Design of College Discrete Mathematics Based on Particle Swarm Optimization. Jiedong Chen1, a

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1 4th Internatonal Conference on Mechatroncs, Materals, Chemstry and Computer Engneerng (ICMMCCE 205) Desgn of College Dscrete Mathematcs Based on Partcle Swarm Optmzaton Jedong Chen, a Automotve Department, Guangdong Industry Techncal College, Chna a @qq.com Keywords: Dscrete mathematcs, Partcle swarm optmzaton, Exchange of nformaton, Mutaton operator Abstract: Amng at exploraton and development capacty, t s usually hard to acheve effectve utlzaton and balance by usng ust one algorthm, whch wll nfluence the solvng accuracy and effcency of the algorthm accordngly. Ths thess apples partcle swarm optmzaton (PSO) to dscrete mult-obectve mathematcal optmzaton, and proposes a dscrete mult-obectve optmzaton algorthm based on PSO. Ths algorthm adopts bnary mechansm to acheve the poston vector of the partcles. Meanwhle, t bulds a non-domnated soluton set to storage the researched non-domnated soluton, so as to ncrease the dversty of non-domnated solutons. The ntroductons of genetc algorthm crossover, mutaton operator replacng partcle velocty and locaton updatng have reduced the computatonal complexty. In the thess, t sets up mathematcal models for these two problems respectvely, and the expermental results show that the algorthm s effectve. Introducton Dscrete mathematcs s defned to be the core backbone currculum of computer scence by IEEE n 977. In 2004, among the fve professonal tranng plans related to computer wth tutoral certfcaton, dscrete mathematcs s the sgnfcant core currculum of three maors, whch are Computer Engneerng (CE), Computer Scence (CS) and System Engneerng (SE) [,2,3]. As the precedng currculum of specalzed courses, such as data structure, fundamentals of complng, database prncple, operatng system and artfcal ntellgence, dscrete mathematcs not only needs to provde necessary basc knowledge, but develop the students abstract thnkng ablty and logcal thnkng ablty, so as to further strengthen ther program desgn ablty. The constraned optmzaton problem of unlnear dscrete varable s the most common problem n engneerng practce. Due to ts non-convexty, there exsts a great deal of dfference between global optmal soluton and locally optonal soluton. In order to solve global optmzaton problem, people try to fnd a stochastc optmzaton algorthm whch has a lower requrement or even no requrement to obectve functon and constrant functon. The evolutonary algorthms [4,5]based on bologcal evolutonsm and heredtsm and the swarm ntellgence algorthm[6]based on botc communty behavor are proved to be effectve. Based on partcle swarm optmzaton, the thess ntroduces some expermental desgns approprately n teachng process of dscrete mathematcs, whch s not only a good verfcaton of basc theory of dscrete mathematcs, but bulds up the students operaton ablty of computer language and foreshadows the learnng of other currculum. Desgn model and dscrete varable of dscrete mathematcs Desgn model of dscrete mathematcs The relaton n the dscrete mathematcs can be represented by matrx method. Matrx can be stored by a two dmensonal array n a computer. The establshment and storage of array are ntroduced n computer language; therefore, ths part wll not be restated n ths thess, but focuses on the dscusson of the mplementaton of the algorthm. In ths thess, we assume Relaton (R) and 205. The authors - Publshed by Atlants Press 200

2 Relaton2 (R2) are both bnary relatons n the set X, where there are n elements n X. And we assume the relatonshp matrx of R and R2 are M and M2. They can not be drectly obtaned from the matrx dagram, therefore, we can realze the udgment of transtvty through the transtve closure of relatons, and the realzaton of transtve closure needs compound operaton of the relatons. Thus, the algorthm desgns of relatonshp complex operaton and closure operaton should be gven frstly. Set R and R2, and calculate relatonshp matrx M of composton relaton R of R and R2: () Suppose =, =; n (2) Calculate M(, ) M(, k) M 2( k, ) through logc multplcaton and logc plus; (3) =+, f n, transfer to (2); f not, transfer to (4); (4) =+, f n, transfer to (2); f not, stop