An Evolutionary Algorithm for Constrained Optimization

Transcription

1 An Evolutionary Algorim or Constrained Optimization Tapabrata Ray, Tai Kang and Seow Kian Chye Center or Advaned Numerial Engineering Simulations Shool o ehanial and Prodution Engineering Nanyang Tehnologial University Singapore mtray@ntu.edu.sg, mktai@ntu.edu.sg, mkseow@ntu.edu.sg Tel : Abstrat In is paper we present an evolutionary algorim or onstrained optimization. The algorim is based on nondominane o solutions separately in e obetive and onstraint spae and uses eetive mating strategies to improve solutions at are weak in eier. Sine e meodology is based on nondominane, saling and aggregation aeting onventional penalty untion meods or onstraint handling does not arise. The algorim inorporates intelligent seletion or ooperative mating. The diversiiation strategy is based on nihing at result in a wide spread o solutions in e parametri spae. Preliminary results o e algorim or onstrained single and multiobetive test problems are presented and ompared to illustrate e eiieny o e algorim in solving onstrained optimization problems. INTRODUCTION Evolutionary omputation EC meods have reeived onsiderable attention over e years as optimization meods or omple untions. EC meods are essentially unonstrained searh tehniques at require a salar measure o quality or itness. The presene o onstraints signiiantly aets e perormane o an optimization algorim, inluding evolutionary searh meods. There have been a number o approahes to handle onstraints inluding reetion o ineasible solutions, penalty untions and eir variants, repair meods, use o deoders, separate treatment o onstraints and obetives and hybrid meods inorporating knowledge o onstraint satisation. A omprehensive review on onstraint handling meods is provided by ihalewiz []. All e meods have limited suess as ey are problem dependent and require a number o additional inputs. Penalty untions using stat dynami or adaptive onepts have been developed over e years. All o em still suer rom ommon problems o aggregation and saling. Repair meods are based on additional untion evaluations, while e deoders and speial operators or onstraint satisation meods are problem speii and annot be used to model a generi onstraint. Separate treatment o onstraints and obetives is an interesting onept at eliminates e problem o saling and aggregation. Constraint handling using a pareto ranking sheme is a relatively new onept having its origin in multiobetive optimization. Fonsea and Flemming [] proposed a pareto ranking sheme to handle multiple obetives. Jimenez and Verdegay [] used a nondominated sorting geneti algorim NSGA ranking sheme to deal wi multiple obetives while a separate evaluation untion was used or ineasible solutions. The NSGA used by Jimenez and Verdegay [] was introdued by Srinivas and Deb [4]. Surry et al. [] applied a pareto ranking sheme among onstraints while itness was used in e obetive untion spae or e optimization o gas supply networks. Fonsea and Flemming [6] proposed a uniied ormulation to handle multiple onstraints and obetives based on pareto ranking sheme. All e above attempts suessully eliminate e drawbaks o aggregation and saling at eist wi e penalty untion meods. In addition, ey do not require any additional input and are problem independent. However, none o e above meods inorporate onepts o ooperative learning rough parent mathing whih is epeted to improve e eiieny o e algorim. An interesting attempt to inorporate e knowledge o onstraint satisation during mating was proposed by Hinterding and ihalewiz [7]. In an attempt to math e beauty wi e brains, onstraint mathing was employed during seletion. A single measure sum o squares o violation was used to ompute a solution s ineasibility. The algorim does not inlude any nihing or diversiiation mehanism to ensure a uniorm spread o points along e pareto rontier or multiobetive

