Imperialist Competitive Algorithm with Variable Parameters to Determine the Global Minimum of Functions with Several Arguments

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1 Fourth Internatonal Conference Modellng and Development of Intellgent Systems October 8 - November, 05 Lucan Blaga Unversty Sbu - Romana Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments Abstract We have mplemented an Imperalst Compettve Algorthm (ICA to a determne the global mnmum for nne functons. Some of these represent benchmarks of the problem, whle others have expressons that we have defned. For a start, we used three of these functons to determne a set of optmal parameters for the ICA. Then we studed the way n whch the behavour of the ICA s affected by these parameters n order to solve the problem put forth for the nne functons. Fnally, we studed the behavour of the ICA as affected by varable parameters for the most dffcult of the nne functons. In our algorthm, the values of some of the parameters change dynamcally. The results ndcate a better behavour of the solutons provded by ths method. The Imperalst Compettve Algorthm: Structure and Parameters At the IEEE Congress on Evolutonary Computaton held on September 7-8, n 007, Esmael Atashpaz Gargar and Carlos Lucas presented a paper descrbng a new type of evolutonary algorthm nspred by hstory []. Called the Imperalst Compettve Algorthm (ICA, t s modelled on the poltcal and hstorcal events from the 7th, 8th and 9th centures. The ICA belongs to the category of meta-heurstc algorthms based on sets of canddate solutons called populatons (along wth genetc algorthms, algorthms of the swarm of partcles type, GSA gravtatonal search algorthms, etc.. The standard structure of an ICA as presented n [] s the followng:. Generatng an ntal set of countres;. Intalzng the mperalst countres; 3. Occupyng the colones; 4. Assmlatng the colones; 5. If a colony has better results than the mperalst country then a. Interchangng the colony wth the mperalst country 6. The mperalst competton: a. Computng the results of the empres b. Occupyng the weakest colony of the weakest empre by another empre c. If the weakest empre has no colones left then. Removng ths empre 40

2 7. If the stoppng requrements are met then Stop Otherwse Repeat the algorthm from step 3. The mperalst country that has the best results n the last teraton s the soluton to the problem. Below s a bref descrpton of the steps of an ICA. A. Generatng an ntal set of countres In the ICA, a country s a possble soluton to the problem studed. When determnng the mnmum for real functons of known expresson wth n varables (n=,3,4,..., a country s represented by a set of n real values. The number of countres s a frst parameter of the ICA. As wth other evolutonary algorthms, the consderable sze of the ntal set leads to a hgher probablty of fndng combnatons wth hgher performance. On the other hand, gven the complexty of the algorthm, rasng the number of countres causes a lnear ncrease n the runtme. In the completed applcatons, dfferent values were chosen for ths parameter: 55, 08 and 0. For the proper generaton, the functon n the C standard lbrary has been used, as t provdes quas-random values. B. Assessment of the ntal set of countres and ther revaluaton The exstence of a functon that allows the evaluaton of the performance of potental solutons, n our case the performance of a country, s a prerequste to solve a problem by means of ths algorthm. In the problem studed, the evaluaton functon s precsely the functon value calculated for the values representng a country. Snce we amed at determnng a mnmum, the evaluaton functon was a penalty: the lower the value of the evaluaton functon, the more effcent a country. C. Intalzaton of mperalst countres The ntal number of mperalst countres (or orgnal number of empres s another parameter characterstc of ths algorthm. In the applcatons that we ran, ths number was 5%, 0% or 5% of the orgnal number of countres. Followng the ntal assessment, the countres wth the best values of the evaluaton functon became mperalst countres. D. Occupyng the colones In ths algorthm, the countres that do not become mperalst countres turn nto colones and fall wthn the scope of mperalst countres. The lower the value of the evaluaton functon of that mperalst country, the hgher the number of colones assgned to an mperalst country. The ensemble comprsng the mperalst country and the occuped colones form an empre. The formulae for determnng the number of colones belongng to each mperalst country, desgnated nrcol, are gven below. nrcol v nrmet 0,,,nrmet - ( where nr met s the ntal number of mperalst countres, and values v are calculated by means of the followng formula v feval( met nr S col ( feval(x s the value of the evaluaton functon of country x, met s the mperalst country wth current number (=0,,,nrmet-, nr col s the ntal number of colones (the dfference between the total number of countres and the ntal number of mperalst countres and S s the sum of the values of the evaluaton functon for all the mperalst countres: 4

