An Optimization Algorithm Based on Binary Difference and Gravitational Evolution

Transcription

1 Internatonal Journal of Computatonal Intellgence Systems Vol. 5 No. 3 (June 0) An Optmzaton Algorthm Based on Bnary fference and Gravtatonal Evoluton Junl L Yang Lou and Yuhu Sh 3. College of Computer Scence Schuan Normal Unversty Chengdu Chna. Informaton Scence and Engneerng College Nngbo Unversty Nngbo 35 Chna 3. epartment of Electrcal and Electronc Engneerng X an Jaotong-Lverpool Unversty Suzhou 53 Chna E-mal: l.unl@vp.63.com louyang@mal.nbu.edu.cn Yuhu.Sh@xtlu.edu.cn Abstract Unversal gravtaton s a natural phenomenon. Inspred by Newton's unversal gravtaton model and based on bnary dfferences strategy we propose an algorthm for global optmzaton problems whch s called the bnary dfference gravtatonal evoluton (BGE) algorthm. BGE s a populaton-based algorthm and the populaton s composed of partcles. Each partcle s treated as a vrtual obect wth two attrbutes of poston and qualty. Some of the best obects n the populaton compose the reference-group and the rest obects compose the floatng-group. The BGE algorthm could fnd the global optmum solutons through two crtcal operatons: the self-update of reference-group and the nteractve-update process between the reference-group and floatng-group utlzng the gravtatonal evoluton method. The parameters of BGE are set by a tral-and-error process and the BGE s proved that t can converge to the global optmal soluton wth probablty. Benchmar functons are used to evaluate the performance of BGE and to compare t wth classc fferental Evoluton. The smulaton results llustrate the encouragng performance of the BGE algorthm wth regards to computng speed and accuracy. Keywords: Optmzaton; Bnary fference; fferental Evoluton; Gravtaton.. Introducton Evolutonary algorthms are a seres of problem-solvng methods that based on smulaton of the natural evoluton system and ts development can be traced bac to 950s. Compared wth the classc optmzaton methods an evolutonary algorthm has many advantages. For example t s unconstraned by the search space lmtatons; t s unconstraned by the functon types; and functon gradent nformaton s not essental etc. Evolutonary algorthms have been wdely used for optmzaton problem. Bologcal evoluton theory nspred the emergence and development of boncs whch motvated many types of evolutonary algorthms such as Genetc Algorthm [-] Smulated Annealng [3] Immune Algorthm [-6] Partcle Swarm Optmzaton [7-9] Ant Colony Optmzaton [0-] and Bacteral Foragng Optmzaton [] etc. All these contrbutons have ganed great achevements on the feld of optmzaton. Chrs and Tsang frstly proposed the concept and frame of GELS (Gravtatonal Emulaton Local Search) [3] n 995 whch was then further developed by Webster []. Balachandar and Kannan [5] proposed RGES (Randomzed Gravtatonal Emulaton Search) algorthm n 007 whch overcame some wea ponts of GELS such as a relatvely slow convergence rate and low qualty of solutons. The three gravtatonal emulaton search methods above were ntally appled to solve combnatoral optmzaton problems such as Travelng Salesman Problem (TSP). Hsao and Chuang et. al. proposed the SGO (Space 83

2 J. L et al. Gravtatonal Optmzaton) [6] whch was based on Ensten s theory and Newton s law of gravtaton. The SGO smulated the process that a number of planets shft n the space to search for the planet wth the most massve. The geometrc transformaton of the space generates a force to mae the planets shftng faster or slower. In 007 Chuang and Jang proposed an algorthm named IRO (Integrated Radaton Optmzaton) [7] based on the phenomenon that the movements of one planet n the gravtatonal feld were compostvely effected by the sum of all the other planets gravtatonal radaton forces. Rashed proposed the GSA (Gravtatonal Search Algorthm) n 007 and t was contnuously mproved