An efficient constraint handling methodology for multi-objective evolutionary algorithms

Transcription

1 Rev. Fac. Ing. Unv. Antoqua N. 49. pp Septembre, 009 An effcent constrant handlng methodology for mult-objectve evolutonary algorthms Una metodología efcente para manejo de restrccones en algortmos evolutvos multobjetvo Maurco Granada Echeverr 1, * Jesús María López Lezama *, Ruben Romero 3 1 Departamento de Ingenería Eléctrca, Unversdad Tecnológca de Perera, Vereda la Julta, Perera, Rsaralda, Colomba Grupo Gmel. Facultad de Ingenería, Unversdad de Antoqua, Calle 67 N , Medellín, Colomba 3 Departmento de Ingenería Eléctrca, Fes-Unesp-Ilha Soltera-Brasl. Avenda Brasl, 56 Centro, , Ilha Soltera SP, Brasl (Recbdo el 9 de octubre de 008. Aceptado el 6 de mayo de 009) Abstract Ths paper presents a new approach for solvng constrant optmzaton problems (COP) based on the phlosophy of lexcographcal goal programmng. A two-phase methodology for solvng COP usng a multobjectve strategy s used. In the frst phase, the objectve functon s completely dsregarded and the entre search effort s drected towards fndng a sngle feasble soluton. In the second phase, the problem s treated as a b-objectve optmzaton problem, turnng the constrant optmzaton nto a two-objectve optmzaton. The two resultng objectves are the orgnal objectve functon and the constrant volaton degree. In the frst phase a methodology based on progressve hardenng of soft constrants s proposed n order to fnd feasble solutons. The performance of the proposed methodology was tested on 11 well-known benchmark functons Keywords: Evolutonary algorthms, mult-objectve algorthms, constrant optmzaton. Resumen Este artículo presenta un nuevo enfoque para resolver problemas de optmzacón restrctos (POR) basado en la flosofía de programacón lexcografta de objetvos. En este caso se utlza una metodología de dos * Autor de correspondenca: teléfono: , fax: , correo electrónco: lezama@udea.edu.co (J. López). 141

2 Rev. Fac. Ing. Unv. Antoqua N. 49. Septembre 009 fases usando una estratega mult-objetvo. En la prmera fase se concentra el esfuerzo en encontrar por lo menos una solucón factble, descartando completamente la funcón objetvo. En la segunda fase se aborda el problema como b-objetvo, convrtendo el problema de optmzacón restrcta a un problema de optmzacón rrestrcto de dos objetvos. Los dos objetvos resultantes son la funcón objetvo orgnal y el grado de volacón de las restrccones. En la prmera fase se propone una metodología basada en el endurecmento progresvo de restrccones blandas para encontrar solucones factbles. El desempeño de la metodología propuesta es valdado a través de 11 casos de prueba bastante conocdos en la lteratura especalzada Palabras clave: Algortmos evolutvos, algortmos mult-objetvo, optmzacón restrcta. Introducton Evolutonary algorthms (EA) have been wdely used n the soluton of optmzaton problems. These technques, compared wth the tradtonal nonlnear programmng methods, handle a smaller amount of nformaton (gradents, and Hessans, among others), are of easy mplementaton, and consttute useful tools for global search. Addtonally, they have a smaller probablty of convergng to a local optmal soluton, and are able to obtan good qualty results n problems of great sze [1]. Many researchers have developed a great amount of EA to solve constrant optmzaton problems (COP). The dfferent methodologes found n the lterature to handle wth COP can be classfed n four man groups: 1) methods based on penalty functons, ) methods based on the preference of feasble solutons nstead of the non feasble ones, 3) hybrd methods, and 4) methods based on mult-objectve optmzaton. Ths last group s currently of great scentfc nterest and becomes the state-of-the-art of the constrant optmzaton algorthms. A detal descrpton of these methodologes s out of the scope of ths paper. For a more n dept readng, the nterested reader s referred to [, 3] and [4]. Most of the real world problems nvolve equalty and nequalty constrants. The general problem formulaton wth contnuous parameters and constrants s defned n [5] as shown n (1): (1) The objectve functon f s defned on the n search space S R, and the set F S defnes the feasble regon. The feasble regon F s restrcted by a set of m constrants ( m 0 ) wth q nequalty constrants g j ( X ), and m q equalty constrants hj ( X ). The applcaton of mult-objectve evolutonary algorthms (MOEA) has addtonal advantages compared to other optmzaton methods, especally when solvng COP. Some of these advantages are [6]: Constrant problems can be handled n a natural fashon. That s, t s not necessary to formulate artfcally penalzed objectve functons, and addtonally penalty parameters are not needed. These parameters ntroduce a subjectve component to the problem soluton. In real world problems, t s unusual to fnd rgd or hard constrants. Therefore, a constrant volaton margn (soft constrants) s permtted as long as an mportant mprovement n the objectve functon s obtaned. 14