3 Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments nrmet 0 S feval( met (3 E. Assmlaton of colones By means of ths operaton characterstc of the ICA, the parameters of the countres of the colony type are transformed so that ther values can be attracted towards the values correspondng to the parameters of the mperalst countres. The way n whch ths transformaton takes place depends on the type of problem that s solved. The formula that we used s the followng: paramcol paramcol p d( parammet paramcol (4 ( where paramcol s the value of the colony parameter followng teraton, parammet s the value of the parameter correspondng to the mperalst country of the colony followng, and d s a random value below par, d[-da/ ; da/]; da s called assmlaton devaton. F. The revoluton operaton Followng the hstorcal model, some of the countres of the colony type undergo an operaton resultng n modfed parameter values. The probablty of a country to undergo ths operaton s called revoluton rate. Here s the way n whch these values can be modfed: - through a random regeneraton: after ths operaton, all parameters of a country are modfed; - by applyng a transformaton opposed to that of assmlatng some of the parameters ( ant - assmlaton : paramcol paramcol p d( parammet paramcol (5 ( d[-d r / ; d r /], d r, n ths case representng the revoluton devaton. G. The mperalst competton At ths stage of the algorthm, the least powerful empre loses the least effcent colony at the expense of another empre. We start by calculatng the performance of each empre usng the followng formula: ( performmp preformmet w performcol (6 j where w s the weght that the performance of a colony contrbutes to the performance of the empre w(0,, performmp and performmet are the performance of the empre and that of the mperalst country (=0,,,nrmet-, and performcol j s the performance of a colony belongng to empre. To determne the empre that wll assmlate the colony, we used the Monte Carlo method. A seres of random values a0, a,, anrmet- / a(0, was generated and the colony wll belong to the empre for whch the value of the expresson below s maxmum: performmp a (7 nrmet j0 performmp j If, followng ths operaton, an empre loses all the colones, the mperalst country of that empre also becomes a colony and s assgned to another randomly selected empre. 4

4 H. Completon of the algorthm There are two condtons that determne the completon of the algorthm and that we mplemented: one empre left (the deal stuaton convergent algorthm; the maxmum number of teratons was reached. As wth other evolutonary algorthms, there are no mathematcal demonstratons to prove that the algorthm converges towards the optmal soluton. In fact, the algorthm refnes the search n the vcnty of the ponts where the evaluaton functons have good values. The compettve mperalst algorthms have a number of parameters that depend on ther structure. For the algorthm structure presented before, the number of parameters s 0. The ICA determnng the global mnmum of functons wth several arguments A major problem that should be consdered when seekng to solve a problem wth the help of an algorthm of ths type s fndng the optmal values for the algorthm parameters. In the specalst lterature, we found no recommendatons for that purpose. To solve ths problem, we consdered two categores of parameters: the frst category ncluded parameters whose values were combned n the tests observng the prncple wth each other : Number of countres: 55, 08 and 0; The values of the ntal set of countres: the algorthm was run for 5,000 ntal sets; The number of mperalst countres: 5%, 0% and 5% of the total number of countres; The method of mplementng revolutons: no revolutons, regeneraton and ant-assmlaton (accordng to formula 4; The maxmum number of teratons: 3,000 (a value hgh enough so that the algorthms end on account of the fact that there s only one empre. The second category ncluded parameters whose values were combned n the tests, followng ths rule: each of the fve values of one of them was combned wth the average values of the others. Ths was the procedure for the followng: the approach step used n assmlaton and revoluton operatons: 0., 0.3, 0.5, 0.7 and 0.9; assmlaton devatons: 0., 0.6,.0,.4,.8; revoluton devaton: 0.,.0,.0, 4.0, 0.0; The percentage that colony contrbutes to the value of the performance of an empre: 0.0, 0.05, 0., 0.5,.0; The rate of revoluton: was chosen as the number of countres whose parameters are modfed so as to be proportonal to the rato between the number of countres and the orgnally defned number of empres, but so that the revolutons are ntated after a number of teratons (marked T equal to, 5, 0, 0 and 50; Thus, we noted how the values of each of these parameters nfluenced the performance of the algorthm. The behavour of the ICA was studed for three functons: f ( x, x sn( 3x ( x f e ( x x x x ( x, x, x3 7 sn( x e sn( x e sn( x3x sn( x 0.5x.x3 x4 f ( x, x, x3, x4 3 (0.5x x x3 x 4 ( x ( x ( x x e 3 x 4 (8 (9 (0 43

