Chapter 2 Real-Coded Adaptive Range Genetic Algorithm

Transcription

1 Chapter Real-Coded Adaptve Range Genetc Algorthm.. Introducton Fndng a global optmum n the contnuous doman s challengng for Genetc Algorthms (GAs. Tradtonal GAs use the bnary representaton that evenly dscretzes a real desgn space. Although such bnary-coded GAs have been successfully appled to a wde range of desgn optmzaton problems, they suffer from dsadvantages, when appled to the real-world problems nvolvng a large number of real desgn varables. Snce bnary substrngs representng each parameter wth the desred precson are concatenated to represent an ndvdual, the resultng strng encodng a large number of desgn varables would wnd up a huge strng length. For example, for 00 varables wth a precson of sx dgts, the strng length s about 000. GAs would perform poorly for such desgn problems. Prevous applcatons have been kept away from ths problem by sacrfcng precson or narrowng down the search regons pror to the optmzaton. However, such approaches mght exclude the regon that actually has the global optmum. Another drawback of the bnary-coded GAs appled to parameter optmzaton problems n contnuous domans comes from dscrepancy between the bnary representaton space and the actual problem space. For example, two ponts close to each other n the representaton space mght be far n the bnary represented problem space. It s stll an open queston to construct an effcent crossover operator that suts to such a modfed problem space. A smple soluton to these problems s the use of the floatng-pont representaton of parameters. In these real-coded GAs, an ndvdual s coded as a vector of real numbers correspondng to the desgn varables. The real-coded GAs are robust, accurate, and effcent because the floatngpont representaton s conceptually closest to the real desgn space, and moreover, the strng length reduces to the number of desgn varables. However, even the real-coded GAs would lead to premature convergence when appled to aerodynamc shape desgns wth a large number of desgn varables. A more sophstcated approach s to dynamcally alter the coarseness of the search space referred to as dynamc codng. In [], Krshnakumar et al. presented Stochastc Genetc Algorthms (Stochastc GAs to effcently solve problems wth a large number of real desgn parameters. The key features of Stochastc GAs are:. Each bnary number represents a regon of the real space nstead of a sngle pont to mantanng good precson wth the small strng length.. Those regons adapt durng the optmzaton process accordng to the /5 th success rule as Evolutonary Strateges (ES to mprove effcency and robustness. The Stochastc GAs have been successfully appled to Integrated Flght Propulson Controller desgns [] and ar combat tactcs optmzaton []. As they explaned, the Stochastc GAs brdge the gap between ES and GAs to handle large desgn problems. Adaptve Range Genetc Algorthms (ARGAs are a new approach usng dynamc codng proposed by Arakawa and Hagwara [3] for bnary-coded GAs to treat contnuous desgn space. The essence of ther dea s to adapt the populaton toward promsng desgn regons durng the optmzaton process, whch enables effcent and robust search n good precson whle keepng the strng length small. Moreover, ARGAs elmnate pror defnton of boundares of the search regons snce ARGAs dstrbute desgn canddates accordng to the normal dstrbutons of the desgn varables n the present populaton. In [4], ARGAs have been appled to pressure vessel desgns and outperformed other optmzaton algorthms. The objectve of ths chapter s to develop robust and effcent GAs applcable to aerodynamc shape desgns. To acheve ths goal, the dea of the dynamc codng s ncorporated wth the used of the floatng-pont representaton. Snce the deas of the Stochastc GAs and the use of the floatngpont representaton are ncompatble, ARGAs for the floatng-pont representaton are developed. The real-coded ARGAs are expected to possess both advantages of the bnary-coded ARGAs and the 9

2 floatng-pont representaton to overcome the problems of havng a large search space that requres contnuous samplng. Frst, to dsplay advantages of the present approach, the proposed approach s appled to a test functon optmzaton problem. Then, an aerodynamc arfol shape optmzaton s demonstrated to ensure the feasblty of the proposed approach n aerodynamc desgn problems... Adaptve Range Genetc Algorthms.. ARGAs for Bnary Representaton When conventonal bnary-coded GAs are appled to real-number optmzaton problems, dscrete values of real desgn varables p are gven by evenly dscretzng pror-defned search regons for each desgn varable [ p,mn, p,max ] accordng to the length of the bnary substrng b,l as where c p = + c = ( b ( p,max p,mn p sl,mn sl l=, l l (. (. In bnary-coded ARGAs, decodng rules for the offsprng are gven by the followng normal dstrbutons: ( µ, σ ( p = πσ N ( µ, σ ( p exp( µ = σ ( p (.3 where the average µ and the standard devaton σ of each desgn varable are determned by the populaton statstcs. Those values are recomputed n every generaton. Then, mappng from a bnary strng nto a real number s gven so that the regon between and n Fg.. s dvded nto equal sze regons accordng to the bnary bt sze as p = µ µ + σ σ ln( + ln( ( ( c sl sl c sl for for c c sl sl (.4 where and are addtonal system parameters defned n [0,]. In the ARGAs, genes of desgn canddates represent relatve locatons n the updated range of the desgn space. Therefore, the offsprng are supposed to represent lkely a range of an optmal value of desgn varables. 0