3 An effcent constrant handlng methodology for mult-objectve evolutonary algorthms MOEA allow obtanng a set of solutons, denomnated Pareto-Optmal-Front (POF), wth the best commtments between the objectve functons nvolved n the problem. Thus, t s possble to fnd solutons that volate constrants margnally. Fgure 1 shows a non-domnated set of solutons n the f v space, where f s the orgnal objectve functon value, and v s the constrant volaton ndex. In ths scheme, one objectve s the volaton degree of constrants and the other s the orgnal objectve functon value. The mnmum feasble soluton (pont A), the mnmum soluton consderng soft constrants wthn a volaton margn ε (pont B), and the orgnal feasble solutons of the sngleobjectve problem are also shown n Fgure 1. All the solutons of the POF that are between the ponts A and B are of great nterest. The phlosophy of the proposed methodology s nspred by the goal programmng methods [7], where the man dea s to fnd solutons that reach a reference (predefned objectve) for one or more objectve functons. If these solutons do not exst, the task wll be to fnd solutons where the dfference wth the reference s mnmum. On the other hand, f there s a soluton wth the same value as the reference objectve functon, the task wll be to dentfy ths soluton. The lexcographcal method s among the goal programmng methods. In ths case the dfferent goals are categorzed wthn many levels of prorty. In Ths way, the problem s frst solved consderng only one goal wth the correspondng constrants of the frst prorty level. If there are multple solutons n the prevous step, another goal programmng problem s formulated consderng the second level of prorty. The goals of the frst level of prorty are used as equalty constrants to assure that the second problem soluton does not volate the frst level constrants. The procedure s repeated sequentally for other prorty levels. Fgure llustrates the operaton prncple of the lexcographcal goal programmng method for a mnmzaton problem wth two objectve functons f 1 and f. If t s consdered that f s 1 more mportant that f, the procedure conssts of frst mnmzng the problem only consderng f 1 and gnorng f. In ths way, the set of solutons s represented by the segments AB and CD for the frst prorty level. The soluton of the second prorty level wll be the one that mnmzes f along the segments AB and CD. In ths case, the soluton of the second prorty level wll be D whch s the global soluton of the problem. If f s more mportant than f 1 the problem soluton changes to pont E. Fgure 1 Search space of a two-objectve problem wth hard and soft constrants Fgure Lexcographcal goal programmng method 143

