Analytics on Fireworks Algorithm Solving Problems with. Shift in Decision Space and Objective Space

Transcription

1 Analytcs on Frewors Algorthm Solvng Problems wth Sht n ecson Space and Objectve Space Sh Cheng, Unversty o Nottngham Nngbo Chna Quande Qn, Shenzhen Unversty, Chna Yuhu Sh, X'an Jaotong-Lverpool Unversty, Chna Qngyu Zhang, Shenzhen Unversty, Chna Rubn Ba, Unversty o Nottngham Nngbo Chna Abstract Frewors algorthms or solvng problems wth optma sht n decson space and/or objectve space are analyzed n ths paper. The standard benchmar problems have several weanesses n the research o swarm ntellgence algorthms or solvng sngle objectve problems. Frst, the optmum s n the center o search range, and s the same at each dmenson o the search space. The optmum sht n decson space and/or objectve space could ncrease the dculty o problem solvng. A mappng strategy, modular arthmetc mappng, s utlzed n the orgnal rewors algorthm to handle solutons out o search range. The solutons are mplctly guded to the center o search range or problems wth symmetrcal search range va ths strategy. The optmzaton perormance o rewors algorthm on sht unctons may be aected by ths strategy. Four nds o mappng strateges, whch nclude mappng by modular arthmetc, mappng to the boundary, mappng to stochastc regon, and mappng to lmted stochastc regon, are compared on problems wth derent dmensons and derent optmum sht range. From epermental results, the rewors algorthms wth mappng to the boundary, or mappng to lmted stochastc regon obtans good perormance on problems wth optmum sht. Ths s probably because the search tendency s ept n these two strateges. The denton o populaton dversty measurement s also proposed n ths paper, rom observaton on populaton dversty changes, the useul normaton o rewors algorthm solvng derent nds o problems could be obtaned. Keywords: evelopmental swarm ntellgence algorthm; Frewors algorthm; Optmum sht; Populaton dversty

2 1. Introducton An optmzaton problem n n R, or smply an optmzaton problem, s a mappng : R n R, where n R s termed as decson space [1] (or parameter space [], problem space), and R s termed as objectve space [3]. Swarm ntellgence s based on a populaton o ndvduals [4]. In swarm ntellgence, an algorthm mantans and successvely mproves a collecton o potental solutons untl some stoppng condton s met. The solutons are ntalzed randomly n the search space. The search normaton s propagated through the nteracton among solutons. Wth solutons convergng and/or dvergng, solutons are guded toward the better and better areas. In swarm ntellgence algorthms, there s a populaton o solutons whch est at the same tme. The premature convergence may happen due to solutons gettng clustered together too ast. The populaton dversty s a measure o eploraton and eplotaton status. Based on the populaton dversty changng measurement, the state o eploraton and eplotaton can be obtaned. The populaton dversty denton s the rst step to gve an accurate observaton o the search state. Many studes o populaton dversty n evolutonary computaton algorthms and swarm ntellgence have been proposed [5-1]. The concept o developmental swarm ntellgence algorthms was proposed n [13]. The developmental swarm ntellgence algorthm should have two nds o ablty: capablty learnng and capacty developng. The Capacty evelopng ocuses on movng the algorthm's search to the area(s) where hgher searchng potental may be possessed, whle the capablty learnng ocuses on ts actually searchng rom the current soluton or sngle pont based optmzaton algorthms and rom the current populaton or populaton-based swarm ntellgence algorthms. The capacty developng s a top-level learnng or macro-level learnng. The capacty developng could be the learnng ablty o an algorthm to adaptvely change ts parameters, structures, and/or ts learnng potental accordng to the search states on the problem to be solved. In other words, the capacty developng s the search strength possessed by an algorthm. The capablty learnng s a bottom-level learnng or mcro-level learnng. The capablty learnng s the ablty or an algorthm to nd better soluton(s) rom current soluton(s) wth the learnng capacty t s possessng [13]. The Frewors algorthm (FWA) [14, 15] and bran storm optmzaton (BSO) [16-18] algorthm are two good eamples o developmental swarm ntellgence (SI)

