Real-coded Quantum Evolutionary Algorithm for Global Numerical Optimization with Continuous Variables

Transcription

1 Chnese Jounal of Eleconcs Vol.20, No.3, July 2011 Real-coded Quanum Evoluonay Algohm fo Global Numecal Opmzaon wh Connuous Vaables GAO Hu 1 and ZHANG Ru 2 (1.School of Taffc and Tanspoaon, Souhwes Jaoong Unvesy, Chengdu , Chna) (2.School of Auomaon, Habn Unvesy of Scence and Technology, Habn , Chna) Absac Ths pape pesens a Novel veson of eal-coded quanum evoluonay algohm (NRQEA) o solve global numecal opmzaon wh connuous vaables. Complemenay muaon opeao, whch s desgned based on he specfc confguaon of eal-coded chomosome and he gaden nfomance of objecve funcon, s used o updae chomosomes and ealze a balance beween exploaon and exploaon. Technque of educng he seach space, whch s mplemened based on he evoluonay pocess of algohm, s adoped o mpove he convegence ae. Smulaon esuls on benchmak funcons show ha he algohm poposed s moe suable fo global numecal opmzaon wh connuous vaables han he compaed algohms, and has he chaacescs of apd convegence, good global seach ably and sably. Key wods Quanum compung, Quanum evoluonay algohm, Real-coded quanum evoluonay algohm, Global numecal opmzaon. I. Inoducon Based on he conceps and pncples of quanum compung, Quanum evoluonay algohm (QEA) s poposed [1,2]. In QEA, qubs chomosome s adoped o manan he dvesy of soluon and ovecome pemaue phenomenon, and quanum oaon gae s used o updae chomosomes and avod sagnaon phenomenon. Howeve, many eseaches have poven ha QEA s moe suable fo combnaoal opmzaon han numecal opmzaon [3 5]. The eason may be ha he soluons of QEA ae epesened decly o ndecly by bnay sng, whch mples ha, fo numecal opmzaon, QEA has nevably he dsadvanages such as he nconvenen pocess of codng and decodng, he lengh exploson of chomosome wh he dmenson and pecson and so on. In addon, he oaon angle of quanum oaon gaes s only deemned by lookng up able, no consdeng he dffeence among chomosomes, whch esuls n a lowe seachng effcency. To make QEA solve numecal opmzaon as effcen as do combnaoal opmzaon, ecenly, hee has been some wok o be done. In Ref.[6], eal-coded chomosome s nvesely mapped o qubs n he soluon space, and eal cossove guded by qubs pobably and chaos muaon s adoped o updae chomosomes. In Ref.[7], mul-chans eal-coded chomosome s used, and quanum oaon gae s appled o updae chomosomes, a he same me, he analycal expesson fo he decon and sze of oaon angle s gven. In a wod, he cucal dea of hose eseaches s ha eal-coded chomosomes efeng o qubs conceps ae used nsead of qubs chomosomes, and evoluonay saeges based on he specfc confguaon of eal-coded chomosomes ae desgned o updae chomosomes. In hs pape, we poposed a Novel veson of eal-coded quanum evoluonay algohms (NRQEA) fo global numecal opmzaon wh connuous vaables. Real-coded chomosomes ae also used nsead of qubs chomosomes n NRQEA, and he man pon of NRQEA s ha a complemenay muaon opeao s desgned o balance he local and global seach effecvely, and a echnque of educng he seach space s adoped o enhance he convegence ae. Smulaon esuls on sx benchmak funcons demonsae he poposed algohm s moe suable fo global numecal opmzaon wh connuous vaables han he compaed algohms. Nex secon descbes he basc mechansm of NRQEA n deal. The smulaon esuls ae poposed n Secon III. Fnally, conclusons ae dawn n Secon IV. II. Basc Mechansm of NRQEA 1. Repesenaon Consde he followng global numecal opmzaon poblem wh connuous vaables: mn f(x) (1) L X U whee X = (x 1, x 2,, x n) T s a vaable veco n R n ; L = (l 1, l 2,, l n) T and U = (u 1, u 2,, u n) T defne he feasble soluon space; f(x) s a n-dmensonal eal-valued Manuscp Receved Nov. 2009; Acceped Dec