3 (0 (00 (00 (0 (00 (0 (000 ( p (B=(000 p (B=( p Fg. Decodng for bnary-coded ARGAs Although the orgnal ARGAs have been successfully appled to real parameter optmzatons, there s stll room for mprovements. The frst one s how to select the system parameters and on whch robustness and effcency of ARGAs largely depend. The second one s the use of constant ntervals even near the center of the normal dstrbutons. Thrd one s that snce genes represent relatve locatons, the offsprngs become constantly away from the centers of the normal dstrbutons when the dstrbutons are updated. Therefore, the actual populaton statstcs does not concde wth the updated populaton statstcs... ARGAs for Floatng-Pont Representaton In real-coded GAs, real values of desgn varable are drectly coded as a real strng r : where p = p r r p, mn,max Or, sometmes normalzed values of the desgn varables are used as: p = ( p, max p,mn r + p,mn (.6 where 0 r To employ the floatng-pont representaton for ARGAs, the real-valued desgn varable p s rewrtten here by a real number r defned n [0,] so that ntegral of the probablty dstrbuton of the normal dstrbuton from to pn s equal to r as: p = σ pn + µ pn r = N( 0,( z dz (.8 where the average µ and the standard devaton σ of each desgn varable are calculated by the top half of the present populaton. Schematc vew of ths codng s llustrated n Fg... It should be noted that the real-coded ARGAs resolve drawbacks of the orgnal ARGAs; no need for selectng and as well as arbtrary resoluton near the average. To prevent nconsstency between the actual and updated populaton statstcs, the present ARGAs update µ and σ every N generatons and then the populaton s rentalzed. Flowchart of the resultng ARGA s shown n Fg. (.5 (.7

4 .3. To mprove robustness of the present ARGAs further, relaxaton factors ω µ and ntroduced to update the average and standard devaton as µ µ new = µ present + ω ( µ samplng µ present new = σ present + ω ( σsamplng σ present σ σ (.9 ω σ are (.0 where ω lower than contrbutes to mprove robustness of the ARGAs. µ samplng and σ samplng are determned by samplng the top half of the present populaton. r pn (r pn Fg.. Decodng for real-coded ARGAs Evaluaton Intal populaton Optmum Every N generatons Selecton Samplng for range adaptaton Range adaptaton Reproducton by crossover+mutaton Reproducton by random dstrbuton Fg..3 Flowchart of the present ARGA

5 .3. Results In ths secton, a real-coded ARGA s appled to two test problems: ( a test functon optmzaton problem, ( an aerodynamc arfol shape optmzaton. In both calculatons, the present ARGA uses non-overlappng system coupled wth eltst strategy where the best and the second best ndvduals are coped nto the next generaton, parental selecton by SUS coupled wth rankng usng Mchalewcz s nonlnear functon, one-pont crossover, unform mutaton that takes place at a probablty of 0. and then adds a random dsturbances to the desgn varable n the amount up to 0% of the doman. User-defned parameters of the real-coded ARGA, ω µ, ω σ and N, are set to, 0.5 and 4, respectvely. These parameters are determned by a parametrc study usng a smple test functon not shown here. Unbased ntal populaton s generated by randomly spreadng solutons over the entre desgn space n consderaton..3. Test Functon Mnmzaton Pror to attack the aerodynamc shape optmzaton, the real-coded ARGAs were appled to a test functon optmzaton problem. The test functon n [5] s a dynamc control as subject to N N + k= 0 k mn( x ( x + u k (. x = x + u k+ k k, k = 0,,..., N, where, x k s a state, and u r = ( u0,..., un s the control vector. The search doman s [-00,00] for each u k. The ntal state x 0 and the sze of the control vector are gven by 00 and 45, respectvely. Here, 500 generatons were allowed wth a populaton sze of 00. Fve trals were run for each GA changng seeds for random numbers to gve dfferent ntal populatons. Fgure.4 compares the optmzaton hstores for both GAs. The bold lne ndcates the analytcally obtaned optmum value. Whle all the trals usng the conventonal GA lead to premature convergence, the ARGA fnd the global optmum at every tral. 0 6 Conventonal GA ARGA FITNESS 0 5 optmum value GENERATION 3