3 algorthms. The good enough optmum could be obtaned through solutons dvergng and convergng n the search space. In FWA algorthm, mmced by the rewors eploraton, the new solutons are generated by the eploraton o ested solutons. Whle n BSO algorthm, the solutons are clustered nto several categores, and new solutons are generated by the mutaton o clusters or ested solutons. The capacty developng,.e., the adaptaton n search, s another common eature n these two algorthms. Swarm ntellgence s based on a populaton o ndvduals. In swarm ntellgence, an algorthm mantans and successvely mproves a collecton o potental solutons untl some stoppng condton s met. The solutons are ntalzed randomly n the search space, and are guded toward the better and better areas through the nteracton among solutons. Mathematcally, the updatng process o populaton o ndvduals over teratons can be looed as a mappng process rom one populaton o ndvduals to another populaton o ndvduals rom one teraton to the net teraton, whch can be represented as P t1 ( P ), where t P t s the populaton o ndvduals at the teraton t, () s the mappng uncton. As a general prncple, the epected tness o a soluton returned should mprove as the search method s gven more computatonal resources n tme and/or space. More desrable, n any sngle run, the qualty o the soluton returned by the method should mprove monotoncally - that s, the qualty o the soluton at tme t 1 should be no worse than the qualty at tme t,.e., tness( t 1) tness( t) or mnmum problems [19, 0]. The capacty developng ablty o FWA means that FWA could dynamcally change ts search ablty on derent problems. Ths ablty s shown on the parameters adaptaton n new soluton generaton. In rewors algorthm, the parameters to control the number and range o new solutons are adaptvely determned by the tness o rewors. The capablty learnng ablty o FWA means that the obtaned solutons are gettng better and better teratvely. In ths paper, the rewors algorthms or solvng problems wth optma sht n decson space and/or objectve space are analyzed. Ths paper s organzed as ollows. Secton revews the basc rewors algorthm. Secton 3 ntroduces and analyzes our nds o mappng strateges. Secton 4 denes two nds o populaton dverstes n rewors and spars, respectvely. Eperments are conducted n Secton 5 ollowed by the dscusson on the rewor dversty and spar dversty changng curves n Secton 6. Fnally, Secton 7 concludes wth some remars and uture research drectons.

4 . Frewors Algorthm Frewors algorthm (FWA) s a swarm ntellgence optmzaton method that mmcs the eploson process o rewors [14, 15, 1, ]. The basc ramewor o rewors algorthm s gven n Fg. 1 []. The procedure o rewors algorthm s gven n Algorthm 1. There are our operators/strateges n FWA, whch are eploson operator, mutaton operator, mappng strategy, and selecton strategy, respectvely. Algorthm 1: The procedure o rewors algorthm 1 Intalze n locatons; Whle have not ound good enough soluton or not reached the pre-determned mamum number o teratons do 3 Set o rewors at locatons; 4 Generate spars through eploson operator; 5 Generate spars through mutaton operator; 6 Obtan the locatons o spars; 7 Mappng the locatons nto easble search space; 8 Evaluate the qualty o the locatons; 9 Select locatons as new rewors; n n Fg. 1. The ramewor o rewors algorthm.

5 .1 Eploson Operator In eploson operator, the number o spars and ampltude o eploson are calculated or each rewor..1.1 Number o Spars Based on the concept that more spars should be generated rom a rewor wth good tness value, the number o spars generated by rewor s dened as ollows: s m n 1 y ma ( ) ( y ( )) ma (1) where m s a parameter that controllng the total number o spars generated by the n rewors, y ma s the mamum (worst or mnmum problem) tness value among the n rewors, yma ma( ( )), 1,,, n, and, whch s a tny constant number, s utlzed to avod zero-dvson-error. To eep the populaton dversty o rewors, spars should be generated rom all rewors. To avod that many spars are generated rom one rewor, the greatest lower bound am and the least upper bound bm are set to spars number s [14, ]: round( a m) s a m sˆ round( b m) s b m () round( s ) otherwse. where a and b are constant parameters, and ab Ampltude o Eploson For a rewor wth good tness value, the spars are generated close to the rewor. Ths could enhance the eplotaton ablty o algorthm. Whle or a rewor wth bad tness value, the spars are generated ar rom the rewor. Ths s amed to enhance the eploraton ablty o algorthm. A Aˆ n 1 ( ) y mn ( ( ) y ) mn (3)

6 where  s the mamum eploson ampltude, and y mn s the mnmum (best or mnmum problem) tness value among the 1,,, n. n rewors, ymn mn( ( )),. Mutaton Operator The Gaussan mutaton s utlzed to generate new spars rom a rewor. For a rewor, randomly select dmensons, N (1,1), z (4) z The Gaussan operator could help solutons jumpng out o local optma..3 Mappng Strategy In the eploson operator and mutaton operator, a new soluton may be generated out o search space. Then the mappng strategy s utlzed to set the soluton nto the easble search range. The mappng by modular arthmetc s used n the orgnal rewors algorthm [14]. The mappng equaton s shown n equaton (5). mod( (5) ) mn ma mn The soluton could be reset to the easble search range va ths strategy..4 Selecton Strategy At the begnnng o each eploson generaton, rewors eploson. The current best locaton value ( å ) å locatons are selected or the, corresponded to the objectve that s best among current locatons, s always ept or the net eploson generaton. The other n 1 locatons are selected based on the dstance n among a soluton and other solutons. The dstance between a soluton and other solutons s dened n Eq. (6): R( ) d(, ), (6) j j j K j K The selecton probablty o a locaton s dened as ollows: R ( ) p ( ) (7) R ( ) jk j