thereafter [8-0]. In GSA a partcle s total gravtaton was a sum of all other partcles gravtatons wth random weghts and the total gravtaton generated the acceleraton to move. GPSO (Partcle Swarm Optmzaton based on Gravtaton) [] was proposed by Kang and Wang et. al n 007 whch ntroduced an acceleraton nto Partcle Swarm Optmzaton. All the four algorthms above were methods based on the poston and dsplacement and nspred by gravtaton each of whch has dfferent update formulas respectvely. In ths paper the proposed algorthm Bnary- fference Gravtatonal Evoluton (BGE) shares the same pont wth the algorthms above that s all these algorthms utlze the concept of gravtaton. However BGE does not smulate the physcal movement processes as above but va a clusterng process based on eltst strategy. In BGE obects n the populaton are clustered nto two dfferent groups: the referenceobect group and the floatng-obect group. Obects of dfferent groups are updated ndependently or cooperatvely and the bnary dfference [5] strategy s adopted to update obects n the updatng processes ndependently or cooperatvely.e. the self-update of reference-group and the nteractve-update group and nteractng process between the two groups. Ths paper s organzed as follows. Secton presents both the detals and ensemble of BGE. Expermental study s gven n Secton 3 where benchmar functons are used to evaluate the performance. Secton gves an analyss on the parameters and the convergence of algorthm s analyzed n Secton 5 followed by the concluson n Secton 6.. Bnary fference Gravtatonal Evoluton Algorthm.. Bnary fference Strategy The fferental Evoluton (E) [-3] algorthm emerged as a very compettve form of evolutonary algorthm more than a decade ago and was frst proposed by R. Storn and K. Prce n 995 whch was a smple and effcent heurstc algorthm for global optmzaton over contnuous spaces and ts feature s a mutaton wth a dfferental strategy. The selecton of dfferental strateges would mae sgnfcant nfluence on the performance of the algorthm. In consderaton of dversty there are at least three ndvduals nvolved n mutaton n E [3-]. To smplfy the mutaton operaton the bnary dfference strategy was ntroduced whch s smlar to the tradtonal but wth only two ndvduals nvolved. As dscussed n [5] bnary dfference usng a sorted populaton could mprove the performance of E especally n the aspect of convergence speed for low-dmensonal problems. Because the obects of the reference-obect group n BGE are sorted by ther qualtes bnary dfference s approprate to be ntroduced nto BGE. In bnary dfference two obects are selected as canddates of whch the one wth better ftness value s selected as the central obect and the obects wth relatve worse ftness values are nvolved n mutaton. The process of producng new obects s as follows. A central obect s temporarly fxed and the worse-ftness obects are updated one after another accordng to the central obect. The new obects are more lely to be close to the central obect. The bnary dfference strategy not only s smpler but also taes advantages of the gravtaton clusterng. The populaton s dvded nto two groups by gravtaton clusterng and the bnary dfference s used for the nteractve update of the two groups. The dmensonal update of an obect wth bnary dfference strategy s as follows: X ( new ) X ( ) csgn c U(0) X ( ) X ( ) where X () and X( ) are two obects n a populaton and X ( new) represents the newly produced ndvdual. The subscrpt means dmenson csgn s a random symbol c s a constant and U (0) s a random value unformly dstrbuted n [0]. Bnary dfference uses a 8

