous optzato ad learg probles. Recetl, hbrdzato of evolutoar algorth wth classcal Gauss-Sedel based Successve Over Relaato (SOR) ethod has successfull bee used to solve large uber of lear equatos; where a ufor adaptato (UA) techque of relaato factor s used. I ths paper, a ew hbrd algorth s proposed whch a te-varat adaptato (TVA) techque of relaato factor s used stead of ufor adaptato techque to solve large uber of lear equatos. The covergece theores of the proposed algorths are proved theoretcall. Ad the perforace of the proposed TVA-based algorth s copared wth the UA-based hbrd algorth the eperetal doa. The proposed algorth outperfors the hbrd oe ters of effcec. KeWords Adaptve Algorth, Evolutoar Algorth, Te-Varat Adaptato, Lear Equatos, Recobato, Mutato. 1. Itroducto Solvg a set of sultaeous lear equatos s a fudaetal proble that occurs dverse applcatos. A lear sste ca be epressed as a atr equato whch each atr or vector eleet belogs to a feld, tpcall the real uber sste. A set of lear equatos ukow 1,2,..., ad uber of equatos s gve b: a b j1 j j, 1,2, (1) Equvaletl, atr for: A b. (2)

2 Where j A ( a ) s the coeffcet atr, ( ) s the vector of ukow varables (.e. soluto vector), b b ) s the rght had costat vector. ( Aog the classcal teratve ethods, to solve sste of lear equatos, Gauss-Sedel based Successve Over Relaato (SOR) ethod s oe of the best ethods. But the speed of covergece depeds o over relaato factor,, wth a ecessar codto for the covergece beg 0 2 [1]. It s ofte ver dffcult to estate the optal relaato factor ad SOR techque s ver sestve to the relaato factor, [2]. O the other had the Evolutoar Algorths (EA) are stochastc algorths whose search ethods odel soe atural pheoea: geetc hertace ad Darwa strfe for survval [3, 4, 5]. Alost all of the works o EA ca be classfed as evolutoar optzato (ether uercal or cobatoral) or evolutoar learg. But Fogel ad Atar [6] used lear equato solvg as test probles for coparg recobato, verso operatos ad Gaussa utato a evolutoar algorth. However, the ephaszed ther stud ot o equato solvg, but rather o coparg the effectveess of recobato relatve to utato. o coparso wth classcal equato-solvg ethods was gve. Recetl, a hbrd evolutoar algorth [7] s developed b tegratg classcal Gauss-Sedel based SOR ethod wth evolutoar coputato techques to solve equatos whch, s self-adapted b usg Ufor Adaptato (UA) techque. For the cause of ufor adaptato, s are ot fel adapted ad there s a tedec of vbratg a stages ad there s o local fe-tug later stages. The dea of self-adaptato was also appled a dfferet felds [11]. Obvous bologcal evdece s that a rapd chage s observed at earl stages of lfe ad a slow chage s observed at latter stages of lfe all kds of aals/plats. These chages are ore ofte occurred dacall depedg o the stuato eposed to the. B ckg ths eerget atural evdece, a specal dac Te-Varat Mutato (TVM) operator s proposed b Hashe [8] ad Mchalewcz et al. [9,10] global optzato probles. I ths paper, a ew hbrd evolutoar algorth s proposed whch a Te- Varat Adaptve (TVA) techque s troduced ag at both provg the fe local tug ad reducg the dsadvatage of ufor adaptato of relaato factors as well as utato. The proposed TVA-based hbrd algorth does ot requre a user to guess or estate the optal relaato factor. The proposed algorth talzes ufor relaato factors a gve doa ad evolves t. The proposed algorth tegrates the Gauss- Sedel-based SOR ethod wth evolutoar algorth, whch uses talzato, recobato, utato, adaptato, ad selecto echass. It akes better use of a populato b eplog dfferet equato-solvg strateges for dfferet dvduals the populato. The these dvduals ca echage forato through recobato ad the error s zed b utato ad selecto echass. Eperetal results show that the proposed TVA-based hbrd algorth ca solve lear equatos wth sall te copared to the UA-based hbrd algorth. 2. The Proposed Hbrd Algorth For the soluto of the lear equatos (1), Gauss-Sedel based SOR ethod [1] s gve b: j

