Theoretical Analysis of Biogeography Based Optimization Aijun ZHU1,2,3 a, Cong HU1,3, Chuanpei XU1,3, Zhi Li1,3

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1 6h Inernaonal Conference on Machnery, Maerals, Envronmen, Boechnology and Compuer (MMEBC 6) Theorecal Analyss of Bogeography Based Opmzaon Aun ZU,,3 a, Cong U,3, Chuanpe XU,3, Zh L,3 School of Elecronc Engneerng and Auomaon, Guln Unversy of Elecronc Technology, Guln 544, P.R.Chna Deparmen of Compuer Scence and Engneerng, Texas A&M Unversy, College Saon, TX, USA 3 Guangx Key Laboraory of Auomac Deecng Technology and Insrumens, Guln 544, P.R.Chna; a emal: zbluebrd@gue.edu.cn Keywords: mea-heursc; global opmzaon; NP hard problem Absrac. Snce he compuaon ably of compuer mproves dramacally, a lo of new mea-heursc mehods arse. All hose algorhms are orgnaed from some mechansms n naure, and are smlar n srucure and wdely used o solve global opmzaon problems. owever, evoluonary algorhm, such as BBO, s lac of src heory foundaon and hard o be analyzed n heory, because comes from heursc dea and has complcaed random behavor. Therefore, In hs paper, we propose a Marov chan model of BBO o analyze he relaonshp beween ndvdual vecor and PopSze, and prove ha a Marov populaon seres{ = ξ,,,,...}(ξ Ω) n BBO s an absorbng Marov chan. Convergence analyss of BBO s obaned, whch s he Marov populaon seres n BBO converge o obecve subspace B wh probably one. Inroducon Snce 96s, a lo of researchers are neresed n evoluonary compuaon wh he developmen of compuer echnology. Some famous algorhms are proposed, such as Genec Algorhm, Evoluonary Programmng, and Evoluonary Sraegy. Snce he compuaon ably of compuer mproves dramacally, a lo of new mea-heursc mehods arse, such as An Colony Opmzaon (ACO) [], Parcle swarm Opmzaon (PSO) [], Dfferenal Evoluon (DE) [3], Grey Wolf Opmzaon (GWO) [4], Bogeography Based Opmzaon (BBO) [5], ybrd Grey Wolf Opmzaon (GWO) [6],e al. All hose algorhms are orgnaed from some mechansms n naure, and are smlar n srucure. We call such algorhms as evoluonary algorhms. In general, evoluonary algorhms are random and heursc opmzaon mehods, whch are wdely used o solve global opmzaon problems. As we use evoluonary algorhms o solve opmzaon problems, s easy o add some heursc deas. Therefore, evoluonary algorhms demonsrae excellen performance n a lo of felds such as daa mnng, engneerng opmzaon an ndusry desgn. owever, evoluonary algorhm, such as BBO, s lac of src heory foundaon and hard o be analyzed n heory [7], because comes from heursc dea and has complcaed random behavor. When we use BBO o solve opmzaon problems, here are shor of heorecal guaranee. Therefore, In hs paper, we propose a Marov chan model of BBO o analyze he relaonshp beween ndvdual vecor and PopSze, and prove ha a Marov populaon seres = {ξ,,,,...}(ξ Ω) n BBO s an absorbng Marov chan. Convergence analyss of BBO s obaned, whch s he Marov populaon seres= {ξ,,,,..., }(ξ Ω) n BBO converge o obecve subspace B wh probably. 6. The auhors - Publshed by Alans Press