calculaton. n If R R, R has the transmsson, where R represents the tmes compound operaton of R. Therefore, we can calculate t through callng the compound operaton of relatons. Dealng wth dscrete varable through mappng functon In ths thess, the three partcle swarm optmzaton algorthms (PSO, PSOPC and HPSO) all use mappng functon method to deal wth the dscrete varables. After sortng wth a certan rule, the dscrete varable set S d, whch has P dscrete varable plate thckness, can be show as follows: Sd { X, X 2,, X, X p}, p () Suppose a mappng functon and use the seral number to replace the dscrete value n Sd, that s, h () = X. Select ts seral number to replace the specfc value of dscrete varables to begn to search, and try to make ts searchng value as contnuous as possble, so as to mprove the effcency of the search. For example, n partcles are n D-dmensonal searchng space, the poston of the partcle can be represented as vector X, that s: 2 d D X { X, X, X, X }, d D, n (2), Namely, X {,2,,p }, through mappng functon h (), the correspondng set of dscrete varable s {X,X 2, X 3,,Xp}. Through mappng, partcle can only search n nteger space of the whole searchng space; that s to say, all the components of vector X are all ntegers. Improved PSO based on dscrete mathematcs model () Optmzed mathematcal model of dscrete mathematcs Mathematcal model of dscrete mult-obectve optmzaton s showed n formula (3). mn f ( x) [ f( x), f 2( x), f3( x), f m ( x),] (3) On the bass of dsperson mult-obectve optmzaton of partcle swarm optmzaton (DMOPSO) mult-obectve partcle swarm optmzaton algorthm, we adopt the mechansm of bnary partcle swarm optmzaton algorthm to calculate dscrete varables. In ths way, partcle search space correspondngly converses to N-dmensonal bnary space, marked as B n. In order to apply partcle swarm optmzaton algorthm dscrete varables optmzaton, we make the poston vector of each partcle n that algorthm belong to bnary space (bnary strngs composed of 0 and ). However, ts velocty vector stll belongs to real number space R n,that s to say, v [ -v max, v max ]. In ths way, the algorthm can preserve the speed formula n the basc partcle swarm algorthm. Whle only modfyng ts poston update formula, the speed and poston updatng formula of each partcle are as follows: v ( t ) wv cl r ( pbest x ) c r ( pbest x ) (4) 20

3 0 else x ( t ) (5) rand S[ v ( t )] Where, rand s a random number between [0, ], and the other parameters n the algorthm are the same as the parameters n the basc partcle swarm optmzaton algorthm. The probablty that a partcle wll be taken as or 0 s determned by the current speed value of the algorthm s: P[ x ( t ) ] S( v ( t )) (6) That s, the probablty selecton parameter S s determned by the partcle speed wthn the range of [0,]. If S s close to, the partcle may be selected as ; f S s close to 0, the partcle s more lkely to be selected as 0. Kennedy and other people has used Sgmod functon to obtan the parameters. Sgmod functon s a common non-lnear acton functon used n neural network, t expresson formula s : s S( v( t)) v e (7) (2) Mnmum poston value rule Update the poston of each element n the arrangement of wth a contnuous PSO: ( t) ( t) xd xd w( xd xd )) cr ( pd xd ) c2r3 ( pd xd ) (8) The new partcle poston of real-value can be obtaned by the teraton of that formula (ntermedate result). Then, wth the prncple that elements of the mnmum value possess the earlest schedulng, conduct an arrangement through transformng the ntermedate result to ts correspondng soluton. The specfc steps are as follows: Assume the updated poston of the partcle to be real vector X { x, x2, xn}, and ts correspondng soluton s arranged as an nteger vector,, }. Wth the prncple that { 2 n elements of the mnmum value possess the earlest schedulng, confrm through X. Assume ntermedate result X (.45, 3.54,2.67, 2.29,-4.02).Its mnmum value s -4.02, whch s correspondng to the ffth element of X. Therefore, the ffth value s the earlest one to be dspatched, whch means =5. The second smallest value s -3.54, t corresponds the second element of. Thus, the second value s the second one to be dspatched, whch means 2 =2. Calculatng n ths way, we obtan the soluton arrangement (5,2,4, l,3). At last, conduct nverse