7 3. Mappng Strateges n Frewors Algorthm 3.1 Mappng by Modular Arthmetc The mappng by modular arthmetc s used n the orgnal rewors algorthm [14]. The soluton could be reset to the easble search range va ths strategy. However, n most cases, the dstance between a soluton, whch eceeds the easble search space, and the search boundary s very close. In other words, the value o or ma s larger than ma, or mn or s smaller than mn, s very tny when soluton eceeds the easble search space. Moreover, or most optmzaton problems, the search space s symmetrcal,.e.,, and the center o search range s 0. We dene that ma mn hal, mn ma and the dstance between soluton and boundary s 0. Then we, hal have soluton, ma hal or up, up eceed the upper bound ma, and soluton, low mn hal or, low eceed the lower bound mn. For the rst case,, up hal, then mod( ), up mn ma mn mod hal hal hal hal hal For the second case,, low hal, then mod( ), low mn ma mn mod hal hal hal hal hal

8 From the analyses, mappng by modular arthmetc strategy always set the soluton to a poston value when the dstance s less than hal search range hal. The closer to 0 the s, the closer the soluton wll be reset to the center o search range. For problem wth symmetrcal search space, s less than ma mn when hal. The result o mod( ma mn ) s. Ths concluson s not accurate or problems wth asymmetrcal search space. The value o may be larger than ma mn even hal. The result o mod( ma mn ) s aected by the ntalzed value o ma or mn. The problems wth asymmetrcal search space are rare n the standard benchmar unctons; and t can be shted to problems wth symmetrcal search space by add or mnus a certan value. For the problems wth asymmetrcal search space, t s dcult to analyze the result o mappng by modular arthmetc strategy. The sgn o ma, and mn, and the relaton between ma, mn and hal should be consdered n the nvestgaton. For eample, the, n the ollowng three scenaros: 1) or ma 40, mn 0, 0 mod(40 0) The new soluton s close to the lower bound mn. ) or ma 30, mn 10, 10 1 mod(30 ( 10)) The new soluton s n the lower range o search space. 3) or ma 10, mn 80, mod(10 80) 118 The new soluton s n the upper range o search space, and t s close to the

9 upper bound ma. To overwhelm the mplct gudng to the center o search space, random mappng n whole search range s proposed to replace mappng by modular arthmetc strategy n [1]. The equaton o random mappng n whole search range s gven n equaton (8) r ( ) (8) mn 1 ma mn where r 1 s an unormly dstrbuted random numbers n the range [0,1). Ths strategy could avod mplct gudng to the center; however, the search tendency s also abandoned. 3. Mappng to the Boundary The conventonal boundary handlng methods try to eep the solutons nsde the easble search space. Search normaton s obtaned only when solutons stay n the search space. I a soluton eceeds the boundary lmt n one dmenson at one teraton, that search normaton wll be abandoned. Instead, a new soluton wll replace the prevous soluton n that dmenson. The classc strategy s to set the soluton at boundary when t eceeds the boundary [3]. The equaton o ths strategy s as ollows: S mn mn ma ma (9) otherwse. Ths strategy resets solutons n a partcular pont the boundary, whch constrans solutons to stay n the search space lmted by boundary. 3.3 Mappng to Stochastc Regon The stochastc strategy can also be used to reset the solutons nto easble range when solutons eceed the search boundary [10, 11, 4]. The equaton s shown n equaton (10), 1 r ( ) 1 r ( ) otherwse. mn 1 ma mn mn (10) ma ma mn ma where r and 1 r are two unormly dstrbuted random numbers n the range [0,1).

10 1 The ( ma mn ) hal s hal o search range on dmenson. By ths strategy, solutons wll be reset wthn the hal search space when solutons eceed the boundary lmt. Ths wll ncreases the algorthm's eploraton, that s, solutons have hgher possbltes to eplore new search areas. However, t decreases the algorthm's ablty o eplotaton at the same tme. 3.4 Mappng to Lmted Stochastc Regon A soluton eceedng the boundary may mean that the global or local optmum may be close to the boundary regon. An algorthm should spend more teratons n ths regon. Wth the consderaton o eepng the ablty o eplotaton, the resettng scope should be taen nto account. The equaton o resettng soluton nto a specal area s as ollows: 1 r c ( ) 1 r c ( ) otherwse. mn 1 ma mn mn (11) ma ma mn ma where s a parameter to control the resettng scope. When the same as the equatons (10), that s, partcles reset wthn a hal space. On the contrary, when c 0, ths strategy s the same as the equaton (9),.e., t s the same as the classc strategy. The closer to s, the more partcles have a hgh possblty to be reset close to the boundary. c 4. Populaton versty 0 the c c 1, ths strategy s In a rewors optmzaton algorthm, the populaton dversty should be measures on the two groups o ndvduals at the same tme. The rst group s the populaton o rewors, whch contans ndvduals. The second group s the populaton o spars, whch contans ndvduals. m n 4.1 Populaton versty n Frewors The rewors are the seeds o new solutons. Populaton dversty n rewors, or rewor dversty, s a measurement o rewors' dstrbuton. The denton o the rewor dversty s gven below, whch s dmenson-wse and based on the L 1 norm.