2 500 Chnese Jounal of Eleconcs 2011 funcon, and egaded as he objecve funcon. In NRQEA, a eal-coded chomosome, whose allele s composed of one componen x of vaable veco X and pobably ampludes (α, β ) T of one qub, = 1, 2,, n, s epesened as: x 1 x 2 x n q = α 1 α 2 α n (2) β 1 β 2 β n whee n s he lengh of chomosome, and les on he dmensons of vaable veco X. 2. Complemenay muaon opeao In NRQEA, sngle-gene muaon, whch means ha only one gene of chomosome s mplemened muaon, s adoped o updae populaon a each eaon. I has been poven ha sngle-gene muaon s supeo o all-gene muaon n ems of seach effcency n Ref.[8]. Assume ha NRQEA manans a populaon {p 1, p 2,, p N } a he -h eaon, whee N s he populaon sze. p j denoes an ndvdual defned as Eq.(2), and fj s s fness value, j = 1, 2,, N. Selec he -h gene (x, α, β) T, = 1, 2,, n, of p j, and updae he value of x usng Gaussan muaon, whch s expessed as: x +1,k = x + (u l )N(0, (σ,k )2 ) (3) whee k {α, β }; N(0, (σ,k )2 ) denoes Gaussan dsbuon of mean 0 and vaance (σ,k )2, and he value of (σ,k )2 s desgned as: { α (σ,k 2, k = α )2 = (4) β 2 /3, k = β x +1,k To avod geneang he nfeasble soluon, he value of s clpped accodng o Eq.(5). Unl he value of x +1,k les n he feasble soluon space, Eq.(5) has o be pefomed epeaedly. { x +1,k x +1,k = 2u x +1,k, x +1,k > u (5) = 2l x +1,k, x +1,k < l If he feasble soluon (x j,1,, x +1,k,, x j,n) T deved fom Eqs.(3) (5) s supeo o he feasble soluon (x j,1,, x,, x j,n) T, hen he vald evoluon s caed ou, and x s se o x +1,k, pobably ampludes (α, β) T s eaned, namely, α +1 = α, β +1 = β. Ohewse, he nvald evoluon s done, he feasble soluon (x j,1,, x,, x j,n) s eaned, and (α, β) T s updaed by quanum oaon gaes as: [ α +1 ] [ ] [ ] cos( θ β +1 = ) sn( θ) α sn( θ) cos( θ) β (6) whee θ s he oaon angle, and he value of θ has an mpoan effec on he pefomance of algohms, we desgn θ as: ( θ = sgn(αβ )θ 0 exp β α 1 ) (7) l whee sgn( ) s he sgn funcon and deemnes he decon of θ, guaanees fuhe he seach decon of convegence o he global opmum; θ 0 s he nal oaon angle; l denoes he mean absolue value of he gadens of objecve funcon a he seach pon Xj. θ 0, (α, β) T and l decde he sze of θ ogehe, conol fuhe he convegence ae. Fom Eqs.(6) and (7), we can see ha, wh he ncease of eaons, he value of α 2 deceases gadually, and hen Fne seach n he neghbouhood of cuen soluon s caed ou. In evese, he value of β 2 nceases gadually, and hen Coase seach n he whole seach space s ealzed. Fg.1 demonsaes clealy he pncple of complemenay muaon opeao n he case of (α, β) T n he I-h quadan. Fg. 1. Pncple of complemenay muaon opeao Based on Fne seach n local seach space and Coase seach n global seach space, NRQEA can ea he balance beween exploaon and exploaon. Le he numbe of Fne seach and Coase seach fo evey ndvdual s m 1 and m 2, especvely. In addon, we ake full advanage