5 Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments For each of these three functons were chosen the felds of defnton [-0; 0] and [-00, 00]. By default, these felds were also those of searchng the mnmum and determnng to what extent the sze of ths feld affects the results. By jonng the values selected for the ICA parameters, a number of 5,400 combnatons resulted. Each of these combnatons was run for the 5,000 sets of countres and for each of the two areas. The tests conducted resulted n a set of optmal values for the ICA parameters,.e.: The method of mplementng the revoluton through regeneraton; The number of countres: 0; The ntal percentage of empres n the total number of countres: 5%. The approach step, p = 0.; The percentage of colones wthn the empre performance, w = 0.5; The devaton from assmlaton, yes =.8; The number of teratons at whch revolutons T = occur; ICAs wth parameters thus determned were used to calculate the mnmum of nne reference functons or wth an expresson determned by us. Startng from the observaton that the ICA does not guarantee fndng the soluton, we ran the algorthm for 00 ntal sets. The number of these sets was selected so that the total runnng tme should be under a mnute, all tests beng run o n an Intel Core 3 mcroprocessor at.93 GHz. Table shows a summary of the results. The values n column nrok represent the number of sets for whch ICA located the global mnmum of the functon (out of the 00 sets used. Functon [-0;0] [-00;00] nrok nrok ( x x f ( x, x sn( 3x ( x e f x x x3 ( x, x, x3 7 sn( x e sn( x e sn( x3x e sn( x 0.5x.x3 x4 f ( x, x, x3, x4 3 (0.5x x x3 x 4 ( x ( x ( x x f ( x, x x ( x (De Jong x, x J0( x x 0. x 0. f ( x (benchmark n [] f x, x x sn4x.x snx, x, y [0,0] (benchmark n [8] 00 6( f ( x, x f 7 x sn( x sn( x x 7 8 ( x x sn( x 3 f9 ( x ( x x. sn( xx. x, f 8 : [0;0] 7 R (Alpne x x ( x, x[,4] (Brown Table. Results obtaned x x 3 x x Vald defnton felds except for functons n the expresson of whch t s explct: f 6, f 8 and f 9 44

6 The followng fgures shows the behavour of the algorthm n determnng the mnmum of functon f: [-00;00][-00;00] for one of the 00 ntal sets wth 08 countres and 5 mperalst countres. Fg. Dstrbuton of colones followng one teraton Fg. Dstrbuton of colones followng0 teratons Fg. 3 Dstrbuton of colones followng 500 teratons The frst fgure llustrates the ntal dstrbuton obtaned by random generaton. The larger dots represent the locaton of the mperalst countres. Wth the same colour are marked the mperalst country and colones that belong to t. After 0 teratons, we can notce the groupng of colones around the mperalst country to whch they belong as a result of assmlaton. We can also see that two potental solutons are closer to the soluton sought for. After 500 teratons, all 5 mperalst countres as well as many of the colones that belong to them are closer to the optmal soluton. Fgures 4, and 5 llustrate the behavour of the algorthm n the case of functon f for one 45