11 1 dv 1 n n n 1 n 1 dv w 1 v where represents the pvot o rewors n dmenson, and v measures rewor dversty or dmenson. Then we dene, 1 [,,,, ] represents the mean o current rewors on each dmenson, and dv 1 [dv,,dv,,dv ], whch measures rewor dversty or each dmenson. w s a weght or dmenson. dv measures the whole rewor group's populaton dversty. 4. Populaton versty n Spars The spars are generated va the operatons on the rewors. The dstrbuton o spars shows the algorthm's eploraton or eplotaton ablty. The denton o the spar dversty s as ollows: 1 m m 1 m 1 m 1 dv dvs w 1 v where represents the pvot o spars n dmenson, and dv measures spar dversty or dmenson. Then we dene, 1 [,,,, ] represents the mean o current spars on each dmenson, and dv 1 [dv,,dv,,dv ], whch measures spar dversty or each dmenson. populaton dversty. dv s measures the whole spar group's Wthout loss o generalty, every dmenson s consdered equally. Settng all weghts 1 w, then the dmenson-wse populaton dversty n rewors and spars can be rewrtten as: 1 1 dv dv dv 1 1

12 5. Epermental Study 5.1 Benchmar Functons The eperments have been conducted to test the rewors algorthm wth derent mappng strateges on the benchmar unctons lsted n Table 1. Consderng the generalty, twelve standard benchmar unctons were selected, whch nclude s unmodal unctons and s multmodal unctons [5, 5]. All unctons are run 50 tmes to ensure a reasonable statstcal result. There are 5000 teratons or 100 dmensonal problems, and 0000 teratons or 00 n every run. For problems wth shted search space, randomly shtng o the locaton o optmum s utlzed n each dmenson or every run. Table 1. The benchmar unctons used n epermental study, where dmenson o each problem, z ( o) generated number n problem's search space global optmum * o, mn, 1 [,,, ], S o s the s an randomly and t s derent n each dmenson, s the mnmum value o the uncton, and S R. Name Functon S mn Parabolc 0( ) z bas0 1 Schweel's P. 1( ) z z bas1 Schweel's P ( ) ( z ) bas 1 1 [ 100,100] [ 10,10] [ 100,100] Step 3( ) ( z 0.5 ) bas3 1 [ 100,100] Quartc Nose 4 4( ) z random[0,1) bas4 1 Zaharov 4 ( ) bas [ 1.8,1.8] [ 100,100]

13 Rosenbroc 1 6( ) [100( z 1 z ) ( z 1) ] bas 6 1 [ 10,10] Rastrgn ( ) 7 [ z 10cos( z) 10] bas 7 1 [ 5.1,5.1] Noncontnuou s ( ) 8 1 Rastrgn [ y 10cos( y) 10] bas 8 [ 5.1,5.1] y 1 z z round( z ) 1 z Acley 9( ) 0ep z [ 3,3] ep cos( z ) 0 e bas 1 9 Grewan 1 10( ) z cos( ) 1 bas z [ 600,600] 10.0 Generalzed Penalzed 1 11( ) {10sn ( y1) ( y 1) 1 [ 50,50] [1 10sn ( y )] ( y 1) } 1 1 uz (,10,100,4) bas 1 y 1 ( z 1) 4 m ( z a) z a, u( z, a,, m) 0 a z a m ( z a) z a In all eperments, the parameters o rewors algorthm are set as ollows: n 5, m 50, a 0.04, b 0.8, Aˆ 40, and mˆ 5 [14]. 11

14 5. Epermental Results on 100 mensonal Problems Several perormance measurements are utlzed n the eperments below. The best, medan, worst, and mean tness values are attaned ater a ed number o teratons. Standard devaton values o the best tness values are also utlzed, whch gves the soluton's dstrbuton. These values gve a measurement o goodness o the algorthm Optma Shted n Objectve Space In ths epermental test, the tness values o all unctons are shted,.e., the optma are shted n the objectve space. Ecept Rosenbroc 6 whch has the optmum 1 n each dmenson, all optma or other optmzed problems are zero, and n the center o each dmenson. Tables II gves result o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. Almost or all unctons, eclude Zaharov 5 and Rosenbroc 6, a soluton close to the global optma could be ound. Ths ndcates the good global search ablty o FWA algorthm on unshted problems or problems only shted n objectve space. The uncton value o the best soluton ound by an algorthm n a run s denoted by ( best ). The error o ths run s denoted as error ( best ) ( å ). Fgure shows the mean error convergence curves o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. In the Fgure, the optmal solutons o all problems have no sht n the decson space,.e., the optmal solutons are n the orgnal poston. Table. Result o rewors algorthm solvng unmodal and multmodal benchmar unctons wth 100 dmensons. All algorthms are run or 50 tmes, where best, medan, worst, and mean ndcate the best, medan, worst, and mean o the best tness values or all runs, respectvely. Func. mn Strategy Best Medan Worst Mean Std. ev module boundary hal lmt module boundary hal lmt