2 Basc of BBO Research wor abou bogeography could be raced o nneeen cenury, when Alfred Wallance[8] and Charles Darwn[9]sared her research. In 96s, Rober MacArhur and Edward Wlson obaned breahrough and hey publshed her achevemen and creaed mahemacal model of bogeography. In 8, Dan Smon [5] proposed a new opmzaon mehod based on bogeography called BBO, whch are wdely appled n Engneerng Opmzaon [-4]. The mahemacal model of bogeography demonsraes he process ha speces mgrae from one sland o anoher. Islands or habas ha are well sued for speces oban hgh aba Suably Index (SI). Theorecal Analyss of BBO As we creae Marov chan model for BBO, we should frs map he sae space of a Marov chan o he operaon space of BBO. If he populaon sze of BBO s one, he sae space of he Marov chan s he soluon space. Tha means every soluon s a sae. If we code he soluon wh d bs bnary sysem, he soluon space s S = {, } d. Furhermore, he number of he saes s S = d. If he populaon sze of BBO s PopSze, whch s greaer han one, suppose he populaon s a ordered and repeaable se. Tha means some ndvduals n he populaon could be he same. The sae space of he Marov chan could be mapped o he populaon space Ω= {,} d PopSze. Furhermore, he number of he populaon saes s Ω= d PopSze. If he populaon s an unordered and repeaable se, he sae space of he Marov chan could be mapped o he populaon space Ω {,} d PopSze. Afer he sae space of he Marov chan s se, arbrary populaon X x and X y, populaon X y s generaed by populaon X x a a probably P( x+ = X y x = Xx), whch s he ranson marx of he Marov chan. Defnon A sae S x of a Marov chan s called an absorbng sae, f could no escape self. Tha means p =. A Marov chan s called an absorbng Marov chan, f has a leas one absorbng sae and an arbrary sae could reach an absorbng sae a one or several seps. The above defnon could be also saed as follows: gven a Marov populaon seres { ξ, =,,,...}( ξ Ω ) and a obecve subspace B Ω, he Marov chan s an absorbng Marov chan, f P( ξ B ξ + B) =, {,,,...}. () In BBO algorhm, here are several operaons such as mgraon and muaon operaon. Marov chan model should be creaed for he respecve operaon. Suppose a soluon space T s d bs bnary sysem, we ge he sze of T s m= d. {,,3,..., m},a soluon x,here s a populaon P wh PopSze ndvduals. We defne a vecor w = ( w, w,..., w,..., wm), {,,3,..., m}, and w ndcaes he number of x n he populaon P. We can oban he relaonshp beween w and PopSze as follows: Proposon An arbrary populaon n BBO s made up of PopSze ndvduals whch m d PopSze d selec from he search space T = { x, x, x3,..., x m }, ha means ψ : R R,here we ge where w ndcaes he number of m PopSze = w () = x n he populaon P, PopSze s he populaon sze.

3 Proof. Le populaon P = { y, y,..., y..., ypopsze}, {,, 3,..., PopSze}, y T, we could fnd xc X and xc = y, c {,,3,..., m}. Therefore, P s rewren as P = { x, x,..., x, x, x,..., x,..., x, x,..., x,,..., x, x,..., x } (3) m m m w w w wm w {,,,..., PopSze}.As we now he populaon sze of P s PopSze, he sum of w {,,,..., PopSz e} s PopSze. Proposon a Marov populaon seres{ ξ, =,,,...}( ξ Ω) n BBO s an absorbng Marov chan. Proof. Gven a Marov populaon seres { ξ, =,,,...}( ξ Ω ) and a obecve subspace B Ω, f he Marov chan s an absorbng Marov chan, he necessary and suffcen condon s as follows: P( ξ B ξ + B) =, {,,,...} Frs prove suffcency: f P( ξ B ξ + B) =, {,,,...} holds, hen means he populaon ξ a momen belongs o obecve subspace B. I s obvous ha he populaon ξ + a momen + mus belongs o obecve subspace B. Accordng he defnon of absorbng Marov chan (defnon ), he populaon seres{ ξ, =,,,...}( ξ Ω) n BBO s an absorbng Marov chan. Then prove necessy: populaon seres { ξ, =,,,...}( ξ Ω ) adops ele sraegy, ha s f he populaon a momen conans he opmal soluon x, wll ae he place of he wors soluon and reman n he nex populaon ξ + a momen +. Evdenly, he populaon ξ + a momen + conans he opmal soluon x, ξ + B.Therefore, P( ξ B ξ + B) =. Lemma Suppose a soluon space wh d bs bnary sysem, θ () s = { : x() s = x()} s whch s defned as Defnon, hen we ge () θ() = θ() = θ3() =... = θ d () = θ d (). / / () θ d θ d θ d θ d θ d /+ /+ /+ 3 () = () = () =... = () = (). Suppose a populaon P = { y, y,..., y..., ypopsze}, {,, 3,..., PopSze}, we order y accordng o he sequence of x, hen we ge y x, f ( =,,..., w) x, f ( = w+, w+,..., w+ w) x3, f( = w+ w +, w+ w +,..., w+ w + w3) = m m m xm, f ( = ( w) +,( w) +,...,( w) + wm) (4) The above formula could be smplfed as y= xg( ), =,,3,..., PopSze Where g ( ) could be expressed as 3

