Abstract. Keywords: Mutation probability, evolutionary computation, optimization, sensitivity, variability. 1. Introduction. 2. Proposed Algorithm

Transcription

1 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue Absrac Redefg Muao Probables for Evoluoary Opmzao Problems Raja Aggarwal Faculy of Egeerg ad Compuer Scece Cocorda Uversy, Moreal, Caada Presely geec operaors evoluoary opmzao compuao employ sochasc or radom fucos. Some doma opmzao s devod of deal saes ad hese muao probables rely heavly o fess crera. Moreover evoluo of deal sysem sae ca be muual wh hbve muaos. Susaed dyamsm refleced hough varably s par of all real sysems ad s o deerre for cause of covergece for fal opmzed sae. There s eed for cosse sysem approach for basg hs dea of roducg dversy elemes as well as corollg dversy hrough devao over geeraos. There s eed o dffereae sysem doma by localzg key sesve varables wh coguous varably me. Preseed research s a aemp o redefe muao probables hs uceray of evoluo bologcal course of opmzao by roducg sesvy ad varably. Epochs as par of hs varably gude muaos ad gover sysem doma classfcao whle varably s asceraed hrough order or dsorder. Proposed formulao defes logcal sequece as a couum for roducg dverses hrough muaos sysem doma elemes. Couum as a logcal sequece comprses varous sraums. Varably of order emaaes from ucleus or sesve varables sysem doma as par of coformal sraum. Ths s mpora as hbve perpeual muaos geeraos could be par of mmue sysems oo. Keywords: Muao probably, evoluoary compuao, opmzao, sesvy, varably. 1. Iroduco All evoluoary problems rely o geec or muaoal operaors for roducg dversy hrough vared probably dsrbuos.e. Hollad [1], Goldberg [2]. All geec algorhms rely o hese operaors for seleco, cross over ad muao roducg varably populao. Evoluoary sraeges, evoluoary programmg ad geec algorhms roduce muao hrough Gaussa operaors or backgroud operaors hrough b verso. Ths leads o offsprg resemblg more o her pares wh crease umber of geeraos Mchaelewcz [3]. Eve heursc rules dcae decay of performace for large varable populao wh larger muao probables (P m >0.05) or small varable populao wh smaller muao probables (P m <0.002). Sce opmzao s based o probablsc raso rules, hese geec operaors eed o be mplemeed ad guded by logcal sysem ad should o base fxed deermsc rules lke mea of dsrbuo as used by Goldberg [2]. Preseed research s a aemp o move away from radom probably fuco dsrbuos o oe based o sysem characerscs wh evoluo deermed hrough dvdual varable evoluo. Thus muao sregh of dvdual varable s dyamcally evolved ad deermed. I s o flueced by covergece velocy. 2. Proposed Algorhm Proposed algorhm draws physcal descrpo from basc fucoal ees of huma begs ad heceforh wll be called Cell- Nucleus opmzao algorhm. Objecve parameer o be opmzed s depede upo sysem doma D whch comprses N varables whch are flexble cosras ad sesves. Sesve varables as govered by sesvy occupy ucleus of sysem doma. Sysem s defed hrough ordered arrageme of sesve varables ucleus depedg upo he level of sesvy. Mos sesve varables or parameers wll occupy he cere mos locaos. These varables maybe ordered or dsordered. Proposed algorhm develops muao sregh hrough varably ad sesvy. Sce prese day muao formulaos rely o varably loe, varable characerzao ad varable self- evoluo s lef ou. Thus I geerae a logcal couum from sesvy o varably ad from varably o probably. Characerzao sesvy s acheved hrough order ad dsorder whereby hgher level of dsorders form Epochs. Sequeal operao depeds upo

2 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue (a) Seleco: Frs of all sesve varables wll be separaed o form he ucleus for sysem doma. (b) Coformal: Here varably hrough Epochs s worked ou. (c) Defo: Fally, muao probably s defed. 2.1 Coexual Defos Some coexual defos are: Sesve Dsace: I refers o he dsace bewee he doma ucleus varable from referece sae o dealzed sesve surface. Sesve Surface: Ths s hem-sphercal surface formed varably for ordered ucleus varables. Ths forms he referece for workg ou sesve dsaces defg muao sregh. SENSITITVE SURFACE DOMAIN BOUNDARY NUCLEUS VARIABLES Fgure 1: Proposed Sysem doma showg ees Fgure 1 shows sysem doma ad sesve varables whch occupy he cere ordered arrageme as govered by sesves. Newly defed sesve dsaces gude muao sregh. 2.2 Oher Defos Varably s measure of epochs me for ucleus varables. Epochs are abrup chages ucleus or sesve varables sae. Fgure 2 shows uscaled paer for hree plaes ucleus varables. A deal or referece sae comprses sesves for core varables o f o sem-crcular profle.