15 module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module E-08 boundary E-08 hal E-08 lmt E-08

16 Fg.. The average perormance o the rewors algorthm solvng unmodal and multmodal problems. The optma o each problem are only shted n objectve space; and the dmenson o each problem s 100. (a) Parabolc 0 (b) Schweel's P. 1 (c) Schweel's P1. (d) Step 3 (e) Quartc Nose 4 () Zaharov 5 (g) Rosenbroc 6 (h) Rastrgn 7 () Noncontnuous Rastrgn 8 (j) Acley 9 () Grewan 10 (l) Generalzed Penalzed Optma Shted n Whole ecson Space For each problem, both the optmal soluton n decson space and tness value n

17 objectve space are shted n ths epermental test. All the optma solutons are shted at a random poston n each dmenson. The shted range or new optmal soluton s the whole decson space,.e., the new optmal soluton å s a random value n the range lower upper [, ]. The lower and upper are the lower and upper search bound or optmzed problem at the th dmenson, respectvely. Tables 3 gves result o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. The optmal results ound on problems wth sht n decson space are sgncant worse than the problems wth orgnal solutons. Ths ndcates that the sht n decson space could aect the algorthm's search perormance. Fgure 3 shows the convergence curves o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. In the Fgure 3, the optmal solutons o all problems are shted to random postons o the whole decson space. Table 3. Result o rewors algorthm solvng unmodal and multmodal benchmar unctons wth 100 dmensons. All algorthms are run or 50 tmes, where best, medan, worst, and mean ndcate the best, medan, worst, and mean o the best tness values or all runs, respectvely. Func. mn Strategy Best Medan Worst Mean Std. ev module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module

18 boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt Fgure 3. The average perormance o the rewors algorthm solvng unmodal and multmodal problems. The optma o each problem are shted n both decson and objectve space; and the dmenson o each problem s 100. (a) Parabolc 0 (b) Schweel's P. 1 (c) Schweel's P1.

19 (d) Step 3 (e) Quartc Nose 4 () Zaharov 5 (g) Rosenbroc 6 (h) Rastrgn 7 () Noncontnuous Rastrgn 8 (j) Acley 9 () Grewan 10 (l) Generalzed Penalzed Optma Shted n Hal ecson Space To test the mpact o derent shted ranges, the optmal solutons are shted n the hal search range. All the optma solutons are shted at a random poston n each dmenson. The shted range or new optmal soluton s the hal decson space,.e., the new optmal soluton å s a random value n the range lower quarter upper quarter [, ]. The lower and upper are the lower and upper search bound or optmzed problem at the quarter o search range n the th dmenson. th dmenson, respectvely. The quarter s a In ths eperments, the optmal solutons are shted away the orgnal postons, but the sht range or each dmenson s only hal o search dmenson. Tables 4 gves

20 result o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. The optmal results ound on problems wth sht n small range are better than the problems wth large shted range. Ths ndcates that the shted range n decson space also could aect the algorthm's search perormance. In general, the larger the shted range s n the decson space, the harder the optmzed problem s. Fgure 4 shows the convergence curves o rewors algorthm wth varants o mappng strateges on problems wth 100 dmensons. In the Fgure 4, the optmal solutons o all problems are shted to random postons, whch s n a hal search range o each dmenson.

21 Table 4. Result o rewors algorthm solvng unmodal and multmodal benchmar unctons wth 100 dmensons. All algorthms are run or 50 tmes, where best, medan, worst, and mean ndcate the best, medan, worst, and mean o the best tness values or all runs, respectvely. Func. mn Strategy Best Medan Worst Mean Std. ev module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module

22 10 11 boundary hal lmt module boundary hal lmt module boundary hal lmt Fgure 4. The average perormance o the rewors algorthm solvng unmodal and multmodal problems. The optma o each problem are shted n both decson and objectve space; and the dmenson o each problem s 100. (a) Parabolc 0 (b) Schweel's P. 1 (c) Schweel's P1. (d) Step 3 (e) Quartc Nose 4 () Zaharov 5 (g) Rosenbroc 6 (h) Rastrgn 7 () Noncontnuous Rastrgn 8

