A Novel Chaos Glowworm Swarm Optmzaton Algorthm for Optmzaton unctons Ka Huang, Yongquan Zhou, College of Mathematcs an Computer Scence Guang Unversty for Natonaltes Nannng, Guang 53, Chna Guang Key Laboratory of Hybr Computaton an IC Desgn Analyss, Nannng Guang 53, Chna yongquanzhou@.com Abstract. Ths paper a novel chaotc glowworm swarm optmzaton algorthm (C s propose. In C algorthm, the chaotc search strateges are ncorporate n to ntalze the frst teraton solutons, so that t can obtan hgh-qualty an evenly strbute ntal solutons, an avos beng trappe n local optma, each glowworm sturbs by chaos n a sturbance range can get more precse global soluton. Compare wth algorthm, eperments wth s test functons shows that convergence qualty an precson are mprove, whch testfy that C are val an feasble. Keywors: Glowworm swarm optmzaton; chaotc search strategy; chaotc glowworm swarm optmzaton; test functons. Introucton Many problems of the natural scences an the engneerng technology can be summarze as global optmzaton problems. About global optmzaton problem, so far, some researchers have alreay propose a varety of optmzaton algorthms, nclung tratonal algorthms base on graent an varous heurstc algorthms. Tratonal algorthms such as DP varable-menson algorthm, gol segmentaton metho, the conjugate recton metho, Powell accelerate metho, nterval metho an so on []. The optmzaton results of these algorthms epen hghly on the ntal contons an the moalty of functons. The moalty of many objectve functons of engneerng problems s complcate, so t s very har to solve ths problem wth tratonal algorthms base on graent. In recent years, wth the evelopment of computatonal ntellgence technology, varous new bonc ntellgent algorthms have been propose, such as partcle swarm algorthm, smulate annealng algorthm, genetc algorthm an so on. Glowworm swarm optmzaton ( [,3] s propose by Krshnana K.N. an Ghose.D n 5. The algorthm has been successfully apple to the sensor's nose test [] an smulaton robot [5], etc. The algorthm s swft an effcent n capturng optmal regons an has hgh commonalty, etc. But t also has some problems, such
as the hghly epenence on ntal solutons, slow convergence, easy to be trappe n local optma an the mprecse solutons. So ths paper propose a chaotc glowworm swarm optmzaton (C to conquer the efects of basc. Smulaton results emonstrate that the mprove algorthm can get hgh-qualty an evenly strbute ntal solutons, whch s effectve to avo beng trappe n nvual local optma. Basc glowworm swarm optmzaton algorthm ( In the, a swarm of glowworms are ntally ranomly strbute n the soluton space. They carry ther own lucfern respectvely whch has equal ntal value. The glowworms emt a lght whose ntensty s proportonal to the assocate lucfern. The lucfern quantty s tghtly assocate wth the poston the glowworms locate n ther movement. The glowworm whose lucfern quantty s hgher has stronger attracton to the other glowworms n ts neghborhoo. The neghborhoo whose sze s ece by raus ( r s calle local-ecson range n. The sze of r ynamcally changes between an r s. r s s calle raal sensor range n. In the movement, each glowworm moves to another glowworm whch s n ts neghborhoo at a certan probablty. Glowworm j who wants to become the neghbor of glowworm must be locate n the neghborhoo of an has hgher lucfern quantty than. Through the movement of glowworms, most glowworms wll converge to the glowworms that have hgher lucfern quantty. Each teraton of s consttute by two stages. The frst one s upate stage, the other one s movement stage. Lucfern upate stage: In ths stage, every glowworm upates ther lucfern by formula (. l ( l ( t J ( ( Where l ( t s the lucfern quantty of at teraton t, (, s the moulus to control lucfern quantty, s the moulus to evaluate objectve value of functon, J t an s the objectve value of functon Movement of glowworm: In ths stage, glowworm selects another glowworm j whch s locate n ts neghborhoo an move to t. The probablty formula s gven by (, the net poston of s ece by (3, the upate of r, whch s n the en of movement stage, s gven by (. Probablty formula use to select a neghbor s as follow: Upate formula of poston: p j l l j l k N ( t k l (