3 1 k 1) k k1 ( k) ( b aj j aj, 1,2, (3) a j1 j ow coeffcet atr A of the equato (2) ca be decoposed as A DL U (4) Where D d ) s the dagoal atr, L l ) s the lower tragular atr ad U u ) ( j s the upper tragular atr. The atr for equato (3) ca be rewrte as: k1 k H V -1 1 ( j Where H ( I D L) {1 I D U} -1 ad V ( I D 1 L) Here, I s the dett atr ad, ) s the relaato factor whch fluece the ( L U covergece rate of the SOR techque greatl; boudar values of. L ad 1 D -1 b ( j (5) U are deoted as lower ad upper Slar to a other evolutoar algorths, the proposed hbrd algorth alwas atas a populato of approate soluto to lear equatos. Each soluto s represeted b a dvdual. The tal populato s geerated radol for the feld. Dfferet dvduals use dfferet relaato factors. Recobato the hbrd algorth volves all dvduals a populato. If the populato sze s, the the recobato wll have parets ad geerate offsprg through lear cobato. Mutato s acheved b perforg oe Gauss-Sedel based SOR terato as gve b (5). Itall s geerated betwee L (=0) ad U (=2) ad the value s adapted stochastcall durg evoluto. The ftess of a dvdual s evaluated based o the error of a approate soluto. For eaple, gve a approate soluto (.e., a dvdual) z, ts error s defed b e ( z) Az b. The relaato factor s adapted after each geerato, depedg o how well a dvdual perfors ( ters of error). The a steps of the TVA-based hbrd evolutoar algorth s descrbed as follows: Step 1: Italzato Geerate, radol fro, a tal populato of approate solutos to the lear equatos (1) usg dfferet relaato factor for each dvdual of the populato. Deote (0) (0) (0) (0) the tal populato as X {,,, } where s the populato sze. Let 1 2 k 0 where k s the geerato couter. Ad talze correspodg relaato factor as: Where d d L 2 d 1 U L for 1 for 1 (6) Step 2: Recobato kc ( kc) ( kc) ( kc) ow geerate X { 1, 2,, } as a teredate populato through the followg recobato: ( k X c) R X ( k ) (7) t

4 Where R ( rj ) s a stochastc atr [12], ad the superscrpt t deotes traspose. Step 3: Mutato k kc The geerate the et teredate populato X fro X as follows: For each kc dvdual k c ( 1 ) populato X produces a offsprg accordg to Eq. (5) ( k ) H V, 1, 2...,. (8) ( kc) ( k) Where s deoted as relaato factor of the th dvdual ad s deoted as th (utated) offsprg, so that ol oe terato s carred out for each utato. Step 4: Adaptato k k Let ad be two offsprg dvduals correspodg relaato factors ad ad e( ) ad e( ) are ther correspodg errors (ftess value). The the relaato factors ad are adapted as follows: (a) If e( ) e( ), () the ove toward b usg 0.5 p )( ) (9) ( ad () ove awa fro usg p ( U ), whe (10) p ( L ), whe p E 0,0.25 T, a error decreasg stochastc paraeter of, ad Where p E ( 0,0.25) T, a error decreasg search paraeter of. Here, (0, 0.25) s the Gaussa dstrbuto wth ea 0 ad stadard devato 0.25, E= a eogeous paraeter of, F= a eogeous paraeter of, s the optal relaato factor, & are adapted relaato factors correspod to ad, ad t γ T (1 ), Te-varat adaptve paraeter, (11) T Here, t = uber of geerato (terato), T= au geerato uber, γ = a eogeous paraeter of T. (b) If e( ) e( ), the adapt ad the order of ad. (c) If e ( ) e ( ), o adaptato. So that = ad the sae wa as above but reverse. Step 5: Selecto ad Reproducto Select the best /2 offsprg dvduals accordg to ther ftess values (errors). The reproduce of the above selected offsprg (.e. each parets dvdual geerates two offsprg). The for the et geerato of dvduals.