23 (j) Acley 9 () Grewan 10 (l) Generalzed Penalzed Epermental Results on 00 mensonal Problems Optma Shted n Objectve Space The dmenson o each problem s ncreased to 00 n ths eperment. The best tness value o each problem s shted n the objectve space. Tables 5 gves result o rewors algorthm wth varants o mappng strateges on problems wth 00 dmensons. Smlar to the results on the problems wth 100 dmensons, almost or all unctons, eclude Zaharov 5 and Rosenbroc 6, a soluton close to the global optma could be ound. Fgure 5 shows the convergence curves o rewors algorthm wth varants o mappng strateges on problems wth 00 dmensons. In the Fgure 5, the optmal solutons o all problems have no sht n the decson space.

24 Table 5. Result o rewors algorthm solvng unmodal and multmodal benchmar unctons wth 00 dmensons. All algorthms are run or 50 tmes, where best, medan, worst, and mean ndcate the best, medan, worst, and mean o the best tness values or all runs, respectvely. Func. mn Strategy Best Medan Worst Mean Std. ev module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module

25 10 11 boundary hal lmt module boundary hal lmt module E-09 boundary E-09 hal E-09 lmt E-09 Fgure 5. The average perormance o the rewors algorthm solvng unmodal and multmodal problems. The optma o each problem are only shted n objectve space; and the dmenson o each problem s 00. (a) Parabolc 0 (b) Schweel's P. 1 (c) Schweel's P1. (d) Step 3 (e) Quartc Nose 4 () Zaharov 5 (g) Rosenbroc 6 (h) Rastrgn 7 () Noncontnuous Rastrgn 8

26 (j) Acley 9 () Grewan 10 (l) Generalzed Penalzed Optma Shted n ecson Space The optmal solutons n the decson space are shted to random postons and the best tness values are also shted n the objectve space. The shted range n the decson space s the whole dmenson. Tables 6 gves result o rewors algorthm wth varants o mappng strateges on problems wth 00 dmensons. All optmzed results are sgncantly aected by the sht n the decson space. Even the uncton evaluaton tmes are doubled n ths eperment, the results obtaned on the problems wth 00 dmensons are worse than the problems wth 100 dmensons under the same optmal solutons sht condtons. Table 6. Result o rewors algorthm solvng unmodal and multmodal benchmar unctons wth 00 dmensons. All algorthms are run or 50 tmes, where best, medan, worst, and mean ndcate the best, medan, worst, and mean o the best tness values or all runs, respectvely. Func. mn Strategy Best Medan Worst Mean Std. ev module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt

27 module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt module boundary hal lmt Fgure 6 shows the convergence curves o rewors algorthm wth varants o mappng strateges on problems wth 00 dmensons. In the Fgure 6, the optmal solutons o all problems are shted to random postons o the whole decson space. Fgure 6. The average perormance o the rewors algorthm solvng unmodal and multmodal problems. The optma o each problem are shted n both decson and objectve space; and the dmenson o each problem s 00.

28 (a) Parabolc 0 (b) Schweel's P. 1 (c) Schweel's P1. (d) Step 3 (e) Quartc Nose 4 () Zaharov 5 (g) Rosenbroc 6 (h) Rastrgn 7 () Noncontnuous Rastrgn 8 (j) Acley 9 () Grewan 10 (l) Generalzed Penalzed Epermental Results scusson From the epermental results on the problems wth derent dmensons, the problems wth derent shted space (objectve space and/or decson space), and the problem wth derent shted search range (the whole dmenson or the hal dmenson), several conclusons could be made as ollows:

29 The optma sht n objectve space has tny or no eect on the dcult o optmzaton problem. The rewors algorthms could nd the equvalent solutons on problems wth or wthout optma sht n objectve space. The optmzaton problems are gettng harder when the optma are shted n the decson space,.e., the optma are more dcult to obtan when optma are derent n each dmenson and/or not located n the center o search space. The dculty o shted optmzaton problems are aected by the dmenson o problems. The dculty s sgncantly ncreased when the number o problems' dmensonalty s ncreased. The perormance o the algorthm on optma sht problems s gettng worse when number o dmensons ncreases. For optma shted problems wth double number o dmensons (rom 100 dmensons to 00 dmenson), even doubled the number o teratons, the perormance s not lnearly mproved. The dculty o optmzaton problems are aected by the sht range n decson space. The problems are gettng harder when the optma are shted n large space. It s more dcult to obtan optma when the optma are shted ar away to the center o search space. 6. Populaton versty scusson The populaton dverstes measurements on the rewors group and spars group are gven n gures 7, 8, 9, and 10. The Fg. 7 and Fg. 8 are curves o populaton dverstes changng or FWA solvng unmodal problem, whle the Fg. 9 and Fg. 10 are curves o populaton dverstes changng or FWA solvng multmodal problem 6. For problems and 6, the rewor dversty and spar dversty are vbrated durng whole search process, and the rewor dversty vbrated n a large range than the spar dversty. In all gures, there s no sgncant derent among FWA algorthms wth derent mappng strateges. Fg. 7. The populaton dverstes o the rewors algorthm solvng shted unmodal problem. The tness value o the problem s only shted n objectve space; and the dmenson o the problem s 00.