j ( t s * (3 j m where R s the poston of, at tme t, n the m-mensonal real space, represents the Euclean norm operator, an s ( represents movng step of glowworm. Upate formula of local-ecson range: r ( t mn{ rs, ma{, r ( t ( n t N ( t }} ( where s the proporton moulus, n t s the moulus use to control the number of neghbors, N s the number of neghbors of. 3 Chaotc glowworm swarm optmzaton algorthm (C 3. Defects of Analyzng efects of n ts search process: ( Intalzaton process of s ranom. Although ranom ntalzaton can guarantee the ntal solutons strbute evenly n the soluton space, the qualty of solutons are unrelable, because a part of solutons apart from the global optmum. If the ntal solutons are not only strbute evenly but also hgh-qualty, t wll contrbute to the qualty an effcency of solutons, an prevent algorthm to be prematurely trappe n local optma n a certan etent. ( The search process of s by constant step. When a glowworm who locates at the local optma has no neghbors, t wll become statonary. All these reasons lea to the naccuracy of. 3. Chaotc search strategy Chaos s wesprea n nature [], a nonlnear phenomenon presents n the vast majorty of nonlnear systems. Chaos s a ranom moton mappe by etermnstc equaton, but t s fferent from the phenomenon of sorer an rregularty. Chaos looks lke a ranom phenomenon, but t has a fne nternal structure. Chaos has such propertes: ( ranom; ( Ergoc: t can search all of the states unrepeate by ts own rules wthn certan range; (3 regularty. So t s avantageous for the optmzaton wth chaos unoubtely. Because of all these avantages, can chaos became a goo strategy for search process to avo beng trappe n local optma an ncrease the ablty of searchng global optmum. Chaos search s usually by Logstc functon, the formula s as follow: z, z, ( z,,,,, (,] (5 Where s a control parameter. When, z, Logstc s totally n chaotc state. In ths tet, assgn, formula(5s use as the chaotc sgnal generator.
3.3 C algorthm Detal steps of C are as follows: Step Intalze,,, s, l, m, N, D, p, n t, rs, an ntalze the mamum teraton number T ma ; Step Intalze glowworms by chaos. Step. Generate a vector of D -menson z z, z, z, z ], [,,,, D z, [,], each menson has tny fference. Step. Use vector z as ntal teraton vector of chaos, accorng to formula (5, z, z, ( z,,(,,..., D;,,..., N can get N number of z, z,, z. N Step.3 Map each menson of a z to the range of the objectve varable by ( b a z,(,, D;,, N, an then calculate the ftness of,,,n, select m nvuals whch are better from the N ntal nvuals. Step 3 generate a vector of D -menson, each menson s between an, such as u u, u,, u. (,,, n Whle(teraton t T o ma or : m Step Upate lucfern of all the glowworm accorng to formula (. Step 5 Calculate the neghbors of each glowworm. Step Select j( j N as the movement recton of by roulette, an upate the poston of by formula (3. Step 7 Upate r by formula(. Step 8 Dsturb to the poston each glowworm locates. Step 8. Generate u u, u,..., u, u u ( u,,,...,. an map each (,,, n menson of Step 8. Get the new poston,, D u to the range of sturbance [ r, r ] ' of ' r r u, u u.. accorng to step 8. by ' ' Step 8.3 Calculate the ftness of. If s better than, then replace, else not. En Whle Step 9 Let t t,complete an teraton, an juge whether the en conton s satsfe,. If satsfe, recor the result an et teraton, or return to step for the net teraton.
Epermental results an analyss. Test functons In orer to emonstrate that C has better convergence spee an can get more precse soluton than. The s functons are as follows: ( 3 ((5 9 (( ( ( (sn (5 ( 3 ( ( ( ( 5( g( h( g ( ( (9 3 3 h ( 3 ( 3 (8 3 8 3 7 5 ( Table. Stanar test functons uncton Search range Optmum ( [ -, ] ( [ -, ] 3( [ -, ] ( [ -.8,.8 ] 5( [ -, ] 3 ( [ -, ] -.53. Epermental parameters Epermental entcal parameters are set as table : Table. Parameters of C n l t...8 5 5 or the functons of,, an, the glowworm number s 5 an mamum teraton s an step s.3 an r s s 3. or the functons of, 3 the glowworm number s an mamum teraton s an step s.3 an rs s.