4 Rev. Fac. Ing. Unv. Antoqua N. 49. Septembre 009 Proposed methodology for COP The proposed methodology for COP s based on the phlosophy of lexcographcal goal programmng. It conssts of turnng a COP nto a b-objectve problem, where one objectve functon f X and the other, s the s the orgnal one ( ) constraned volaton degree ( ) v X. In other words, one objectve functon consders optmalty and the other one consders feasblty. The algorthm s composed by two phases. In the frst stage, the orgnal objectve functon s completely dscarded and the optmzaton problem s concentrated on mnmzng the constraned volaton degree of the solutons. Thus, the algorthm mght fnd a feasble soluton because the search s concentrated only on the mnmzaton of the constraned volaton degree. The second phase conssts of optmzng smultaneously the orgnal objectve functon and the constraned volaton degree usng a multobjectve strategy. Phase I: Constrant enforcement algorthm In ths phase, the objectve functon s completely dscarded and all the algorthm effort s drected towards fndng at least one feasble soluton. For each alternatve of the populaton, a ftness functon, accordng to v ( X ), s assgned. Then, an eltst strategy s used to assure that the soluton wth smaller v( X ) s ncluded n the followng generaton. Ths phase allows obtanng a soluton that satsfes all the constrants (usable soluton n the real world). Ths technque s approprate to solve hghly constraned problems, where fndng a feasble soluton can be dffcult. In order to fnd the constraned volaton degree of an alternatve X n the constrant j, the frst step of the proposed strategy conssts of turnng the equalty constrants nto soft constrants usng a tolerance δ. Thus, the constrant volaton degree of the alternatve wll be gven by () () Where denotes the absolute value. In order to gve the same mportance degree to all constrants, each alternatve volated must be normalzed dvdng t by the greatest volaton value of the populaton. In ths case the greatest volaton value for each constrant j s calculated usng (3). (3) The maxmum volaton value for each constrant n the whole populaton s used to normalze each volated constrant calculated n (). Fnally, to produce a scalar number that represents the constrant volaton degree for each alternatve of the populaton (n a range between 0 and 1), the normalzed values are added and then dvded by the total number of m constrants as shown n (4). (4) Obtanng the ftness functon for phase one In order to llustrate the calculus of the ftness functon, the constrant optmzaton problem presented n [8] and defned by the set of equatons (5) s consdered. st (5) Where r1, r, K, r7 are the problem constrants. Table 1 shows a populaton of 5 alternatves randomly generated. The small letters (x 1, x,...,x 5 ) stand for the varables and the captal 144

5 An effcent constrant handlng methodology for mult-objectve evolutonary algorthms letters (X 1, X,...,X 5 ) stand for the soluton alternatves. Durng the generaton of the populaton t s guaranteed that constrants r 6 and r 7 are satsfed (lmts of the decson varables). Thus, the problem s only lmted by the frst 5 constrants ( r1, r, K, r5 ). Evaluatng each alternatve of the populaton for each of the constrants n problem (5) and dscardng the objectve functon completely, the data regstered n table are obtaned. Applyng () and assumng a tolerance δ = the data presented n table 3 are obtaned, and the term cmax ( j ) s calculated usng (3). Table 1 Randomly generated populaton x 1 x x 3 x 4 X X X X X Table Volaton values for each alternatve and each constrant r 1 r r 3 r 4 r 5 X X X X X Then, when a new alternatve s generated, a comparson between the constrant volatons for each alternatve, and the maxmum volatons calculated n (3) allows keepng the values of the vector c max ( j ) updated. It s advsable to generate an addtonal vector max ( j) contanng the ndex of the alternatves that produce each cmax ( j). Thus, for example, the maxmum volaton of constrant 1 s caused by ndvdual 5 ( X ) as 5 shown table 3. Table 3 Constrant volatons consderng δ = C 1 C C 3 C 4 C 5 X X X X X C max (j) I max (j) Fnally, applyng expresson (4) to the data shown n table 3, a scalar vector v( X ) that quantfes the nfeasble degree of each ndvdual of the populaton s obtaned as shown n (6). (6) The vector v( X ) corresponds to the ftness functon of phase one, whch wll be used n the selecton process. For the feasble soluton search, a tradtonal genetc algorthm (GA) wth real codfcaton s used ncorporatng progressve hardenng of soft constrants. Progressve hardenng of soft constrants (PHSC) Phase one Fgure 1 shows the soft constrants handled through a volaton margn ε. The technque used to fnd the feasble solutons conssts of consderng an nterval of the volaton margn ( ε mn ε ε max ). In ths way, the ntal objectve of the GA s to mnmze the parameter v( X ) of each alternatve, calculated wth expresson (4) consderng ε max. The algorthm s ntally run wth a hgh volaton margn, and therefore, the GA reaches ts objectve wth a low computatonal effort. Next, the volaton margn s reduced every tme the GA reaches a partal objectve, untl a 145