30 (a) rewor (b) rewor (c) rewor (d) rewor (e) spar () spar (g) spar (h) spar Fg. 8. The populaton dverstes o the rewors algorthm solvng shted unmodal problem. The optma o the problem are shted n both decson and objectve space; and the dmenson o the problem s 00. (a) rewor (b) rewor (c) rewor (d) rewor (e) spar () spar

31 (g) spar (h) spar The populaton dverstes are vbrated n the derent search ranges. The range o rewor dversty may be related to the settng o value. The range o spar dversty may be related to the step sze o new soluton generaton. Fg. 9. The populaton dverstes o the rewors algorthm solvng shted multmodal problem 6. The tness value o the problem s only shted n objectve space; and the dmenson o the problem s 00. (a) 6 rewor (b) 6 rewor (c) 6 rewor (d) 6 rewor (e) 6 spar () 6 spar (g) 6 spar (h) 6 spar

32 Fg. 10. The populaton dverstes o the rewors algorthm solvng shted multmodal problem 6. The optma o the problem are shted n both decson and objectve space; and the dmenson o the problem s 00. (a) 6 rewor (b) 6 rewor (c) 6 rewor (d) 6 rewor (e) 6 spar () 6 spar (g) 6 spar (h) 6 spar 7. Conclusons The rewors algorthms or solvng problems wth optma sht n decson space and/or objectve space were analyzed n ths paper. The benchmar problems have several weanesses n the orgnal research o swarm ntellgence algorthms or solvng sngle objectve problems. The optmum s n the center o search range, and s the same at each dmenson o search space. The optmum sht n decson space and/or objectve space could ncrease the dculty o problem solvng. The solutons are mplctly guded to the center o search range or problems wth symmetrcal search range va mappng by modular arthmetc strategy. The optmzaton perormance o rewors algorthm on sht unctons may be aected

33 by ths strategy. Four nds o mappng strateges, whch nclude mappng by modular arthmetc, mappng to the boundary, mappng to stochastc regon, and mappng to lmted stochastc regon, were compared on problems wth derent dmensons and derent optmum sht range. From epermental results, the rewors algorthms wth mappng to the boundary, or mappng to lmted stochastc regon had good perormance on problems wth optmum sht. Ths s probably because the search tendency s ept n these two strateges. The dentons o populaton dverstes measurement were also proposed n ths paper, rom observaton on populaton dversty changes, the useul normaton o rewors algorthm solvng derent nds o problems could be obtaned. In ths paper, populaton dverstes were montored on FWA or solvng sngle-objectve problems. Mult-objectve problems have derent goal o optmzaton, t does not nd one sngle soluton, but many. The concept o convergence has derent meanngs between sngle and multple objectve problems []. Populaton dversty s also mportant when applyng FWA to solve mult-objectve problems [7, 8]. enng populaton dversty or a mult-objectve rewors algorthms and montor ts changng durng search process s our uture research wor. Acnowledgment Ths wor s partally supported by Natonal Natural Scence Foundaton o Chna under Grant Number , , ; Natonal Scence Foundaton o SZU under grant 836; the Foundaton or stngushed Young Talents n Hgher Educaton o Guangdong, Chna, under grant 01WYM_0116; and the MOE Youth Foundaton Project o Humantes and Socal Scences at Unverstes n Chna under grant 13YJC63013; and The Youth Foundaton Project o Humantes and Socal Scences n Shenzhen Unversty under grant 14QNFC8; and by Nngbo Scence & Technology Bureau (Scence and Technology Project Number 01B10055). Reerence [1] S. F. Adra, T. J. odd, I. A. Grn, and P. J. Flemng, Convergence acceleraton operator or multobjectve optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 1, no. 4, pp , August 009. [] Y. Jn and B. Sendho, A systems approach to evolutonary multobjectve structural optmzaton and beyond, IEEE Computatonal Intellgence Magazne, vol. 4, no. 3, pp. 6-76, August 009. [3] R. K. Sundaram, A Frst Course n Optmzaton Theory. Cambrdge Unversty