5 Step 6: Terato If { e(z) : zx} < (Threshold error), the stop the algorth ad get uque soluto. If { e(z) : zx}, the stop the algorth but fal to get a soluto. Otherwse go to Step Covergece Theores The followg theore establshes the covergece of the hbrd algorth. Theore-1: If there est a ε 0 ε 1 such that, for the or of H, H ε 1, the ( k ) L *, where * s the soluto to the lear sste of equatos.e., A* b. k Proof: The dvduals the populato at geerato k are the error betwee the approate soluto The accordg to the recobato Sce kc (k) r, 1,2.,..., j j1 j ( kc) ( kc) (k) e * r j j * r j j1 j1 1 ad r j 0, 1,2.,..., ( kc) e k r j j * r k j j * ( kc) k e r j j * ( kc) k e < a { e : j 1,... } Aga accordg to utato for 1, 2,... k kc H V ad also sce * H * V k k c H ( the * *) ( k), 1,2,. Let (k) ad eact soluto *.e. (k ) r k * j j (k ) e be ( k). e = * k * ( kc H * ) k c ( kc) H ( *) H e k H a { e j ; j 1,... } k ( k) j e ε a { e ; j 1,... } ow accordg to the selecto echas, we have for 1,... k1 k e e k ε a{ e j ; j 1,... } k1 Ths ples that the sequece {a { e j ; j 1,...,}; k 0,1,2. } s strctl ootoc decreasg ad thus coverget.

6 The rate of covergece s accelerated b TVA-based hbrd ethod. The followg theore justfes the adaptato techque for relaato factors used proposed TVA-based hbrd evolutoar algorth. Theore 2: Let ρ( ) be the spectral radus of atr relaato factor, ad ad ad respectvel. Assue ) ootoc creasg whe * H ad let * be the optal be the relaato factors of the selected par dvduals ρ( s ootoc decreasg whe * ad ρ( ) s. Also cosder ρ( ) ρ( ). The () ρ( ) ρ( ), whe (0.5 p ) ( ) ad () ρ ( ) ρ( ), whe p ( U ) f p ( ) f Proof: We frst assue that ρ( ) s ootoc decreasg whe * ad ρ( ) s ootoc creasg whe *. Let ad be the two relaato factors of a radol selected par dvduals ad ad also let ρ ( ) ρ ( ). The there wll be four cases: Case-1 Both, * : Sce ρ( ) s ootoc decreasg whe * ad as assue ρ ( ) ρ ( ) ; so. The we get ad so ( ) ρ( ), where L ( 0.5 p )( ). Aga sce, so we get * ad therefore ρ( ) ρ( ) where p ( U ). Case-2 Both, * : Sce ρ( ) s ootoc creasg whe * ad as assue ρ ( ) ρ ( ) ; so. The we get ad so ρ( ) ρ( ), where ( 0.5 p )( ). Aga sce, so we get * ad therefore ρ( ) ρ( ), where p ). Case-3 If ( L * : Sce ρ( ) s ootoc decreasg whe * ad as assue ρ ( ) ρ ( ) ; so we get * ad therefore ρ( ) ρ( ), where 0.5 p )( ). Aga sce * so we get ( therefore ρ( ) ρ( ), where p ). ( U * ad [ote that ths case wll go a bt awa fro * wth a ver sall probablt] Case-4 If * : Sce ρ( ) s ootoc creasg whe * ad as assue ρ ( ) ρ ( ) ; so we get * ad therefore ρ( ) ρ( ), where 0.5 p )( ). Aga sce * so we get *, ad ( therefore ρ( ) ρ( ), where p ). [ote that ths case ( L wll go a bt awa fro * wth a ver sall probablt]