of he change ae of he objecve funcon a calculang θ. A he seep aea n he soluon space, he change ae of he objecve funcon s bgge, hus θ s nceased popely o make he algohms seach he soluon space a a smalle sep, whch can avod mssng he global opmum. On he conay, a he fla aea n he soluon space, he change ae of he objecve funcon s smalle, hus θ s deceased popely o make he algohms seach he soluon space a a bgge sep, whch can acceleae he seach pocess of algohms. Consdeng ha sngle-gene muaon s adoped n NRQEA, he value of l n Eq.(7) can be desgned as 1 l = m 1 + m 2 m 1 +m 2 m=1 f(x +1 j ) m f(xj) m x (8) x +1 whee m s he numbe of muaon. 3. Dscee cossove opeao In NRQEA, Dscee cossove s pefomed a peod τ c o enhance he nfomaon necouse among he ndvduals and make use of he bee gene obaned. Selec k excellen ndvduals n populaon by he ode of fness value, k < N, and hen, fo p u, u = 1,, k, selec anohe ndvdual p v, v = 1,, N, v u, a andom n populaon, le p u and p v as paens, exchange evey coespondng gene of hem by 0.5 pobably, and geneae new ndvdual c. NRQEA pus n pacce dscee cossove fo each excellen ndvdual seleced m 3 mes. If he fness value of c s

3 Real-coded Quanum Evoluonay Algohm fo Global Numecal Opmzaon wh Connuous Vaables 501 supeo o ha of p u, hen p u s se o c. Ohewse, p u s eaned. I s wohwhle o menon ha, when he componens of vaable veco X n Eq.(1) ae songly coelaed, dscee cossove opeao would play an mpoan ole on pevenng algohms fom beng apped n he local opma. 4. Technque of educng seach space Geneally, he smalle he seach space coveng he global opmum, he ease he global opmum beng found. In ohe wods, he sze of he seach space has an mpoan effec on he seach effcency of algohms [9]. So a echnque of educng he seach space accodng o he evoluonay pocess of algohms s pesened, namely, he seach space s educed gadually when he bes soluon obaned can no be fuhe mpoved n successve τ eaons. Le X = (x 1, x 2,, x n) T epesens he ange of he - h educng he seach space, and x [l, u ], = 1, 2,, n, whee l and u lm he ange of x and s defned as: { l = mn{x 1,, x 2,,, x N,} ζ(u 1 u = max{x 1,, x 2,,, x N,} + ζ(u 1 l 1 ) l 1 ) whee ζ s a andom numbe unfomly dsbued beween 0 and 1. To guaanee ha he seach space s smalle and smalle, he boun of l and u s lmed especvely as: { l = l 1, f l < l 1 (10) l = l, f l l 1 { u = u 1, f u > u 1 (11) u = u, f u u 1 5. Pocedues of NRQEA The pocedues of NRQEA can be descbed as follows: Sep 1 Deemne he paamees: N, θ 0, m 1, m 2, τ c, k, m 3, and τ, and nalze a populaon p = {p 1, p 2,, p N }. Sep 2 Evaluae fness value of each ndvdual and selec he bes ndvdual b. (9) Sep 3 Updae populaon p by complemenay muaon opeao. Fo he ndvdual p j n populaon, j = 1,, N. Sep 3.1 Updae he vaable x of he -h gene n un usng Eqs.(3) (5), = 1,, n, and Fne seach and Coase seach s pefomed m 1 and m 2 mes, especvely. Selec he bes ndvdual b. Sep 3.2 Updae he pobably ampludes (α, β) T of he -h gene n un usng Eqs.(6) (8). Unl all ndvduals n populaon ae updaed, Sep 3 s pefomed epeaedly. Sep 4 Cay ou he dscee cossove epealy m 3 mes fo k excellen ndvduals seleced n p when he dscee cossove condon s sasfed, especvely. Sep 5 Reduce he seach space usng Eqs.