7 Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments of the tests n whch 08 countres and 5 mperalst countres were used. The frst fgure llustrates the ntal dstrbuton obtaned by random generaton. The larger dots represent the locaton of the countres. Wth the same colour are marked the mperalst country and colones that belong to t. Followng 7 teratons, we can notce the groupng of the colones around the mperalst country they belong to as a result of assmlaton. Followng 400 teratons, there s a group consstng of two mperalst countres located very close to each other and a group of three mperalst countres located very closet o them. Many of the colones that belong to them are located n ther vcnty. Followng 83 teratons, three empres were left, ther mperalst countres beng located very close to them. Fg. 4 Dstrbuton of colones followng one teraton Fg. 5 Dstrbuton of colones followng teraton no Behavour of ICA wth varable parameters 3. ICA wth dynamc weght In the smulatons performed, there were many stuatons n whch the standard algorthm converged very slowly or was even stuck because the performance of last two, sometmes even three or four empres left n the algorthm, became very close dfferences of the order of 0.% of the values of the performance. The result was that, from a certan teraton, one and the same colony shfts from one empre to the other. Here s the explanaton: when a colony shfts from one empre be t I to the other be t I, (snce performance (I < performance (I, t undergoes an assmlaton process by the new mperalst country. The mmedate effect s a worsenng of the performance of the colony; ths value affects the performance of the empre whch took over the colony, so that n the next teraton performance (I < performance (I and the weakest colony s precsely the one whch has been taken over. Thus, the colony returns to the frst empre, but wth a performance value that was worsened by applyng a new assmlaton operaton by I. Once more, performance (I < performance (I and the weakest colony that wll be taken over by I s obvously the one that has just returned to I. Ths algorthm follows a loop: one colony shfts between the two empres, the others are assmlated n an ncreasngly hgher percentage, and the chances that colones wth better performance than that of the mperalst country may appear decrease wth each teraton. Such blockage has been removed by usng weghts whose values change dynamcally as follows: ntally, each colony s assgned the same 46

8 weght, havng a predetermned value; whenever a colony changes the mperalst country, the value of the weght of that colony s reduced by multplyng t wth a below par value. We called ths value weght contracton (cw. Thus, equaton (7 turned nto: performmp preformmet w performcol ( j j where wj s the weght assocated to colony j. Another change to the standard algorthm that was performed n order to avod blockage was the followng: the weakest colony s determned havng as a crteron the product between the evaluaton functon and the weght of the colony n that teraton (and not only the value gven by the evaluaton functon for that colony. Fgures 0 and llustrate the comparatve evoluton of empre performances when the ICA s appled to functon f: [-00;00][-00;00] for an ntal set of 08 countres and 5 mperalst countres. Fgures and 3 llustrate the evoluton of the number of colones wthn each empre under the same condtons. It can be noted that the ICA wth varable weght s completed after 53 teratons, but the performance of the algorthm s not affected. Fg. 6 Empre performances n the case of the ICA wth fxed weght Fg. 7 Empre performances n the case of the ICA wth varable weght cw=0.5 Fg. 8 Evoluton of the number of colones n the case of the ICA wth fxed weght Fg. 9 Evoluton of the number of colones n the case of the ICA wth varable weght cw=0.5 47

9 Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments In the followng, we present the results obtaned by applyng the ICA wth varable weght to determne the mnmum of functon f wth the defnton feld [-00 ; 00]3, practcally the most dffcult of the functons studed. Table shows the behavour of the ICA n three cases: hgh fxed weght, low fxed weght and varable weght. The number of sets was 00 for the ICA wth w=0.5 and cw= (fxed weght. In the other cases, ths number was nversely proportonal to the average number of teratons. After that, the algorthm stopped (we desgnated ths quantty nrit so that the total runnng tme may be the same for each of the three cases analyzed. f : [-00;00] 3 R 55 countres 08 countres 0 countres Tp ICA mn(f nrok nrit mn(f nrok nrit mn(f nrok nrit w=0.5, cw= w=0.0, cw= w=0.5, cw= Table. ICA wth fxed weght vs. ICA wth varable weght The most effcent varant of those shown n the table accordng to the number of cases where the mnmum of the functon s obtaned s that of the algorthm wth varable weght and 0 countres n the ntal set. 3. ICA wth varable revoluton rate The ICA behavour was studed when the number of countres where revolutons occur vares, ths number ncreasng durng the sequence of the algorthm. Ths s because, as the algorthm progresses, the number of colones that wll be n the vcnty of the mperalst countres s ncreasngly hgher, due to the convergent formulae used n the assmlaton operaton. Hence, the dea of havng an ncreasng number of countres nvolved n the revolutons. In the tests conducted, the revoluton rate, desgnated prob r below, had the followng values: number of countres prob r ( ntal number of empres number of countres prob r 0 (3 ntal number of empres number of countres teraton prob r ntal number of emprea 50 (4 number of countres teraton prob r ntal number of empres 00 (5 As t can be notced, n the frst three formulae, the revoluton rate s fxed and, n the last two formulae, t s varable. The other parameters were kept constant at the values determned n secton. The tests were also carred out n order to determne the mnmum of functon f: [- 00;00]3 R. The results are shown n Table 3. 48