3 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue Fgure 2: Trlogy of plaes 3. Mahemacal Formulao Le here be varables ha defe objecve parameer or fuco o be opmzed. The doma D s { 1,2,3... } D..., = (1) Ou of hese varables, m varables are sesve as govered by equao (3). The ucleus ad cyoplasm varables se wll be defed by { 1,2,3 m} {,2,3,... m } S =,..., C =..., 1 (2) Sesve varables are seleced hrough sesvy. S [ E( y x )] ( y) Var Var = (3)

4 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue Sysem Doma Three ypes of sysem domas depedg upo he sysem varables sesves ca be defed as (a) Supersesve sysem C = { } (4) (b) Sesve sysem S D S C = 0 (5) (c) Immue sysem C = D S = 0 (6) Supersesve sysem mark o-exsece of cyoplasm varables whle mmue sysem mark saga sysem. Mos of he sysem are reaed as sesve sysem as defed by Equao 5. Sce sysem self evolves me, objec parameer o be opmzed a saaeous me s represeed as O(). Furher chage or varably he values of doma space varables be represeed by Δ x ad chage or varably value of objecve parameer wll be Δ O(). Smlarly we ca defe he cosra se as O,m m ( ) O( ) O,max max (7) Varably he dvdual doma varables arbues o he dffereces he saaeous ad mea value. Thus [( ( ) ) ] 2 Δ ( ) = E μ (8) Varably facor s defed as he rao of he varao he value of he objecve parameer o be opmsed o he cumulave varao all he varables of he doma sysem. V f ( ) = ΔO( ) = 1 Δ ( ) (9)

5 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue Correlag Epoch ad Sesves Sce Epoch s measure of hgher sesvy or varably, we cosderg epochs E() o be hrd order varably he objecve parameer. Epochs corbug o muao are measured hrough hgher varably measured sesve dsaces. E( ) 3 O( ) = η 3 (10) (A) SENSITIVE SURFACE PROFILE (B) MUTATION PROFILE Fgure 3: Sesve Profle comparso 3.3 Muao operaor sregh Muao operaor sregh s guded by sesve dsaces ad sesve varables. Sce sesvy s draw from varably, crease Epochs wll acceuae muao sregh. Thus dyamc muao sregh evolves for dvdual parameer ad s o deermsc for rapd covergece. Proposed Muao operaor sregh s defed as + 1 = + 2 ( μ, ) ξ. N σ D s ξ = ( ) 0.5C 1 c, m (12) (11)

6 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue c m = c, (13), 1 + = 2 ( X s, X ) ( X s, X ) = 1 1/ 2 D s (14) 2 Where Modfed dsace parameer s govered by referece sesble dsaces muaoal profle wh me. c, refers o umber for he sesve varable geerag muaoal profle. Number of varables udergog sesve chages decreases wh me ad hus modfed parameer for sesve varables also reduces wh me as show Equao Aalyss ad Coclusos Fgure 4 compares sac muao sreghs depedg upo umber of varables defg objecve fuco o be opmzed. Fgure 5 compares dyamc averaged muao sreghs from Back [7] ad proposed operaor. Codog parameer ad chage sesve varables are fucos of me. Fgure 4: Sac muao sreghs comparso

7 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue Fgure 5: Dyamc muao sreghs comparso No-uform, ormal, polyomal muaos have all roduced radom dsrbuo fuco opmzg egeerg sysems. Eve mea evoluo approach of Fogel [4], emprcal muao raes by Schaffer e al. [5] & Hesser ad Maer [6] uses eher sac muao probably or rase muaoal sregh based o exogeous parameers. Fess fucos Fogel [4] hemselves do o accou for hbve muao. There s o cosderao for sesves localzed varables gudg muao sregh for evoluoary problems. Proposed formulao has re-defed muao sregh cosderao for evoluoary opmzao problems by cosderg dyamc self-evolvg muaoal parameers based o cosderaos of sesvy ad varably. Algorhm brgs o cosderao role played by sesve dsaces ad sesve varables for opmzao logcal sysem whch evolves hrough me. Nomeclaure C 1 E c, N N S X Var V f Codog parameer Epoch Number of sesve varables or parameers Number of sesve varables udergog muaoal chages Toal umber of varables or parameers Normal dsrbuo Sesvy of h varable Sesve dsace Varace Varably facor

8 EgOp 2008 Ieraoal Coferece o Egeerg Opmzao Ro de Jaero, Brazl, Jue μ σ ξ Mea Sadard Devao Muao Operaor Refereces 1. Hollad, J. H. Adapao aural ad arfcal sysems. The Uversy of Mchga Press, A Arbor, MI, Goldberg, D. E. Geec algorhms search, opmzao ad mache learg, Addso Wesley, Readg, MA, Mchalewcz, Z. Geec Algorhms & Daa srucures Evoluoary programs. Berl: Sprger-Verlag, Fogel, D. B. A aalyss of evoluoary programmg, I Fogel ad Amar, pp , Schaffer, J. D. Carua, R. A. Eshelma, L. J. Das, R. A sudy of corol parameers affecg ole performace of geec Algorhms for fuco opmzao, I Schaffer, pp 51-60, Hesser, J. Maer R. Towards a opmal muao probably geec algorhms. I Schwefel ad Maer, pp , Back, T. The eraco of muao rae, seleco, ad self adapao wh geec algorhm. I Maer ad Maderck, pp , 1992.