2 problems. oreover, a single aggregate measure o ineasibility ails to inorporate e knowledge o individual onstraint satisation/violation and in addition leads to salability and aggregation problems. The proposed evolutionary algorim eliminates all e drawbaks as disussed above. Solutions are ranked separately based on obetives, onstraints and ombined matries desribed in Setion. This proess eliminates e problems o saling and aggregation. oreover, sine e onstraints are handled separately, e true obetive untion is optimized raer an some transormed evaluation untion. The rank o a solution in e obetive, onstraint or e ombined matri is used or intelligent mating between solutions to improve ose at are weak in eier onstraint satisation or obetive perormane. The mating proess wiin e proposed evolutionary algorim inorporates e knowledge o every individual onstraint satisation/violation and obetive perormane. Strategies to handle highly onstrained and moderately onstrained problems are outlined. Setion provides a detailed desription o e algorim. Four eamples omprising o ree onstrained single obetive and one multiobetive onstrained optimization problem are presented to illustrate e perormane o e algorim, identiying a aster onvergene rough ooperative learning. PROPOSED AGORITH The proposed evolutionary algorim is desribed in e ontet o multiobetive optimization. A single obetive problem is handled in e same ormulation by assigning k=. A general onstrained multiobetive optimization problem in minimization sense is presented as: inimize = [ K ] subet to g k i ai, i =,, K, q h = b, =,, K, r where is a vetor o k obetives to be minimized subet to q inequality and r equality onstraints. = K ] is e vetor o n design variables. [ n The OBJECTIVE matri or a population o solutions assumes e orm OBJECTIVE = It is ommon pratie to transorm e equality onstraints wi a tolerane δ to a set o inequalities and use a uniied ormulation or all onstraints: O k k k h b δ whih is same as h b δ and h b δ Thus r equality onstraints will give rise to r inequalities, and e total number o inequalities or e problem is denoted by s, where s=qr. For eah solution, denotes e onstraint satisation K where vetor given by = [ ] s 0 i i onstraint satisied, i =,,..., s ai gi i i onstraint violated, i =,,..., q i = bi δ hi i i onstraint violated, i = q, q,..., q r bi δ hi i i onstraint violated, i = q r, q r,..., s For e above i s, i = 0 indiates e i onstraint is satisied, whereas i > 0 indiates e violation o e onstraint. The CONSTRAINT matri or a population o solutions assumes e orm s = s CONSTRAINT O s A COBINED matri at is a ombination o obetive and onstraint matri assumes e orm COBINED=. PARETO RANKING From a population o solutions, all nondominated solutions are assigned a rank o. The rank individuals are removed rom e population and e new set o nondominated solutions is assigned a rank o. The proess is ontinued until every solution in e population is assigned a rank. Rank= in any o e obetive, onstraint or ombined matries indiate at e solution is nondominated. The pareto rank o eah solution in e population is omputed individually in e OBJECTIVE, CONSTRAINT and COBINED matri and are stored in vetors RankOb, RankCon and RankCom respetively. Having desribed e general ormulation o e onstraint and e obetive untion matries and e onept o pareto ranking, e pseudo ode o e algorim is introdued. O k k k O s s s

3 Algorim Initialize solutions to orm a population Do { Compute Pareto Ranking based on OBJECTIVE matri to yield a vetor RankOb Compute Pareto Ranking based on CONSTRAINT matri to yield a vetor RankCon Compute Pareto Ranking based on COBINED matri to yield a vetor RankCom ultiobetive Optimization: Selet individuals rom e population in is generation i RankCom = & Feasible and put em into e population or e net generation. Single Obetive Optimization: Selet individuals rom e population in is generation i RankCom is better an allowable rank & Feasible and put em into e population or e net generation Allowable rank=maimum rank o an individual in e population/. To generate e remaining members o e population or e net generation Do { Selet an individual A and its rom e population at is generation. ate A wi its. Put parents and hildren into e population or e net generation. } while e population is not ull. Remove dupliate points in parametri spae and shrink population } while e maimum number o generations is not attained.. INITIAIZATION The initialization is based on a random generation o starting solutions using uniorm random number generator and e variable bounds side onstraints. A solution = K ] is generated as ollows: i [ n = - * R upper bound lower bound lower bound where lower bound and upper bound are e lower and upper bounds o e i variable and R is a random number between 0 and.. SEECTION PROBABIITY The probability o seletion o an individual is based on e vetors RankOb, RankCon or RankCom and denoted as ProbOb, ProbCon and ProbCom respetively. As an eample, e vetor ProbOb is omputed as ollows: The vetor RankOb wi element values varying rom to P ma is transormed to a itness vetor FitOb wi elements varying rom P ma to using a linear saling P ma denotes e rank o e worst solution. The probability o seletion ProbOb o an individual is en omputed based on is itness vetor FitOb using e roulette wheel seletion sheme. The proess ensures at solutions at are itter have a higher probability o being seleted..4 CHOOSING A PARTNER FOR ATING A mating is perormed between a solution A and its B or C. The proess o seletion is dependent on e type o e onstrained problem. Problems are lassiied into e ollowing:. Unonstrained problem Obetive- Obetive ating. oderately onstrained problem Obetive- Constraint ating. Highly onstrained problem Constraint- Constraint ating For an unonstrained problem, e seletion o A, B and C is based on ProbOb. For a moderately onstrained problem, seletion o A is based on ProbOb while e seletion o B and C is based on ProbCon. Suh a mating between solutions at are good in obetive untion wi at o solutions at are good in onstraint satisation is analogous to mating between e beauty and e brains. For a highly onstrained problem, seletion o A, B and C is based on ProbCon. Sine inding a easible solution is quite diiult or highly onstrained problems, e seletion o mating s is based on e solution s ability towards onstraint satisation. The proess o seletion or a moderatelyonstrained problem is outlined below or a greater understanding o e seletion proess..4. oderately Constrained: Partner seletion Selet irst individual A based on ProbOb Selet potential mating andidate B based on ProbCon Selet potential mating andidate C based on ProbCon Partner o A is eier B or C, depending upon Condition, or.