10 prob r (formula Number of countres mn(f nrok mn(f nrok mn(f nrok Table 3. ICA wth fxed revoluton rate vs. ICA wth varable rate A better behavour of the ICA wth varable revoluton rate can be notced n tests wth a hgh number of countres (0 accordng to formula (6. Ths formula corresponds to the case where there s a more rapd ncrease n the number of countres subject to the operaton of revoluton dependng on the teraton reached by the algorthm. 4 Conclusons The ICA proved to be an effcent algorthm n determnng the mnmum of functons wth several arguments. To determne an optmal set of parameters for the algorthm, we ran 7,000,000 tests. The ICA wth ths optmal set of parameters was used to determne the global mnmum for nne functons; for seven of them, we used two felds of defnton. In 4 of the 6 cases that resulted, the ICA determned the mnmum n 00% of the 00 tests conducted. The lowest probablty of obtanng the mnmum was 78%. In secton 3, we studed the behavour of the ICA wth two of ts characterstc parameters dynamcally modfed. The studes were conducted for the functon consdered to be the most dffcult. The frst study amed at modfyng the parameter that determnes the weght wth whch a colony nfluences the performance of the whole empre to whch t belongs. Ths parameter was modfed wth the ntal purpose of removng the blockages of the ICA because of very poor performance colones. Thus modfed, the ICA presented superor convergence: f for the ICA wth fxed weght the algorthm stopped after 79 teratons on average, the ICA wth varable weght stopped after 49 teratons on average. Thus, wthn the same tme nterval, the ICA wth fxed weght located the global mnmum for the functon studed for 78 ntal sets and the ICA wth varable weght located the mnmum n 56 ntal sets. The second study was concerned wth the behavour of the ICA when the number of countres where revolutons take place vares, ncreasng durng the sequence of the algorthm. Ths s because, as the algorthm progresses, the number of colones that wll be n the vcnty of mperalst countres s ncreasngly hgher, due to the convergent formulae used n the assmlaton operaton. Hence, the dea of havng an ncreasng number of countres nvolved n the revolutons. In the tests conducted, we used two formulae for ths parameter. For one of them, the ICA located the mnmum of functon f n 86 of the 00 ntal sets, whch represented an mprovement of the performance of the algorthm wth 0.5%. References [] Gargar Atashpaz, E., Caro, L.: Imperalst Compettve Algorthm: An Algorthm for Optmzaton Inspred by Imperalstc Competton, IEEE Congress on Evolutonary Computaton, CEC,

11 Imperalst Compettve Algorthm wth Varable Parameters to Determne the Global Mnmum of Functons wth Several Arguments [] Behesht, Y., Shamsuddn, S. M. Hj.: A Revew of Populaton-based Meta-Heurstc Algorthms, Int. J. Advance. Soft Comput. Appl., Vol. 5, No., March 03 ISSN [3] Forouharfard, S., Zandeh, M.: An mperalst compettve algorthm to schedule of recevng and shppng trucks n cross-dockng systems, Internatonal Journal of Advanced Manufacturng Technology, 00 [4] Hojjat, E., Shahrar, L.: Graph Colourng Problem Based on Dscrete Imperalst Compettve Algorthm, CoRR, 03. [5] Lu, J.Y.-C., Yuan Z., S., Chang-Ten C.: On the Convergence of Imperalst Compettve Algorthm, 7th Asa Modellng Symposum, 03. [6] Soltan-Sarvestan, M. A., Badamchzadeh, M. A., Soltan-Sarvestan, Sh., Javanray, D.: Investgaton of Revoluton Operator n Imperalst Compettve Algorthm, 4th Internatonal Conference on Computer and Electrcal Engneerng (ICCEE 0. [7] Curea S., Trfa V.: Imperalst Compettve Algorthm wth Varable Parameters for the Optmzaton of a Fuzzy Controller, Internatonal Conference on System Theory, Control and Computng ICSTCC, Snaa, 04. [8] Schwefel, H.-P.: Numercal optmzaton of computer models, Wley, 98, ISBN3: , LC: QA40.5.S383. STELIAN CIUREA Lucan Blaga Unversty of Sbu Faculty of Engneerng, Department of Computer and Electrcal Engneerng E. Coran Str, No. 4, Sbu-55005, ROMANIA, E-mal: stelan.curea@ulbsbu.ro 50