34 Press, [4] J. Kennedy, R. Eberhart, and Y. Sh, Swarm Intellgence. Morgan Kaumann Publsher, 001. [5] M. L. Mauldn, Mantanng dversty n genetc search, n Proceedngs o the Natonal Conerence on Artcal Intellgence (AAAI 1984), August 1984, pp [6] E. K. Bure, S. Gustason, and G. Kendall, A survey and analyss o dversty measures n genetc programmng, n Proceedngs o the Genetc and Evolutonary Computaton Conerence (GECCO 00). San Francsco, CA, USA: Morgan Kaumann Publshers Inc., 00, pp [7] Y. Sh and R. Eberhart, Populaton dversty o partcle swarms, n Proceedngs o the 008 Congress on Evolutonary Computaton (CEC008), 008, pp [8], Montorng o partcle swarm optmzaton, Fronters o Computer Scence, vol. 3, no. 1, pp , March 009. [9] S. Cheng and Y. Sh, versty control n partcle swarm optmzaton, n Proceedngs o 011 IEEE Symposum on Swarm Intellgence (SIS 011), Pars, France, Aprl 011, pp [10] S. Cheng, Y. Sh, and Q. Qn, Epermental Study on Boundary Constrants Handlng n Partcle Swarm Optmzaton: From Populaton versty Perspectve, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol., no. 3, pp , July-September 011. [11] S. Cheng, Populaton dversty n partcle swarm optmzaton: enton, observaton, control, and applcaton, Ph.. dssertaton, epartment o Electrcal Engneerng and Electroncs, Unversty o Lverpool, May 013. [1] S. Cheng, Y. Sh, and Q. Qn, A study o normalzed populaton dversty n partcle swarm optmzaton, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol. 4, no. 1, pp. 1-34, January-March 013. [13] Y. Sh, evelopmental swarm ntellgence: evelopmental learnng perspectve o swarm ntellgence algorthms, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol. 5, no. 1, pp , January-March 014. [14] Y. Tan and Y. Zhu, Frewors algorthm or optmzaton, n Advances n Swarm Intellgence, ser. Lecture Notes n Computer Scence, Y. Tan, Y. Sh, and K. C.

35 Tan, Eds. Sprnger Berln Hedelberg, 010, vol. 6145, pp [15] Y. Tan, C. Yu, S. Zheng, and K. ng, Introducton to rewors algorthm, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol. 4, no. 4, pp , October-ecember 013. [16] Y. Sh, Bran storm optmzaton algorthm, n Advances n Swarm Intellgence, ser. Lecture Notes n Computer Scence, Y. Tan, Y. Sh, Y. Cha, and G. Wang, Eds. Sprnger Berln/Hedelberg, 011, vol. 678, pp [17], An optmzaton algorthm based on branstormng process, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol., no. 4, pp. 35-6, October-ecember 011. [18] S. Cheng, Y. Sh, Q. Qn, Q. Zhang, and R. Ba, Populaton dversty mantenance n bran storm optmzaton algorthm, Journal o Artcal Intellgence and Sot Computng Research, p. n press, 014. [19] S. G. Fcc, Monotonc soluton concepts n coevoluton, n Genetc and Evolutonary Computaton Conerence (GECCO 005), June 005, pp [0] S. Cheng, Y. Sh, Q. Qn, and S. Gao, Soluton clusterng analyss n bran storm optmzaton algorthm, n Proceedngs o The 013 IEEE Symposum on Swarm Intellgence, (SIS 013). Sngapore: IEEE, 013, pp [1] S. Zheng, A. Janece, and Y. Tan, Enhanced rewors algorthm, n Proceedngs o 013 IEEE Congress on Evolutonary Computaton, (CEC 013). Cancun, Meco: IEEE, 013, pp [] Y. Tan, Frewors Algorthm: A Novel Swarm Intellgence Optmzaton Method. Sprnger, 015. [3] W. Zhang, X.-F. Xe, and.-c. B, Handlng boundary constrants or numercal optmzaton by partcle swarm lyng n perodc search space, n Proceedngs o the 004 Congress on Evolutonary Computaton, 004, pp [4] R. Eberhart and Y. Sh, Computatonal Intellgence: Concepts to Implementatons. Morgan Kaumann Publsher, 007. [5] X. Yao, Y. Lu, and G. Ln, Evolutonary programmng made aster, IEEE Transactons on Evolutonary Computaton, vol. 3, no., pp. 8-10, July [6] J. J. Lang, A. K. Qn, P. N. Suganthan, and S. Basar, Comprehensve

36 learnng partcle swarm optmzer or global optmzaton o multmodal unctons, IEEE Transactons on Evolutonary Computaton, vol. 10, no. 3, pp , June 006. [7] S. F. Adra and P. J. Flemng, versty management n evolutonary many-objectve optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 15, no., pp , Aprl 011. [8] S. Cheng, Y. Sh, and Q. Qn, Populaton dversty o partcle swarm optmzer solvng sngle and mult-objectve problems, Internatonal Journal o Swarm Intellgence Research (IJSIR), vol. 3, no. 4, pp. 3-60, 01.