6 Rev. Fac. Ing. Unv. Antoqua N. 49. Septembre 009 constrant volaton margn smaller or equal to ε mn s fnally reached. In ths pont, the GA has found a feasble soluton. The expresson ε ( 1 ) = τ ε s used as a reducton strategy of the volaton margn. ε mn corresponds to the tolerance used to evaluate the equalty constrants fulfllment (a typcal value s εmn = δ = ). ε max s a bat value that allows the GA to easly fnd a populaton wth a reasonable nfeasblty margn. From ths populaton, the optmzaton process gudes the search towards feasble regons of better qualty untl fndng a feasble soluton wth the desred precson degree. Fgure 3 shows the search process of feasble solutons for problem (5), startng wth a random populaton wth ε max = 0.4 and τ = 0.5. The whte crcles correspond to the ntal populaton, the astersks ndcate the evoluton of the populaton after several generatons, and the vertcal dashed lnes ndcate the current volaton margn. Fgure 3 Search process of feasble solutons. Evoluton of the populaton alternatves for dfferent ε Phase II: Optmzaton algorthm for constrant problems Phase II s actvated when at least one feasble soluton has been found by phase one. In phase one the ftness functon corresponds to the constrant volaton degree, and the evoluton of the alternatves consders the qualty of each nondomnated set to whch each alternatve belongs. In phase II the constrant volaton and the orgnal objectve functon must be mnmzed smultaneously wthn a modfed objectve space as shown n fgure 1 (space f-v). The feasble alternatve wth the best objectve functon wll be the current ncumbent of the space search. A GA and an eltst operator based on a nondomnated sortng (NSGA-II [9]) are used n ths paper for solvng the b-objectve problem. In addton, to preserve dversty n the alternatves belongng to the non-domnated solutons set, a nches scheme s used, takng nto consderaton the normalzed Eucldean dstance between two objectve vectors. Ths dstance s known as crowdng dstance metrc. A detaled descrpton of the calculus of ths dstance s presented n [7], and [10]. The mult-objectve theory ntroduces the concept of domnance, whch defnes that a soluton X domnates another soluton 1 X f both condtons 1) and ) are true: 1) The soluton X s not worse than 1 X n all objectves. ) The soluton X s strctly better than 1 X n at least one objectve. If any of the above condtons s volated, the soluton X does not domnate the soluton 1 X. It s possble to apply ths defnton n an teratve way to any set of solutons belongng to a multobjectve optmzaton problem to establsh the domnant and non-domnant sets of alternatves. The set of domnant solutons through all the objectve space s called the Pareto-optmal front. Therefore, the GA (or any other evolutonary approach) ams to move the current front n each teraton towards regons of better qualty. Fgure 4 shows the optmal front evoluton for problem (5) usng NSGA-II approach. There are a bunch of dfferent strateges that can be used to ntensfy the exploraton n the target search regon. Some of these strateges as reported n the specalzed lterature can be: the guded domnaton approach, domnance prncple 146