7 If e( ) e( ), adapt ad ad worse relaato factor factor the sae wa as above but reverse the role of. Cosderg all the above stuatos, we a coclude that the coparatvel s alwas proved ad the coparatvel better relaato also s proved wth a ver hgh probablt each geerato. Hece, the proposed te-varat based adaptato echas crease the rate of covergece. 4. uercal Eperet I order to evaluate the effectveess of the proposed TVA-based hbrd algorth, uercal eperet have bee carred out o a uber of probles to solve the sstes of lear Eq. (2) of the for: A b. Frst we solve the proble of the above equato b settg the paraeters: a 2, ad b for atr A s 1,2,, ad a j j for j, j 1,2,, ; the deso of the coeffcet Error, e() E-6 1E-8 1E-10 1E Geerato, t Fgure 1: Curve (a) represets the Evoluto Hstor of TVA-based hbrd algorth ad curve (b) represets the Evoluto Hstor of UA-based hbrd algorth (a) (b) (.e. 100 ). The populato sze s set at 2; U ad L are set at 2 ad 0 respectvel so that tal s becoe 0.5 ad 1.5. The eogeous paraeters γ, E ad F are set at 40.0, 0.1 ad 0.01 respectvel ad au geerato, T, s set at Each dvdual s talzed fro the doa 100 (-30, 30) radol ad uforl. The 13 proble s to be solved wth a error saller tha 10 (threshold error). The progra s ru 10 tes usg 10 dfferet saple paths ad the averaged the. Fgure 1 shows the uercal results ( graphcal for) acheved b the UA-based hbrd algorth [7] ad proposed TVA-based hbrd algorth. It s observed Fg. 1 that the rate of covergece of TVA-based algorth s uch better tha that of UA-based algorth ad TVA-based algorth ehbts a fe local tug. Table I shows s test probles, labeled fro P1 to P6, wth deso, 100. For each test proble P: = 1,2 6, the coeffcet atr A ad costat vector b are all geerated uforl ad radol wth gve doas (show 2d colu wth correspodg rows of Table I) ad also tal populato X are all geerated uforl ad radol

8 wth doa [-30,30]. Ital relaato factors are set at 0.5 ad 1.5 for all the cases. For dfferet probles (P1 P6) correspodg threshold errors,, ad correspodg au geeratos (teratos), T, are show the Table I. Table I Coparso betwee TVA-based ad UA-based algorths for several radol geerated test probles Label of Test Probles Doa of the Paraeters of the test Probles Threshold Error, Mau Geerato, T For UA-based * Elapsed Te ( Secod) Geerato (elapsed) For TVA-Based * Elapsed Te ( Secod) Geerato (elapsed) P1 a= 2; aj =j; b= P2 a(-70,70); aj (-2,2) b(-2,2) P3 a(-70,70); aj (0,4) b(0,70) P4 a(1,100); aj (-2,2) b = 2 P5 a = 200; aj (-30,30) b(-400,400) P6 a(-70,70); aj (0,4) b(0,70) * Elapsed Tes are show, colu fve ad colu eght, just for relatve coparso of two algorths. ote: 1. The eleets of coeffcet atr A, b, ad tal populato X are detcal for each coparso. 2. Algorths are pleeted Borlad C++ evroet usg Petu IV PC (1.2GHz). Table I shows the coparso of the uber of geerato (terato) ad relatve elapsed te used b the UA-based hbrd algorth ad b proposed TVA-based hbrd algorth to the gve precseess, (see colu three of the Table I). Oe observato ca be ade edatel fro the Table I, ecept for proble P3 where the UA-based ethod perfored ear to sae as TVA-based ethod, TVA-based hbrd algorth perfored uch better tha the UA-based hbrd algorth for all other probles. 1.4 Varato of Geerato Fgure 2: Self-adaptato of the UA-based Algorth Varato of Geerato Fgure 3: Self-adaptato of the TVA-based algorth