operaton to (5,2,4, l,3), and we can obtan the nteger vector X (4,2,5,3, l) of the correspondng new partcle poston. (3) Intalzed partcle swarm Varous solutons producng method nvolves two stages: one s to use Greedy Randomzed Adaptve Search Procedure (GRAsp) to produce ntal soluton; the other s to use local search algorthm to explore the feld of ts ntal soluton and fnd the most advantages local pont. Repeat these two stages and fnally produce P sze dfferent soluton arrangement, whch makes up soluton set P P sze should be larger than the sze of swarm b of partcle swarm. Next, select b l optmal soluton from ths P sze dfferent soluton to put nto the ntal swarm of partcle swarm, and then delete them from soluton set P. In the end, repeat the followng steps for b: tmes (b 2 =b-b l ): A. Newly calculate the mnmum value of the dstance between each of the surplus soluton of P and all solutons n the ntal swarm of current partcle swarm. B. Select the soluton wth the mnmum dstance from the soluton of P, and put t nto the ntal partcle swarm. C. Delete the selected soluton from P. Chart s the pseudocode of GRAsp, whose process of obtanng the soluton s as follows: Assume the soluton s composed of many soluton elements (for example, the soluton of LOP problem s the nteger between -n). Accordng to a sort of heurstc crtera, calcuate an evaluaton 202

4 value for each soluton elements out of the partal solutons. It represents the good degree of addng ths soluton element to partal soluton n the current stuaton, and the partal soluton elements wth hgh evaluaton value consttutes a restrcted canddate lst. And then, randomly select a soluton element from RCL to partal soluton. Keep on repeatng ths process, untl the structure of the soluton s fnshed. And the partal soluton s the null soluton at the begnnng. Chart. GRASP Pseudocode (4) Local search algorthm Adopt the local search algorthm, whch s on the bass of hsert mover. The defnaton plays the role n hsert mover operaton INSERT-OVE(, ) of soluton arrangement { 2, n}, for example: frstly, t deletes the element from soluton arrangement n, the '+~n element of soluton arrangement moves forward for one step. Then, nsert ths element nto the poston of soluton arrangement, the orgnal ~n- element of soluton arrangement moves backward for one step, thereby, t produces a new soluton arrangement (,,,,,,,, f n) (9) (,,,, n) f The feld of nqury of the hsert mover operaton SERT-MOVE (, ) s: N { : INSERT _ MOVE(, ), for n and 2, } (0) 2 n N 2 can be dvded nto n sub-dvson (sub-feld). All the sub-felds are related to the soluton J element. The defnton of N 2 s: N J 2 { : INSERT _ MOVE(, ), for n} () The sum of all element values n correspondng lne of the soluton element on E s regarded to be the attractve value of sub-feld. Rank these attractve value, so, when makng local search, frstly explore the sub-feld wth the bggest attractve value. Process of dscrete mult-obectve optmzaton algorthm based on PSO The key problem of PSO s the selecton of P best and G best. Among the problem of sngle obect optmzaton, we can make the choces through obectve functonal value. However, t s hard to make sure whch partcle s better through obectve functonal value n mult-obectve Optmzaton Problem, thus, no way can be found to select the ndvdual optmal value and local optmal value. Mantanng non-domnated soluton set s to provde dversfed P best for the teratve search of partcle swarm. At the same tme, mantan and further strenghten the dversty of the soluton n non-domnated solutons through selectng an approprate P best for each partcle. The thess uses non-domnated soluton set to store the non-domnated solutons produced by teratve search. And conduct updatng mantenance for non-domnated soluton set tmely. Durng the process of optmzaton routne, the P best of the partcle conducts teraton update by domnatng set. G best 203