7 An effcent constrant handlng methodology for mult-objectve evolutonary algorthms by weghts, and n general, modfcatons on the crowng dstance metrc. In ths paper the guded domnaton approach descrbed n [7] was mplemented. Thus, a dfferent domnance concept for mnmzaton problems s formulated. A weghted functon of the objectves s defned as shown n (7) by any soluton than the one allowed by the tradtonal defnton. Besdes, by choosng approprate values of the coeffcents a 1 and a 1, a secton of the Paretooptmal regon can be emphaszed (see Fgure 5). In ths paper, n order to ntensfy the exploraton of an nterestng regon of the Pareto-optmal front (as shown n Fgure 4) the followng coeffcents were used: a 1 = 0 and a 1 = Fgure 4 Pareto-Optmal-Front and target search regon Where a j represents the mprovement n the j-th objectve functon for a one-unt loss n the -th objectve functon. Then, the new domnance concept s: A soluton X 1 domnates another soluton X f Ω ( f ( x1 )) Ω ( f ( x) ) for all =1,,,M and the strct nequalty s satsfed at least for one objectve. In ths problem, there are two objectve functons (M=). The two weghted functons are shown n (8) and (9). (, ) Ω f f = a f + f (8) (9) Thus, the modfed defnton of domnance allows a larger regon to become domnated Fgure 5 The non-domnated porton of the Paretooptmal regon Genetc Algorthm The mult-objectve technque (NSGA-II) requres the ncorporaton of a GA that mproves the Pareto-optmal front qualty durng the teratve process. A GA wth the followng characterstcs s used: Real codfcaton: bnary chan codfcaton s not used, whch mples a modfcaton n the recombnaton and mutaton operators. Lnear crossover: the mplemented crossover operator creates three solutons (offsprng) n 1,t each generaton t from two parent solutons X and,t X, as shown n expressons (10), (11) and (1). 1 X + 1 X (10) 1, t, t 3 X 1 X (11) 1, t, t 1 X + 3 X (1) 1, t, t 147

8 Rev. Fac. Ing. Unv. Antoqua N. 49. Septembre 009 Out of these three solutons, one s elmnated by tournament. Random mutaton: the mutaton scheme conssts ( 1, 1) of creatng a random alternatve Y t+ consderng ( 1, ) ( ) ( ) all search space t + 1 Y ( U L = r X X ). Where r s a random number between [0,1] and superscrpts U and L ndcate the superor and nferor search space lmt, respectvely. Test cases and results The mult-objectve NSGA-II method and the PHSC approach proposed n ths paper were appled to 11 test cases reported n [8] and [11]. In Table 4 a summary of the 11 test cases s presented. LI, NE, and NI represent the number of lnear nequaltes, nonlnear equatons and nonlnear nequaltes, respectvely, n s the number of decson varables nvolved and a s the actve constrant. The mutaton rate, for all cases, s 1% and the recombnaton rate s 90%. The populaton sze for all cases s 15 ndvduals. The maxmum generaton number s 5000 and 30 runs for each case were executed. For cases G and G3, k=30 was used. For all casesε max = 0.4, andε mn = The obtaned results are shown n table 5. It can be observed that for most of the cases, the proposed methodology reaches values of the objectve functon equal to those reported n the specalzed lterature. Partcularly, for case G5 (represented n ths paper by the set of equatons (5)) there were found three alternatves, all of them wth objectve values better than the best value reported n the lterature. Table 6 presents detal nformaton of the varables and constrant values for the best alternatves found by the proposed approach for problem G5. In general terms, the use of PHSC showed a better performance of the algorthm n phase I. Partcularly, the G5 case s hghly constraned and to fnd a feasble soluton s a dffcult task. Venkatraman reports, for the G5 case, an average number of generatons of to fnd the frst feasble soluton on 50 runs wth ε = Usng PHSC an average of generatons s obtaned on 30 runs wth ε = The use of a smaller tolerance allows obtanng a greater number of nondomnated solutons n the target search regon. Another strongly constrant case s G10, for ths case the average number of generatons reported by Venkatraman s However, the average number obtaned applyng PHSC was Table 4 Summary of eleven test cases. (for G and G3 t s assumed k=30) Test case n Type of functon LI NE NI a G1 13 quadratc G k nonlnear G3 k polynomal G4 5 quadratc G5 4 cubc G6 cubc 0 0 G7 10 quadratc G8 nonlnear G9 7 polynomal G10 8 lnear G11 quadratc