.3 Test Envronment The an C are coe n MATLAB8a an mplemente on Intel Core Duo CPU E5.GHz PC wth G RAM uner wnows XP operaton system.. Smulaton results an analyss We test C an by two methos: the frst one s that we conser t * convergent whle fun bes fun s satsfe uner the gven precse, or we conser t not convergent. or the low-menson functons of,, an 5, the s. or the hgh-menson functons of 3, the s *. fun best s the optmum we get, whle fun s the theory optmum; the secon one s that C an are mplemente respectvely to obtan the best, worst an average objectve functon values, so we can compare the two algorthm through those ata an followng convergent curve graph. rom table 3 an table, we can see that s trappe n local optma for, whle C s well convergent. or an, C nees less teraton to converge an has more precse soluton than. Table 3. The comparson of the requre teratons uner settng precse n tmes eperments The lest The teratons The most The average amount of are teratons Converg unc- Algor teratons are tme of neee for are neee -ence ton -thm neee for global for rate eperments optma are epermen eperments obtan -ts 89 379 3.387 e+ 7/ ( C 89 3.85e+ / 87.97e+ 5/ ( C 35.8 / A/N A/N A/N / 3( C 3 95.e+ / 5 9 9.8 / ( C 8 9 5.3 / 353 3 5/ 5( C 3 3.e+ / A/N A/N A/N / ( C 7 7.99e+ /
Table. The test results compare between an C unct on Algorthm C ( ( C Best objectve value.3395877e- 8.39795e- 7.3938897e- 7 Worst objectve value.77537537 e- 3.838877837.733337 e-9.8957793 ( 3 C.3895533.3398.873593.9959857395 ( 5 ( ( ftness value 5 5 C 3.3589583585e- 9.595893595e- 9.88799935993 e-7.8393399 e-7 C 3.5935355 3.5997779 3.738 3.8755 C -.53959839 57.399777 C -.537598 7 8.8793953 e+3 ftness value 8-5 Average objectve value 3.35388 8e-7.3975989 3.95835373837 55e-.893 3.3983785.3537 8 7.95933873 83e-8 9.533379959 78e-8 3.8 73 3.993579 9 -.538955 7.835935 99e+3 C 3 teratons g.. Comparson of curve graph for 3 teratons g. 3. Comparson of curve graph for ftness value.5.5 C ftness value.8... C 5 5 teratons g.. Comparson of curve graph for 3 3 teratons g. 5. Comparson of curve graph for
5 C 5 C ftness value 3 ftness value 3 3 teratons - 3 teratons g.. Comparson of curve graph for 5 g. 7. Comparson of curve graph for rom above convergence graph, we can see that C has better convergence spee an hgher precse than. 5 Conclusons Ths paper propose a chaotc glowworm swarm optmzaton algorthm. Strategy of chaotc ntalzaton ncorporate to can ncrease the qualty of ntal solutons an avo beng n local optma n a certan etent an enhance the ablty of capturng the global optmum. Moreover, chaotc sturbance ae to can enhance the convergence spee an the precse of soluton. Acknowlegement Ths work s supporte by Grants 998 from Guang Scence ounaton. References. Csenest. Numercal eperences wth a new generalze subnterval selecton crteron for nterval global optmzaton. Relable Computng, vol.9,no., 9-5(3. Krshnanan, K.N. Ghose, D. Glowworm swarm optmzaton: a new metho for optmzng mult-moal functons.computatonal Intellgence Stues, vol., no., 93-9(9 3. Krshnan an, K.N. Glowworm swarm optmzaton: a multmoal functon optmzaton paragm wth applcatons to multple sgnal source localzaton tasks. Inan: Department of Aerospace Engneerng, Inan Insttute of Scence (7. Krshnanan K.N. an Ghose D. A glowworm swarm optmzaton base mult-robot system for sgnal source localzaton. Desgn an Control of Intellgent Robotc Systems, 53-7(9 5. Krshnanan K.N. an Ghose D. Chasng multple moble sgnal sources: a glowworm swarm optmzaton approach. In Thr Inan Internatonal Conference on Artfcal Intellgence (IICAI 7, Inan, (7. Lu B,Wan L,Jn Y H,et al. Improve partcle swarm optmzaton combne wth chaos. Chaos, Soltons an ractals, 5,-7(5