5 s randomly selected from non-domnated soluton set, so as to ensure that non-domnated soluton set s gradually close to the optmal soluton of Pareto. The process of dscrete mult-obect algorthm based on partcle swarm optmzaton s: () Intalze partcle swarm P, and set the max teratons M, partcle swarm scale N, partcle dmensonalty of DMOPSO algorthm. Randomly produce the poston x and speed v of each partcle. The ntal value of teratons counter s set to be 0. (2) Set teratons counter to be, and calculate each partcle s ftness value f k (x ) of multple obectve functon f k (x) ftness k, ) f k ( x ) (2) ( Calculate the ftness value of the partcle n the lght of formula (3). On the bass of domnaton concepton, rank the ftness value from top to bottom. Keep the frst N partcle of the partcle swarm n domnant sub-set NP, and the rest N 2 partcle n non-domnant sub-set P. The postons of these partcle s Non-domnated soluton. Z ftness( ) ftness( k, ) (3) Z k (3) Accordng to formula (7), update the speed and poston of N partcle n domnated sub-set NP. Determne the best ntal poston P best[k,] of each partcle as ther own ntal poston, and the global optmum G best s randomly selected from non-domnated set N 2. In ths way, t can ncrease the dversty of solutons and avod producng the stuaton of local optmum. (4) Domnate thought n terms of the ftness value, and update the P best[k,] of the partcles n non-domnatng subgroup P. Z pbest ( ) pbest( k, ) (4) Z k (5) Mantan the non-domnatng subgroup P. For the newly produced non-domnatng soluton x (t+), we wll adopt the followng strateges: When the newly produced soluton s domnated by the members of non-domnatng subgroup P, prevent the new soluton from onng P; when the newly produced soluton domnates part members of P, remove the members whch s under the domnaton n P, and add the newly produced soluton nto ths non-domnatng subgroup P; When the newly produced soluton and the members of non-domnatng subgroup P are not domnated by each other, add the newly produced soluton nto P drectly. When the sze of non-domnatng subgroup P reaches or excees the maxmum sze specfed n advance, sort by the ftness value, and delete the non domnated solutons whch s domnated by all the other members. The ftness value s calculated by (3). (6) Check f t reaches the maxmum number of teratons, f so, stop searchng; f not, contnue to search by transferng to (2). Test based on dscrete mult-obectve optmzaton algorthm of PSO In order to verfy the performance of the algorthm n ths thess, we adopt the TSP problem provded by TSPLIB whch s Internatonal general as the control standard of known optmal soluton for test data. The expermental envronment s Celeron.7G CPU, 256M RAM, Wn2000 OS and VC6.0. The expermental contrast s dvded nto two parts. The frst part s to compare wth smlar partcle swarm algorthm; and the second part s to compare wth other evolutonary algorthms. Through constructng the data, t shows some expermental results of the algorthm on solvng the large-szed TSP problem provded by TSPLIB n ths thess. Through choosng to keep the dstance of the nearest cty every tme wth greedy thought, t wll form better sub fragment, so as to ncrease the local search speed rapdly n the ntal perod of search. However, a few good contnuous sub segments take advantage n the group soon, whch leads to the focus of partcles. The search behavor at ths tme s not conducve to the global detecton of partcle swarm optmzaton on optmal soluton. However, n ths thess, t keeps the better sub fragment 204

6 whle usng the falure mechansm of poor fragment, that s to say, contnuously destroys the sub fragments whch are easy to produce nferor solutons and produces new sub fragment, whch makes the partcles show a trend of dversfcaton. Therefore, t can make the algorthm come to the optmal soluton wth a few number of teratons, meanwhle, provde a very good soluton to the defect that the greedy algorthm s usually easy to be attracted by the local extreme value and hard to escape. Table. Contrast experment for solvng TSP problem n PSO algorthm Soluton space sze (4-)!