(9) (11) when he bes soluon obaned can no be fuhe mpoved n successve τ eaons, and hen ean he bes ndvdual and nalze ohe ndvduals of p n he educed seach space. Sep 6 sasfed. Loop o Sep 2 unl a emnaon ceon s III. Expemens and Resuls To evaluae he poposed NRQEA, we compaed s pefomance agans QEA [4] and RCQEA [10] on sx benchmak funcons. All algohms ae pogammed by Malab6.5 and un on Thnkpad IBM-T43 wh a 1.86GHz CPU and 512MHz memoy. 1. Benchmak funcons Sx benchmak funcons whch ae aken fom Ref.[4] wee used fo he pefomance ess. All he benchmak funcons ae mnmzaon poblems. The deals of hese funcons ae lsed n Table 1. Funcon Sphee Ackley Table 1. Deals of sx benchmak funcons Dmenson Seach Global Fomulaon (n) space mnumu f 1 (x) = n =1 x2 30 ( 100, 100) 0 ( ) 1 f 2 (x) = 20 exp 0.2 n =1 n x2 ( ) 30 ( 32, 32) 0 1 exp n =1 n cos(2πx ) e Gewank f 3 (x) = 1 n n = x2 =1 Rasgn f 4 (x) = 10n + n Schwefel f 5 (x) = n n Rosenbock ( ) x cos ( 600, 600) 0 =1 (x2 10 cos(2πx )) 30 ( 5.12, 5.12) 0 =1 x sn( x ) 30 ( 500, 500) 0 f 6 (x) = n 1 =1 (100(x +1 x 2 )2 + (x 1) 2 ) 30 ( 30, 30) 0 2. Paamees selecon The paamees selecon fo QEA and RCQEA ae efeed n Refs.[4, 10]. The paamees selecon fo NRQEA ae obaned fom lage numbes of expemens. QEA: he populaon sze s 10, local goup sze s 5, local mgaon peod s equal o 50, and global mgaon peod s equal o 100, ohe paamees ae se as follows: θ 0 = 0.05π, and ε = Consdeng he esoluons of vaables, he numbes of Q-bs fo sx benchmak funcons s se o 18 bs (pe vaable), especvely. RCQEA: he populaon sze s equal o 10, ohe paamees ae se as follows: θ 0 = 0.4π, γ = 0.05, m 1 = 6, m 2 = 2, τ c = 100, k = 5, and m 3 = 6. NRQEA: he populaon sze s equal o 10, ohe paamees ae se as follows: θ 0 = 0.1π, m 1 = 6, m 2 = 2, τ c = 100, k = 5, m 3 = 6, and τ = 10.

4 502 Chnese Jounal of Eleconcs 2011 Fo he compason pupose, he populaon sze of all algohms s equal, and he mos paamees of RCQEA and NRQEA ae also same. The emnaon condon wh he maxmum numbe of eaons s used. To ge d of he andomsy, QEA, RCQEA and NRQEA ae all un 50 mes. 3. Resuls and dscusson Table 2 shows he pefomance compasons among QEA, RCQEA and NRQEA based on he sascal esuls. Fom he bes and he wos, we can daw concluson ha NRQEA has a bee ably o seach he global opmum han QEA and RCQEA, and can avod effecvely fallng no he local opmum egons. Fom he mean and he devaon, we can see ha NRQEA has bee pefomance han QEA and RCQEA n ems of a seady convegence and a bee obusness. Table 2. Resuls compason of 50 uns fo sx benchmak funcons f Algohm Mean Bes Wos Devaon QEA 1.81E E E E+2 f 1 RCQEA 1.74E E E E 3 NRQEA 8.81E E E E 10 QEA f 2 RCQEA 3.51E E E E 3 NRQEA 1.09E E E E 5 QEA f 3 RCQEA 4.26E E E E 2 NRQEA 2.27E E E 7 QEA E f 4 RCQEA E NRQEA 1.57E E E 11 QEA 3.73E E E E+2 f 5 RCQEA 8.97E E E-2 NRQEA 2.21E E E E 5 QEA 3.88E E E E+5 f 6 RCQEA NRQEA 6.74E E E E 2 Fg.2 llusaes he aveage convegence cuves of QEA, RCQEA and NRQEA fo sx funcons. Fom Fg.2, we can clealy see ha NRQEA pefoms sgnfcanly bee han QEA and RCQEA n ems of convegence ae and opmum esuls. The fallng slope of NRQEA s seepe han ha of QEA and RCQEA, whch shows ha he convegence ae of NRQEA s fase han ha of QEA and RCQEA. And s wohwhle o menon ha NRQEA manans a nealy consan convegence ae all he me, and appoaches apdly owads he global opmum. A he same me, he opmum esul of NRQEA excels