3 Bnary fference Gravtatonal Evoluton random and varable factor csgn c U(0) to eep dversty of the populaton... Gravtatonal Groupng Mode Gravtaton s an attractve force between two obects whch exsts between any two obects wth masses. The value of gravtaton s n proporton to the mass of ether obect whle n nverse proporton to the dstance between them. Calculaton of gravtaton can be descrbed as follows: m m F G () r where G 6.67 s the gravtatonal constant m and m are qualtes of two obects and r represents the dstance between two obects. Wthout loss of generalty we consder only the maxmzaton problem as optmzaton problem consdered n ths paper: Z max f ( X ) XS The search space s S { X l x u... n} where X ( x x x ) T n. l u( n) are the lower and upper bound. f () s the obectve functon. We treat a partcle n the search space S as an obect usng gravtaton for optmzaton. For any two obects A and A n the space S ther physcal postons are represented by X( A ) and X( A ) and ther qualtes are represented by ma ( ) and ma ( ). Then the gravtaton of A and A s smply descrbed as: F m( A ) m( A ) h( f ( A )) h( f ( A )) r K r K 0 0 where r means dstance measurement and K0 s a small-valued constant to ensure denomnator unequal to zero n Formula (). h() s a one-dmensonal scale () transformaton functon satsfed x ( ) hx ( ) 0 whch strctly ncrease monotoncally. For convenence h() s set as the absolute value here n accordance wth the concept that qualty s a nonnegatve value. m( ) h( f ( )) represents the qualty of an obect whch s scale transformed from obectve functon. Formula () s essentally equal to Formula () though the form s changed. In ths paper we calculate the gravtaton measurements but not the true gravtaton values. For convenence the ftness of an obect we mentoned n ths paper ndcates the qualty whch s a scale transformaton of the obectve functon..3. Bnary fference Gravtatonal Evoluton Summary The populaton denoted by RN s assemble of all obects whch are clustered nto two groups.e. the reference-obect group R and the floatng-obect group N. Followngs are the symbols used n the BGE descrptons. X( R ) represents the poston of the th reference-obect R X( R ) represents ts th dmenson and mr ( ) represents the qualty of the reference-obect R. Smlarly X( N ) represents the poston X( N ) represents ts th dmenson and mn ( ) represents the qualty of the floatng-obect N. U( a b) s a random value unformly dstrbuted n nterval [ ab. ] Bnary fference Gravtatonal Evoluton algorthm can be summarzed as follows: Step The reference-obect group R and the floatng-obect group N are randomly generated: R { R n} N { N n} where n and n are two nteger numbers and represent the group szes of two groups respectvely. Step If the halt condtons are satsfed then halt. Otherwse resort the populaton RN whch should satsfy the followng rules: m( R ) m( R ) n m( R) m( N ) n n Step3 Bnary dfference strategy s appled to any two reference-obects R ( n ) and R ( n ) and a new obect C / ( n) s generated: 85

4 J. L et al. U (0) 0.5 csgn n U (0) 0.5 U ( l u ) f X ( R ) X ( R ) XC ( / ) X ( R ) csgn c U (0) X ( R ) X ( R ) otherwse where XC ( / ) XC ( / ) and mc ( / ) represent the poston the th dmenson and the qualty of the newly generated obect C / respectvely. X ( R ) X ( R ) represents the dstance of two obects. C / replaces Rn f m( C ) m( R ) and then / resort the reference-obect group R whch should satsfy the followng rules: n m( R ) m( R ) n ; Step The gravtaton measurement F/ and dstance measurement r / (Eucldean dstance measurement s used here) between any R ( n ) and N ( n ) s calculated as follows: r X ( R ) X ( N ) m( R) m( N ) F/ r/ K0 / Then a bdrectonal selecton runs: () For every floatng-obect N ( n ) to select a reference-obect R ndex _( ) whch has the maxmal gravtaton wth N where max F / max( F / n ) max ndex _( ) mn({ F / F / n}) () For every reference-obect R ( n ) to select a floatng-obect N ndex _ ( ) whch has the mnmum gravtaton wth R ( n ) and then t s thorough elmnated whch means t would be replaced and do not partcpate n generatng the next generatons where set( ) { ndex _( ) n} mn F / mn({ F / set( )}) mn ndex _ ( ) mn({ F / F / set( )}) The elmnated obect N ndex _ ( ) does not exst f set(). Step5 There are two stuatons for update of floatng-obects based on step. () For the floatng-obects whch have not been elmnated { N ndex _( ) n n} bnary dfference s appled to update them: U (0) 0.5 csgn n U (0) 0.5 U ( l u ) f X ( Rndex _( ) ) X ( N ) X( Rndex _( ) ) X( N ) csgn c U (0) X ( Rndex _( ) ) X ( N ) otherwse () For the elmnated floatng-obects { N ndex _ ( ) n} we randomly select a reference-obect to replace t. Then go to Step... Algorthm Illustratons... Sortng Operaton The populaton s dvded nto the reference-obect group and the floatng-obect group n BGE where the reference-obects have the better ftness values thus they are obects wth bgger qualtes. The referenceobects are arranged accordng to ther qualtes n the group. Reference-obects Orderng Rules 3 m() m() m(3) m() Fg. Orderng of Reference-obect Group The sortng operaton s appled to the referenceobect group only and the rules of sortng s llustrated n Fgure where m() represent qualty of an obect and obects are sorted n a proper sequence by ther values of qualtes. The other part of populaton.e. the floatng-obect group does not need to be sorted and the only requrement we need to ensure s that any floatng-obect s ftness value should never be better than the worst reference-obect s. 86

5 Bnary fference Gravtatonal Evoluton... Method of Update There are two types of update methods n BGE.e. the self-update of reference-group and the nteractveupdate process between the reference-group and floatng-group. Reference-obects Update of Referenceobects Themselves C / C /3 C /5 C /6 Fg. An Example of Reference-obects Self-update As t s showed n Fgure the self-update process of the reference-obects occurred after obects are sorted. Reference-obects and and 3 and 3 are used to produce new obects one par after another by bnary dfference. Thus any two referenceobects are selected n the process above. The rule of new obects selecton s that we select the superor wth bgger qualty and elmnate the nferor. The purpose of a reference-obect s to eep a hgh level on qualty as well as ftness whch to a certan degree gnores the dversty of the reference-group but speed up the convergence rate only. regeneraton s called forced-update whch means beng replaced and not partcpate n generatng the next generatons. The forced-update step of nteractveupdate contrbutes a lot n the goal of eepng populaton dversty...3. Physcal Meanngs As showed n both Formula () and () f two obects have a large value of gravtaton that may be caused by ether a small dstance n the denomnator or bg qualty of each obect n the numerator. The second case s what we expected whle the former may lead to a trap of local optmum. To solve ths problem we ntroduce a threshold parameter the value of whch wll be dscussed n another paragraph.... Advantages of Gravtaton Clusterng: Reference-obects 3 Floatng-obect Fg. 3 An Example of Gravtaton Clusterng After the self-update process of reference-group an nteractve-update of reference-obects and floatngobects are performed. Fgure 3 shows an example of gravtaton clusterng the essental of whch s the nteractve-update. Compared wth reference-obects and 3 floatng-obect has a bgger gravtaton measurement wth reference-obect so floatngobect selects reference-obect the lnes of reference-obect and floatng-obect shows ther relatonshp. The smlar progress occurs on other obects. For the reference-obect there has four floatng-obects ( 3 and 8) selected among whch floatng-obect 3 (wth a dot) has the mnmum gravtaton measurement wth reference-obect then floatng-obect 3 would be elmnated from populaton. Smlarly other reference-obects elmnate floatngobects as above to fnsh the nteractve-update process between the reference-group and floatng-group. The postons of elmnated floatng-obects are regenerated n a random way as ntalzaton. The Fg. Searchng Method Fgure shows an example of one dmenson as shown reference -obects R R and floatng-obects N N accordng to Formula () F m( R ) m( N) / ( r K0) F m( R ) m( N) / ( r K0). Both F and F are large (when F s small the relatonshp of R and N s gnored) where F s caused by bg qualtes of R and t s good for explorng on new regon whch ncreases dversty. On the contrary F may be caused by the small dstance r. Ths s good for explotng n the current regon whch ncreases accuracy but may cause a rs of beng trapped n local optmums. 