/2=3,3,50,400 Partcle number 00 Average evoluton algebra 20, Average search space 20,000*00=2,000,000,000*00 =00,000 20*00=2,000 sze Rato of Search space 2,000,000/3,3,50, 00,000/3,3,50 and soluton space 400=0.064%,400=3.2E-5 Best soluton 3->->0->9->8->0->7->2->6->->5->4->3->2 Value of best soluton (Equal to the best known result n the world) 2,000/3,3,50,400 =6.42E-7 Table 2. shows the contrast results of the algorthm n ths thess, PSO (Standard PSO algorth m ) provded by lterature [49], RPAC (random perturbaton ant colony and SCPSO (Partcle swarm optmzaton based on subgroup hybrdzaton mechansm ). The experment data adopts the manage meng of three classc TSP problems provded by TSPLIB, whch are gr24,ayes29 and gr48.the grou p sze choose to use 00 (500 n other algorthm groups), average value s obtaned by 200 teraton per tme and keeps 30 tmes repeatng totally.in the experment process of ths thess, both gr24 and bayes29 problem obtan the known optmal soluton durng the 30 tmes repeatng; moreover, gr48 r eaches up to 93.3% for the tmes of obtanng optmum soluton durng the results of 30 tmes repeat ng, but no optmum soluton s obtaned for other comparson algorthms. From Table 2., we can se e that the convergence rate of the algorthm n ths thess and the qualty of the soluton are obvousl y better than the other several contrasted algorthms, even wth some smaller group szes. TSP Problem Table 2. Contrast experment 2 for solvng TSP problem n dfferent PSO algorthms TSP PSO RPAC SCPSO Average Optmum Average Optmzaton Average Soluton of Soluton Soluton Soluton Soluton Algorthm Best Soluton of Algorthm n ths paper n ths paper gr Bayes gr Take gr24 problem as an example, Chart 2 shows the convergence curve whch compares the dfference between adoptng genetc algorthm to crossover operator and mult group hybrd n orgnal PSO algorthm. Compared wth genetc algorthm to crossover operator of POS algorthm and orgnal PSO algorthm, the qualty and convergence speed of adoptng mult group hybrd of PSO algorthm are both better than the frst two algorthms, through usng some better solutons fragments from other subgroups. However, n mult subgroup hybrd PSO algorthm the subgroups probably conduct ther own evoluton ndependently at the very begnnng. So, only when the ftness change of all the subgroups stop, onc exchange among the subgroups begn to conduct, whch doesn t make the convergence speed of the algorthm effectvely mproved. At the same tme, for the onc exchange among the subgroups begn to conduct only when the ftness change of all the subgroups tends to stop, the dversty of each subgroup t s not deal overall, although t surely has some mprovement. The algorthm n ths thess adopts nformaton exchange strategy and dynamc 205

7 dvson search strategy; therefore, t turns out to be better than the other three comparson algorthms no matter n the convergence rate of the algorthm or n the guarantee of the partcle dversty. Chart 2. Convergence curve chart of comparson between the algorthm n ths thess and other PSO algorthms Summary In recent years, PSO has acqured good results on contnuous optmzaton problem, however, t s stll a brand new attempt on solvng dscrete optmzaton problem. In ths chapter, the mproved partcle swarm optmzaton proposed n the thess s successfully appled to TSP, whch s a classc dscrete optmzaton problem. Accordng to PSO, through desgnng an nformaton exchange strategy among partcles wth greedy thoughts, the new algorthm makes the ndvdual partcle acqure more useful nformaton of other partcles, so as to present dfferent moton behavors durng the process of evolutonary. Improvng PSO through combnng wth the dynamc dvson search strategy wll effectvely balance the search effcency and accuracy. Through a large number of experments, t shows that the mproved PSO n ths thess can be effectvely appled to TSP problem. The solvng accuracy and effcency are stll relatvely satsfactory. References [] He Yun. Swarm Intellgent Optmzaton and ts Appled Research n Chemstry and Chemcal Engneerng. Hangzhou: Zheang Unversty, [2] Cu Guangzhao, L Xaoguang, etc.. Optmzaton of DNA Codng Sequence Based on Modfed Partcle Swarm Genetc Algorthm. Journal of Computer Scence. Vol, 2, 200, p3-36. [3] Yu Xueng, Ma Xaofe, Xa Bn. Dynamc Partcle Swarm Optmzaton. Computer Engneerng. Vol 4, 200, p [4] Pan Feng, Tu Xuyan, Chen Je, etc.. Hybrd Partcle Swarm Optmzaton-HPSO. Computer Engneerng. Vol, 3(), 2005, p [5] Zeng Janchao, Je Jng, Cu Zhhua. Partcle Swarm Optmzaton. Beng: Scence Press, [6] Yang Hongru, Gao Hongyuan, Pang Zhengwe, etc.. Mult-user Detector Based on Dscrete Partcle Swarm Optmzaton. Harbn Insttute of Technology Press. Vol 37(9), 2005, p [7] Yang We, L Qqang. Revew of Partcle Swarm Optmzaton. Chnese Engneerng Scence, Vol 6(5), 2004, p