sgnfcanly ha of QEA and RC- QEA, whch fuhe shows ha NRQEA has a moe poweful ably o seach he global opmum han QEA and RCQEA. Fom he above expemens, we can see ha, he pefomance of RCQEA s sgnfcanly bee han QEA, he eason should be ha he epesenaon of eal numbe s moe suable fo numecal opmzaon han ha of bnay sng and he desgned evoluonay saeges can ea he balance beween exploaon and exploaon, moeove, he pefomance of he NRQEA s sgnfcanly supeo o ha of RC- QEA, he eason should be ha he mpoved complemenay muaon opeao consdes well he chaacesc of objecve funcon and he echnque of educng he seach space makes easy o fnd he bee seach aea. Fg. 2. The aveage convegence cuves of dffeen algohms IV. Concluson Ths pape has poposed NRQEA, whose coe s ha an mpoved evoluonay saegy usng he gaden nfomaon of objecve funcon s desgned, and a echnque of educng he seach space accodng o he evoluonay pocess of algohms s adoped. The expemens esuls demonsae ha NRQEA has he poweful ably o seach global opmal soluons and he ably o effecvely avod fallng no local opmum soluons, and NRQEA s hghly effcen fo solvng numecal opmzaon wh connuous vaables. In he fuue, how o mpove he pefomance of NRQEA s ou man wok. In addon, moe es and analyss of dffeen poblems need o be done on NRQEA o wden he applcaon scope of NRQEA. Refeences [1] T. Hey, Quanum compung: An noducon, Compung and Conol Engneeng Jounal, Vol.10, No.3, pp , [2] A. Naayanan, M. Mooe, Quanum-nsped genec algohms, Poc. of IEEE In. Conf. on Evoluonay Compuaon, Nagoya, Japan, pp.61 66, [3] K.H. Han, J.H. Km, Quanum-nsped evoluonay algohm fo a class of combnaoal opmzaon, IEEE Tansacon on Evoluonay Compuaon, Vol.6, No.6, pp , [4] K.H. Han, J.H. Km, Quanum-nsped evoluonay algohms wh a new emnaon ceon, H ε gae, and wophase scheme, IEEE Tansacons on Evoluonay Compuaon, Vol.8, No.2, pp , [5] Yang Junan, Zhuang Zhenquan, Sh Lang, Mul-Unvese

5 Real-coded Quanum Evoluonay Algohm fo Global Numecal Opmzaon wh Connuous Vaables 503 paallel quanum genec algohm, Aca Eleconca Snca, Vol.32, No.6, pp , (n Chnese) [6] Cheng Hu, Zhang Jashu, Zhang Chao, Real coded chaoc quanum-nsped genec algohm, Conol and Decson, Vol.20, No.11, pp , (n Chnese) [7] L Panch, L Shyong, Quanum-nsped evoluonay algohm fo connuous spaces opmzaon, Chnese Jounal of Eleconcs, Vol.17, No.1, pp , [8] Wang Xangzhong, Yu Shouy, Impoved evoluon saeges fo hgh-dmensonal opmzaon, Conol Theoy and Applcaons, Vol.23, No.1, pp , (n Chnese) [9] Gong Dunwe, Sun Xaoyan, Mul-populaon genec algohms wh vaaonal seach aeas, Conol Theoy and Applcaons, Vol.23, No.2, pp , (n Chnese) [10] Gao Hu, Xu Guanghu, Zhang Ru e al., Real-coded quanum evoluonay algohm, Conol and Decson, Vol.23, No.01, pp.87 90, (n Chnese) GAO Hu was bon n He eceved Ph.D. degee n oad and alway engneeng fom Habn Insue of Technology n Snce 2008, he has been wokng as a posdoco n School of Taffc and Tanspoaon of Souhwes Jaoong Unvesy. Hs eseach neess nclude nellgen compuaon, nellgen conol and nellgen affc sysem. (Emal: wsjdx gh@sna.com) ZHANG Ru was bon n She eceved Ph.D. degee n conol heoy and conol engneeng fom Habn Unvesy of Scence and Technology (HUST) n Now she s a pofesso n School of auomaon a HUST. He eseach neess nclude powe quanly analyss, powe lne communcaon and nellgen opmzaon.