9 An effcent constrant handlng methodology for mult-objectve evolutonary algorthms The algorthm performance n phase II s smlar to the one reported by Venkatraman n [11]. Nevertheless, for the G5 case the proposed method was able to fnd, n 30 runs, 3 solutons wth a better objectve functon than the one reported n [11]. These solutons belong to the domnated front and have an acceptable constrant volaton degree (see table 6). Comparng the best alternatve reported wth the 3 alternatves found, t can be notced that the alternatves 1 and 3 exactly satsfy constrants R3=0 and R4=0. All alternatves satsfy the nequalty constrants R1 and R. Table 5 Comparson of best results. ANG= average number of generatons when the frst feasble soluton s found. Functon Best known soluton Venkatraman and Yen [1] Kozel and Mchalewcz [] Proposed methodology ANG Venkatraman and Yen ANG Proposed methodology Mn. G Max. G Max. G Mn. G Mn. G NR Mn. G Mn. G Max. G Mn. G Mn. G Mn. G Table 6 Varables and constrant values for the best alternatves of problem G Objectve functon value Varables Constrants Alternatve 1 Alternatve Alternatve 3 Best alternatve reported x x x x R R R R R

10 Rev. Fac. Ing. Unv. Antoqua N. 49. Septembre 009 Conclusons In ths paper a new methodology to deal wth constrant optmzaton problems was presented. The man contrbuton of the proposed methodology conssts of an effcent constrant handlng approach usng progressve hardenng of soft constrants along wth an ntensve exploraton of a target search regon of the Pareto-optmal front. The mult-objectve NSGA-II method along wth the proposed methodology was mplemented on 11 test cases wdely studed n the specalzed lterature. Results showed that the proposed methodology s compettve wth the stateof-the-art constrant optmzaton algorthms. In partcular, for test case G5, three dfferent alternatves better than the one reported n the lterature were found. For the other test cases the algorthm found the best soluton already reported. However, n some cases, a consderable reducton of the number of generatons was acheved. Future work wll consder other recombnaton and mutaton strateges usng real codfcaton, such as blend crossover, smulated bnary crossover, smplex crossover, non-unform mutaton and polynomal mutaton, among others. The use of algorthms that ncorporate the lateral dversty concept, such as NSGAIIcontrolled, allows a search of greater qualty n the target search regon and can be mplemented wth the purpose of mprovng some results. Ths phlosophy can be appled to hghly constraned problems. References 1. D. Powell, M. Skolnck. Usng genetc algorthms n engneerng desgn optmzaton wth nonlnear constrants. Proceedngs of the 5 th Internatonal conference on Genetc Algorthms. Urbana- Champagn pp J. Km, H. Myung. Evolutonary programmng technques for constrant optmzaton problems. IEEE Transactons on Evolutonary Computaton. Vol pp K. Deb. An effcent constrant handlng method for genetc algorthms. Computatonal Methods Appled on Mechancal Engneerng. Vol pp A. Kurr, J. Guterrez. Penalty functon methods for constraned optmzaton wth genetc algorthms: A statstcal analyss. Proceedngs of the nd Mexcan nternatonal conference on artfcal ntellgence. Mérda Mx. 00. pp Z. Mchalewcz, M. Schoemauer. Evolutonary algorthms for constraned parameter optmzaton problems. Evolutonary Computaton. Vol pp C. Zxng, W. Yong. A mult-objectve optmzatonbased evolutonary algorthm for constraned optmzaton. IEEE Trans. Evolutonary Computaton. Vol pp K. Deb. Mult-Objectve Optmzaton usng Evolutonary Algorthms. Department of mechancal Engneerng. Indan Insttute of technology, Kanpur, Inda. ª ed. John Wley and Sons. New York pp S. Kozel, Z. Mchalewcz. Evolutonary algorthms, homomorphous mappngs and constraned parameter optmzaton. Evolutonary Computaton. Vol pp K. Deb, A. Pratap, S. Agarwal, T. Meyarvan. A fast and eltst mult-objectve genetc algorthm: NSGA- II. IEEE Transactons on Evolutonary Computaton. Vol pp C. A. Peñuela, M. Granada. Optmzacón multobjetvo usando un algortmo genétco y un operador eltsta basado en un ordenamento nodomnado (NSGA-II). Revsta Scenta et Technca. Vol pp S. Venkatraman, G. Yen. A generc framework for constraned optmzaton usng genetc algorthms. IEEE Transactons on Evolutonary Computaton. Vol pp