9 Fg. 2 shows the ature of self-adaptato of the UA-based hbrd algorth ad Fg. 3 shows the ature of self-adaptato of the TVA-based hbrd algorth. It s observed Fg. 2 ad Fg. 3 that the self-adaptato process of relaato factors TVA-based hbrd algorth s uch better tha that of UA-based hbrd algorth. Fg. 3 shows that how tal = 0.5, s adapted to ts ear optu value ad reaches to a best posto for whch rate of covergece s accelerated. O the other had Fg. 2 shows that tall = 0.5, b self-adaptato process, does ot graduall reaches to a best posto. It s oted that both UA-based ad TVA-based algorths are pleeted Borlad C++ evroet. 5. Cocludg Rearks I ths paper, a Te-varat adaptve (TVA)-based hbrd evolutoar algorth has bee proposed for solvg lear sstes of equatos. The TVA-based hbrd algorth tegrates the classcal Gauss-Sedel based SOR ethod wth evolutoar coputato techques. The te-varat based adaptato s troduced for adaptato of relaato factors, whch akes the algorth ore atural ad accelerates ts rate of covergece. The recobato operator the algorth ed two parets b a kd of averagg, whch s slar to the teredate recobato ofte used evoluto strateges [4, 5]. The utato operator s equvalet to oe terato the Gauss-Sedel based SOR ethod. The utato s stochastc sce the relaato factor s adapted stochastcall. The proposed TVA-based relaato factor adaptato techque acts as a local fe tuer ad helps to escape fro the dsadvatage of ufor adaptato. uercal eperets wth the test probles have show that the proposed TVA-based hbrd algorth perfors better tha the UA-based hbrd algorth. Also TVA-based hbrd algorth s ore effcet ad robust tha the UA-based hbrd algorth. Refereces [1] A. Gourd ad M. Bouahrat, Appled uercal Methods, Pretce Hall of Ida, ew Delh, pp , [2] L. A. Hagaa, ad D. M. Youg, Appled Iteratve, Methods, ew York. Acadec press, [3] M. Schoeauer, ad Z. Mchalewcz, Evolutoar Coputato, Cotrol ad Cberetcs Vol.26, o.3, pp , [4] T. Bäck ad H. P Schwefel, A overvew of Evolutoar Algorths for Paraeter Optzato, Evolutoar Coputato, Vol.1, o.1, pp. 1-23, [5] T. Bäck, U. Hael ad H. -P. Schwefel, Evolutoar Coputato: Coets o the Hstor ad Curret State, IEEE Tras. o Evolutoar Coputato, Vol.1, o. 1, pp. 3-17, [6] D. B. Fogel ad J. W. Atar, Coparg Geetc Operators wth Gaussa Mutatos Sulated Evolutoar Process Usg Lear Sstes, Bol Cber, Vol. 63, o- 2, pp , 1990.

10 [7] Ju He, J. Xu, ad X. Yao, Solvg Equatos b Hbrd Evolutoar Coputato Techques, Trasactos o Evolutoar Coputato, Vol.4, o. 3, pp , Sept [8] M. M. A. Hashe, Global Optzato Through a ew Class of Evolutoar Algorth, Ph.D. dssertato, Saga Uverst, Japa, pp , Septeber [9] Z.Mchalewcz, Evolutoar Coputato Techque for olear Prograg Proble, Iteratoal Tras. o Operato Research, Vol. 1, o. 2, pp , [10] Z. Mchalewcz, ad. F. Atta, Evolutoar Optzato of Costraed Probles, Proceedg of the 3 rd Aual Coferece o Evolutoar Prograg, Rver Edge, 3, World Scetfc; pp , [11] R. Saloo ad J. L. Va Hee, Acceleratg Back Propagato Through Dac Self-adaptato, eural etworks Vol. 9, o-4, pp , [12] E. Krszg, Advaced Egeerg Matheatcs, 7 th edto Joh Whe & Sos, ew York; pp , 1993.