3. Experments 3.. Comparson Methods We choose fferental Evoluton to compare BGE wth because both use dfference vector to generate new ndvduals. The parameters of the two algorthms are set as follows: 87

6 J. L et al. E: The parameters are set as that proposed by Yang et al.[6] that s the populaton sze NP 0 where s the dmenson of varable n soluton space CR 0.9 F 0.5 and a mutaton strategy of E/rand/. The haltng condton s:. The number of ftness functon evaluatons overstepped Supposed Mnf s the global optmum value f best s the searched optmum value and 6 0 s the accuracy value f Mnf fbest the runnng stops. BGE: The reference-obect group sze n 0 the floatng-obect group sze n 0 c K 0 0 as that n E Benchmar Functons and 9.And 0. The haltng condton s set the same There are 9 benchmar functons presented n table whch are used to evaluate the performance of BGE and to compare t wth the classc fferental Evoluton. Table. Benchmar Functons Test Functon oman Range Optmal Pont Sphere Model [-0000] f x Mn f f(00 0) 0 Schwefel's Problem. f x x [-00] Mn f f (00 0) 0 Schwefel's Problem. f3 max{ x } [-0000] Mn f 3 f (00 0) 0 Step Functon f ( x 0.5 ) Quartc Functon.e. Nose f x random[0) 5 Generalzed Schwefel's Problem.6 f6 ( xsn( x )) Generalzed Rastrgn Functon 7 f x 0cos( x ) 0 Generalzed Penalzed Functon I f y y 8 {0sn ( ) ( ) [ 0sn ( y )] ( y ) } where ux ( 000 ) [-0000] Mn f f (00 0) 0 [-.8.8] Mn f 5 f (00 0) Mn f6 [ ] f ( ) [-5.5.] Mn f 7 f (00 0) 0 [-5050] Mn f 8 f ( ) m ( x a) x a u( x a m) 0 a x a m ( x a) x a y ( x ) Generalzed Penalzed Functon II 9 0.{sn (3 ) ( ) f x x [ sn (3 x )] ( x ) [ sn ( x)]} where ux ( 500) m ( x a) x a u( x a m) 0 a x a m ( x a) x a 3.3. Smulaton Results [-5050] Mn f 9 f ( ) 0 We run each algorthm 50 tmes ndependently on each benchmar functon above. The dmenson of problems s set to be 0. Successful run means the dfference between the obtaned soluton and the true value to be 6 less than 0 before the number of functon evaluatons reaches the maxmum value whch s set to be Test Functon Table. Comparson of results on 50-tme ndependent random testng wth benchmar functons BGE E Best/Worst Std FEs Best/Worst Std FEs f 3.77e-7/9.86e-6 e e-6/9.98e-6 e-6 0 f.79e-6/9.97e-6 e e-6/9.9e-6 e f3 0/ / f 0/ / f5.0e-6/3.9e- e e-5/.0e- e f / e / e f7.9e-9/9.83e-6 e e-6/0.99 e f8 5.8e-8/9.96e-6 e e-6/9.9e-6 e f9.e-9/9.93e-6 e e-6/9.98e-6 e-6 09 As shown n Table where Best /Worst mean the best and worst solutons n the testng based on the average of 50 tmes of ndependent runs; Std and FEs s short for standard devaton and functon evaluatons. Accordng to the data n Table fferental Evoluton cannot always fnd the optmal value of f 7 for the worst soluton n the 50 runs s dssatsfyng the precson. We defne when the obtaned soluton whch has the precson less then 0 as a successful run. There are 6 successful runs n 50 total runs whle 9 88