4 g ( ) = mn( z) subec o w. z (5) Proposon 3 Marov ransfer marx T afer one generaon n BBO s an PopSze PopSze marx, where PopSze s PopSze = ' S ( ) PopSze = PopSze PopSze ' PopSze+ S ( ) = { R : {,,,..., m}, = m, = PopSze. Proof. As PopSze =, PopSze s m because here are m soluons n he soluon space. The + rgh of he above equaon equals. Because + = PopSze =. Because PopSze NP = m, we ge m (6) = PopSze, we ge = and =. The rgh of m m equaon () equals = m. As PopSze =, equaon () holds. m Suppose equaon () holds, as PopSze = Pop. Then as PopSze = Pop +, we ge m K =....,,...,,, Pop K m Pop K Pop Pop K Pop + = + + = m m! K K! Because of = and =, we oban,,..., Pop, K!!... Pop! K! Pop, Pop!! m! K! m! = =.!!...! K!!!!!...!!! Pop Pop Pop Pop The above equaon could be rewren as =... = '. S ( ) = Lemma The opmal value n a seres of generaons n BBO algorhm s non-ncreasng,.e., S( X( + )) S( X( )) (7) Proof. Durng he updae of every generaon n BBO algorhm, he bes ndvdual x bes n -h generaon s sored and used o replace he wors ndvdual n (+)-h generaon. Therefore, here s always he bes ndvdual n he (+)-h generaon, whch belongs o he -h generaon. Theorem. Gven a Marov populaon seres { ξ, =,,,...}( ξ Ω ) n BBO algorhm and an obecve subspace B Ω, where Ω s he populaon space. B = { Y = { y, y,..., y }: {,,..., NP}, y B }. The Marov populaon seres NP 4

5 { ξ, =,,,..., }( ξ Ω ) converge o obecve subspace B wh probably,.e., lm P( ξ B ξ ) = (8) Proof. Suppose x s he opmal soluon o he cos funcon cos ( ) : f x x, accordng o he Proposon ( he Marov populaon seres{ ξ, =,,,...}( ξ Ω) n BBO s an absorbng Marov chan ), we can ge x x, =,,3,... Therefore, lm P( ξ B ξ ) = ; f x x,suppose >, x x, >, x x,hen ξ B and ξ B. Therefore, P( ξ, ξ ) >, P( ξ, ξ ) >,we ge ξ ξ ; f x x, suppose >, x x and >, x x, hen P( ξ, ξ ) =,ha s ξ! ξ.because B s a normally reurned, rreducble and non-cycle closed se, ξ, π( Y), π( Y) s a lm probably dsrbuon. we oban π ( Y), Y B lm P( ξ = Y ξ) =, Y B Tha sξ whch mus be n B,herefore lm P( ξ B ξ ) =. Concluson Evoluonary algorhm, such as BBO, s lac of src heory foundaon and hard o be analyzed n heory due o he complcaed random behavor. Therefore, we propose a Marov chan model of BBO o analyze he relaonshp beween ndvdual vecor and PopSze, and prove ha a Marov populaon seres n BBO s an absorbng Marov chan. Convergence analyss of BBO s obaned, whch s he Marov populaon seres n BBO converge o obecve subspace B. Acnowledgemen Ths wor s suppored by he Naonal Naural Scence Foundaon of Chna (656, 6563), Scence and Technology Research Proec of Guangx Deparmen of Educaon (KY5YB), Guangx Key Laboraory of Auomac Deecng Technology and Insrumens (YQ6), Talen Proec of Guln unversy of elecronc echnology(uf58y), Chna Scholarshp Councl(58455). Thans Professor Fabrzo Lombard, who s wh Norheasern Unversy, USA, he has gven some valuable suggesons. Grea apprecaon o Rab N Mahapara, who s wh Texas A&M Unversy. References [] M. Dorgo, M. Braar, T. Suzle. An colony opmzaon, Compu Inell Magaz, 6, : [] J. Kennedy, R. Eberhar, Parcle swarm opmzaon. IEEE Inernaonal Conference on Neural Newors, 995:

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