6 Fg..4 Optmzaton hstores for the test functon optmzaton problem.3. Aerodynamc Arfol Shape Optmzaton To demonstrate performance of the real-coded ARGAs n comparson wth the conventonal real-coded GAs, aerodynamc arfol shape optmzatons were carred out. The objectve functon was the lft-to-drag rato to be maxmzed where the free stream Mach number and the angle of attack were set to 0.8 and degrees, respectvely. The arfol thckness was constraned so that the maxmum thckness was greater than % of the chord length. The aerodynamc performance of each desgn was evaluated by the Naver-Stokes solver descrbed n secton 4.4. PARSEC arfol (see subsecton was used wth the tralng-edge thckness and ts ordnate (at X= frozen to 0. Fgure.5 compares optmzaton hstores of three trals usng the real-coded ARGA and the conventonal real-coded GA. Both of the populaton sze and the number of generatons were 00. The present ARGA outperformed the conventonal GA startng from all ntal populatons. FITNESS (TO BE MAXIMIZED Conventonal GA ARGA GENERATION Fg..5 Optmzaton hstores for the aerodynamc arfol shape optmzaton Fgure.6 compares the best arfol shapes desgned by the conventonal GA and ARGA and the correspondng pressure coeffcent dstrbutons. The surface pressure dstrbuton of the desgn optmzed by ARGA s almost dentcal to that of NASA supercrtcal arfols. Ths ndcates the feasblty of the present ARGA n aerodynamc desgns. 4

7 Z/C conventonal GA GA - ARGA X/C Fg..6 Desgned arfol shape and the correspondng pressure dstrbuton.4. Summary To develop robust and effcent EAs applcable to aerodynamc shape desgns, the real-coded ARGAs have been developed by ncorporatng the dea of the bnary-coded ARGAs wth the use of the floatng-pont representaton. The resultng real-coded ARGAs are expected to possess both advantages of the bnary-coded ARGAs and the use of the floatng-pont representaton to overcome the problems of havng a large search space that requres contnuous samplng. Frst, the effcency and the robustness of the proposed approach have been demonstrated by usng a typcal test functon. Then the proposed approach has been appled to an aerodynamc arfol shape optmzaton problem. The results confrm that the real-coded ARGAs consstently fnd better solutons than the conventonal real-coded GAs do. The desgn result s consdered to be the global optmal and thus ensures the feasblty of the real-coded ARGAs n aerodynamc desgns Cp 5

8 References [] Krshnakumar, K., Swamnathan, R., Garg, S. and Narayanaswamy, S., Solvng Large Parameter Optmzaton Problems Usng Genetc Algorthms, Proceedngs of the Gudance, Navgaton, and Control Conference, Baltmore, MD, 995. [] Mulgund, S., Harper, K., Krshnakumar, K. and Zacharas. G., Ar Combat Tactcs Optmzaton Usng Stochastc Genetc Algorthms, Proceedngs of 998 IEEE Internatonal Conference on Systems, Man, and Cybernetcs, San Dego, CA, Oct. 998, pp [3] Arakawa, M. and Hagwara, I., Development of Adaptve Real Range (ARRange Genetc Algorthms, JSME Internatonal Journal, Seres C, Vol. 4, No. 4, 998, pp [4] Arakawa, M. and Hagwara, H., Nonlnear Integer, Dscrete and Contnuous Optmzaton Usng Adaptve Range Genetc Algorthms, n CD-ROM Proceedngs of the 34th Desgn Automaton Conference, Anahem, Calforna, Jun.997. [5] Jankow, C. Z. and Mchalewcz, Z., An Expermental Comparson of Bnary and Floatng Pont Representatons n Genetc Algorthms, Proceedngs of the Fourth Internatonal Conference on Genetc Algorthms, Morgan Kaufmann Publshers, Inc., San Mateo, CA, 99, pp