7 Bnary fference Gravtatonal Evoluton optmzng f 7 utlzng E whle for other functons n ths test the successful runs are all 50.e. E can solve the rest problems and the proposed BGE can optmze all the functons wth relatve smaller Std and FEs values whch means a hgher precson and fast convergence speed Table shows BGE has a better performance than E due to a smaller number of functon evaluatons of whch a maxmal dsparty was that E got a more than 0-tme functon evaluatons than BGE that means the latter s 0 tmes faster than the former. Fgure 5 shows the comparson between E and BGE whch llustrate the ftness values decreased as teraton ncreasng for optmzng the functons. (c) Fg. 5 Comparson of BGE and E n Testng of (a)f (b)f (c)f8. As can be observed clearly from the fgures BGE reached the global optmum requrng less number of teraton than E. urng the optmzng process the blue sold lne declnes faster than the red dotted lne whch llustrates that the BGE converges faster than E does.. Expermental Study of Parameters (a).. The group szes n and n The values of group sze n and n may nfluence the performance of BGE a lot. Here we dscuss these two parameters usng Schaffer Problem (SF) as an example whch was a classcal deceptve problem. sn x 0.5 f x SF 0.00( x ) Mn f f (00 0) 0 SF SF (b) We optmze f SF wth -dmenson and 5-dmenson utlzng BGE to test dfferent settngs of group sze n and n. The halt condton s set as ether f f SF 6 Mn 0 or the number of ftness functon evaluatons overstepped the Max-FEs whch was set to be for -dmenson problem and for 5-dmenson problem respectvely. Other parameters of 9 0 BGE are set as follows 0 c K 0 0. n and n are expermentally set to be

8 J. L et al. respectvely. For each par of n and n 50 runs are conducted ndependently and a total of tmes of runs are conducted for the two problems. Fgure 6 shows the nfluences on optmzaton f SF of -dmenson and Fgure. 7 shows that of 5-dmenson. Accordng to the halt condton we adopt success as 6 operaton stopped when f Mn f SF 0 other than the number of ftness functon exceeds Max-FEs. It can be seen from Fgure 6 that for a -dmenson problem when n n t has a hgher success rate whle a lower rate when n n. From Fgure 6(b) when n n t has a lower number of average functon evaluaton whle an opposte result when n n. As an observaton from Fgure 6 a selecton of n ns good for the -dmenson Schaffer Problem. The same observaton can be obtaned from Fgure 7 for the 5- dmenson problem. Intutvely accordng to the algorthm procedure when n n the reference-obects update taes hgh percentage of the whole update progress whch leads a fast convergence. However t wll lose dversty especally when dealng wth deceptve problem. When n n ths s another status that the dversty s contented enough but wth a lttle slow convergence. BGE s an algorthm that can balance between a hgh convergence rate and a hgh dversty. The floatng-obect group sze could be arbtrarly bg but a too large group sze s tme consumng. The concluson s n should be approprate small n could be slghtly larger than n and ther relatonshp should meet n n. The specfc values are defned dependng on crcumstances. (b) Fg. 6 The Effect of Alternatve Group Szes on Optmzaton Schaffer Problem (SF) of - dmenson (a) Result of Success Rate (b) Result of Average Functon Evaluatons (a) (b) Fg. 7 The Effect of Alternatve Group Szes on Optmzaton Schaffer Problem (SF) of 5- dmenson (a) Result of Success Rate (b) Result of Average Functon Evaluatons.. The threshold parameter (a) The threshold parameter s dscussed n ths part whch nfluences the algorthm performance a lot as well. BGE s tested on Shfted Sphere Functon ( f SSF ). 90

9 Bnary fference Gravtatonal Evoluton SSF f z z x o 00 x 00 u l 3 l ( u l ) o u l 3 l ( u l ) 3 Mn f f ( o o o ) 0 SSF SSF Here we separately test the problem wth a dmenson of and 00 respectvely. The parameters are 0 set as follows: n 0 n 0 c K 0 0. The halt condtons are set as ether f f SSF the number of ftness functon evaluatons overstepped ncreases gradually by Mn 0 or 0 0 and for each settng of 50 runs are conducted ndependently. Table. 3 The result of benchmar testng on Shfted Sphere Functon wth dfferent values of Test Shfted Sphere Functon functons Std FEs Std FEs Std FEs Std FEs 0 - e e e e e e e e e e e e e e e e e e e e e-8 69 e e e e e-9 9 e e e e e-9 69 e e-8 68 e e-9 79 e e-8 60 e-8 80 e e As shown n Table 3 wth the value of decreasng the standard devaton Std and the average number of ftness functon evaluatons FEs decrease as well whch means a better performance n optmzaton. From the statstc results shown n Table 3 t can be observed that 8 0 n general a settng of 0 ~ 0 preferably ft for most problems whch can be used as a common settng. As for low-dmensonal problems a settng of nearby the precson value leads to nce performances whle for a hgh-dmensonal problem the smaller s set the better result would be got. For example for Shfted 5 Sphere Functon wth 00-dmenson when 0 the FEs value decreases to 5983 better than when Analyss of Algorthm Convergence BGE s a type of randomzed optmzaton algorthms. The condtons of provng the convergence of a randomzed algorthm were frstly proposed by Sols and Wets [7]. They have gven the theorems to prove whether an algorthm has converged to the global optmal wth probablty whch can be summarzed as follows: Hypothess [7] f f ( ( z )) f ( z) S then f ( ( z )) f ( ). where s a functon to generate potental solutons s a random vector generated from the probablty space n (R B ) and f s the obectve functon. S whch s the subspace of R n represents the constrant space of the problem. s probablty measurement on B whch s the doman of R n subset. Hypothess [7] f A s a Borel subset of S satsfes ( A) 0 then ( ( A)) 0 0 where ( A) s a n -dmensonal closure of subset A and ( A) s a probablty ndcatng the rate that generates A. Theorem [7] Suppose f s a fathomable functon S s a fathomable subset of R n { z } 0 s a soluton sequence generated by the randomzed algorthm. If both Hypothess and Hypothess are satsfed smultaneously then lm P[ z ] where R represents the set of the global optmal solutons. Accordng to the theorem f both Hypothess and Hypothess are satsfed smultaneously for BGE t can be confrmed that the proposed BGE algorthm converges to the global optmal soluton wth probablty. The convergence proof of BGE s gven as follows: In the BGE algorthm the return value before the tth teraton s the functon value of x ( t ).e. f ( x ( t )) and f ( x ( t )) represents the functon value of the tth teraton value x () t where f( x ) represents R 9

10 J. L et al. the obectve functon. The functon of Hypothess s defned as: x ( t ) f f ( x ( t )) f ( x ( t)) ( x( t ) x( t)) x ( t) f f ( x ( t )) f ( x ( t)) It can be nferred Hypothess s satsfed for BGE. For Hypothess all that s needed s to prove the S - szed sample space contans S thus S M t S where M t represents the support set of the th ndvdual s sample space n tth teraton. Suppose there are N teratons n the search and the range of the th teraton s S whch s the support set as well. Therefore the unon space of the populaton (a set of ndvduals) s N S. The range of an ndvdual s adustable and when range covers the boundary of the soluton space though there are only a few ndvduals t can enable N S S Then Hypothess s satsfed for BGE algorthm. In concluson BGE converges to the global optmal soluton wth probablty accordng to the theorem. 6. Concluson In ths paper the BGE algorthm based on two crtcal models.e. bnary dfference and gravtatonal evoluton for global optmzaton was proposed. The proposed algorthm BGE was compared wth the fferental Evoluton by testng both algorthms on benchmar functons. Smulaton results show that the BGE can explore the soluton space more effectvely than E to obtan the global soluton and the BGE requres a much smaller sze populaton than E does. The parameters of BGE are studed and set by traland-error and the convergence analyss was also conducted to show BGE can converge to the global optmal soluton wth probablty. Certanly there s stll room to further mprove BGE. For nstance the number of parameters should be reduced to smplfy the algorthm and the clusterng and gravtatonal models should be studed to solve hgh-dmensonal problems whch are our future research wor. Acnowledgement Ths paper s partally supported by Natonal Natural Scence Foundaton of Chna under Grant Numbers ; and the Natural Scence Foundaton of Zheang Provnce under Grant Number Y References. J. H. Holland. Adaptaton n Natural and Artfcal Systems Unversty of Mchgan Press Ann Arbor MI Y. G. X T. Y. Cha and W. M. Yun Survey on Genetc Algorthm Control Theory and Applcatons vol.3 no.6 pp S. Krpatrc C.. Gelatt M. P. Vecch Optmzaton by Smulated Annealng Scence vol.0 no.598 pp K. Mor M. Tsuyama and T. Fuuda Immune Algorthm wth Searchng versty and ts Applcaton to Resource Allocaton Problem T.IEE Japan vol. 3-C no.0 pp M. J. L A. Luo and T. S. Tong Artfcal Immune Algorthm and Its Applcatons Control Theory and Applcatons vol. no L. Wang J. Pan and L. C. Jao The Immune Algorthm Acta Electronca Snca vol. 8 no R. C. Eberhart and Y. H. Sh Computatonal Intellgence: Concepts to Implementatons Elsever Sngapore J. Kennedy and R. C. Eberhart Partcle Swarm Optmzaton Proc. of the IEEE Int. Conf. on Neural Networs Pscataway NJ pp X. P. Zhang Y. P. u G. Q. Qn and Z. Qn Adaptve Partcle Swarm Algorthm wth ynamcally Changng Inerta Weght Journal of X an Jaotong Unversty vol. 39 no. 0 pp M. orgo and V. Manezzo A. Colorn. Ant System: Optmzaton by a Colony of Cooperatng Agents IEEE Transactons on Systems Man and Cybernetcs Part B vol.6 no. pp M. orgo M. Brattar and T. Stützle Ant Colony Optmzaton-Artfcal Ants as a Computatonal Intellgence Technque IEEE Computatonal Intellgence Magazne K. M. Passno Bacteral Foragng Optmzaton Internatonal Journal of Swarm Intellgence Research vol. no. pp V. Chrs E. Tsang Guded Local Search Techncal Report CSM-7 epartment of Computer Scence Unversty of Essex UK B. L. Webster Solvng Combnatoral Optmzaton Problems Usng a New Algorthm Based on Gravtatonal Attracton Ph.. thess Melbourne Florda Insttute of Technology S. R. Balachandar and K. Kannan Randomzed Gravtatonal Emulaton Search Algorthm for Symmetrc 9

11 Bnary fference Gravtatonal Evoluton Travelng Salesman Problem Appled Mathematcs and Computaton vol.9 pp Y. T. Hsao C. L. Chuang J. A. Jang and C. C. Chen A Novel Optmzaton Algorthm: Space Gravtatonal Optmzaton Proc. of 005 IEEE Internatonal Conference on Systems Man and Cybernetcs vol.3 pp C. L. Chuang and J. A. Jang Integrated Radaton Optmzaton: Inspred by the Gravtatonal Radaton n the Curvature of Space-tme Proc. of 007 IEEE Congress on Evolutonary Computaton pp E. Rashed Gravtatonal Search Algorthm Ph.. thess Shahd Bahonar Unversty of Kerman Kerman Iran R. Esmat N. P. Hossen and S. Saed GSA: A Gravtatonal Search Algorthm Informaton Scences vol.79 pp E. Rashed N. P. Hossen and S. Saed BGSA: Bnary Gravtatonal Search Algorthm. Natural Computng. vol.9 pp Q. Kang L. Wang and Q. Wu A Novel Self-organzng Partcle Swarm Optmzaton Based on Gravtaton Feld Model Proc. of the 007 Amercan Control Conference New Yor USA R. Storn and K. Pcce fferental Evoluton A Smple and Effcent Heurstc for Global Optmzaton over Contnuous Spaces Journal of Global Optmzaton vol. pp Q. W. Yang L. Ca and Y. C. Xue A Survey of fferental Evoluton Algorthms Pattern Recognton and Artfcal Intellgence vol. no. pp R. Mendes and A. S. Mohas yne: a fferental Evoluton for ynamc Optmzaton Problems Proc of the Congress on Evolutonary Optmzaton Ednburgh UK.vol.3 pp Y. Lou J. L. L and Y. S. Wang. A Bnary-fferental Evoluton Algorthm Based on Orderng of Indvduals Proc. of the Internatonal Conference on Natural Computaton. vol.5 pp Z. Y. Yang K. Tang and X. Yao Self-adaptve fferental Evoluton wth Neghborhood Search Proc. of the 008 IEEE Congress on Evolutonary Computaton pp F. J. Sols and R. T. Wets Mnmzaton by Random Search Technques. Mathematcs of Operatons Research vol.6 no. pp