4 Condition : I B and C are bo easible I RankOb_B < RankOb_C en : Partner is B else : Partner is C. I RankOb_B = RankOb_C en : Choose e one wi e minimum adaptive nihe ount to be eplained in Setion.6. where, RankOb_B denotes e rank o solution B in e vetor RankOb. Condition : I B and C are bo ineasible I RankCon_B < RankCon_C en : Partner is B else : Partner is C. I RankCon_B = RankCon_C en : Choose e one wi minimum overlapping onstraint satisation wi A to be eplained in Setion.7. Condition : < A is base and Q <0.: New var. = R A lower bound A is base and Q 0.: New var. = R A Partner is base and Q <0.: New var. = R A Partner is base and Q 0.: New var. = R Condition : upper bound > A is base and Q <0.: New var. = R A A is base and Q 0.: New var. = R upper bound A Partner is base and Q <0.: New var. = R lower bound Partner is base and Q 0.: New var. = R A Condition : I one is easible and e oer is not. I B is easible while C is not en : Partner is B else : Partner is C.. ATING Every mating generates additional solutions unlike onventional proess o rossover generating two hildren. Out o e ree solutions, one is generated by uniorm rossover between A and its while e oer two are generated using random mi and move. Every mating will plae parents and additional solutions to e population or e net generation. The proess o random mi and move is as ollows: For i=: n Ation : Randomly pik A or its and denote it as base. Ation : Randomly pik a number Q or diretion <0. is negative, positive oerwise Ation : Randomly pik a number R between 0 and. Condition : Q <0.9: New var. = = i, A Q 0.9: New var. = R upper bound lower bound End The proess o random mi and move will ensure at any easible variable value an be generated even i it does not eist in eier A or its. Generation o a large number o initial solutions to maintain all possible variable values is not onsidered avorable as ose solutions are generated wiout any knowledge o e searh proess and adds on to a omputational overhead. The proposed meod as illustrated an be used wi relatively small population size as e proess o generating solutions omes along wi random mi and move..6 ADAPTIVE NICHE COUNT Adaptive nihe ount o a solution is e number o solutions in at population whih are wiin e average distane metri and is omputed as ollows: For i=: Compute e Eulidean distanes between it and all oer - solutions Compute e average Eulidean distane

5 Count e number o solutions at are wiin e average distane End A solution wi a small nihe ount as ompared to anoer physially means at ere are ew solutions in its neighborhood. Suh solutions are preerred over oers and is e diversiiation strategy used in e algorim. hene an obetive-onstraint-mating sheme was employed or e solution. The optimum solution is 78.0,.0, 9.99, 4.0, wi an obetive untion value o Two onstraints upper bound o e irst inequality and e lower bound o e ird inequality are ative at e optimum..7 NON-OVERAPPING CONSTRAINT SATISFACTION The strategy is based on e philosophy at a solution is allowed to mate wi anoer i one omplements e oer towards onstraint satisation. Suh a mating between e beauty and e brains is inorporated wi a hope to generate solutions wi better onstraint satisation. The onept o non-overlapping onstraint satisation is inorporated as ollows: Wi reerene to e CONSTRAINT matri disussed earlier, eah o e solutions A, B and C has an assoiated onstraint satisation vetor A, B and C respetively. The sets {S A },{S B } and {S C } denote e set o onstraints satisied by solution A, B and C respetively. The seletion o eier B or C is based on e ollowing ondition: I {S A } {S B } > {S A } {S C } en : e is C. I {S A } {S B } < {S A } {S C } en : e is B. I { {S A } {S B } = {S A } {S C } en : e is randomly hosen between B and C..8 POPUATION SHRINKING Ater eah new population is ull, a sreening is done to remove idential points in e parametri variable spae to give room or new and dierent solutions. RESUTS AND DISCUSSION The perormane o e algorim is reported or our onstrained optimization problems. Eamples, and are single obetive problems while Eample 4 is a multiobetive problem. Eample : The irst eample is a onstrained single obetive optimization problem. It has ive variables, a single quadrati obetive untion and is subeted to si nonlinear inequalities [8]. The ratio o easible points to sampled number o points or a,000,000 point random sampling was reported to be 0. [8]. The above ratio indiates at e problem is moderately onstrained and inimize 78 = , , , i =,4,. Table : Comparison o Results Number o Funtion Value Funtion Evaluations Reported [8] 0, worst avg best Present,70,64, Reported [8],400, worst -06. avg best Present,07,48, The obetive untion values as obtained rom suessive trials using e present algorim are ompared wi e best, worst and e average o 0 trials reported in Reerene [8]. It an be observed rom Table at e proposed algorim obtained omparable obetive untion values using a signiiantly smaller number o untion evaluations. Eample : The seond eample is a onstrained single obetive optimization problem. It has two variables, a single ubi obetive untion and is subeted to two 4 Subet to i 4

6 nonlinear inequalities [8]. The ratio o easible points to sampled number o points or a,000,000 point random sampling was reported to be [8]. The above ratio indiates at e problem is highly onstrained and hene a onstraint-onstraint-mating sheme was employed or e solution. It is also interesting to note at an obetive-onstraint mating sheme ails to identiy any easible solution ater 74 untion evaluations. inimize Subet to = 0 00, The optimum solution is 4.09, wi an obetive untion value o The irst two onstraints are ative at e optimum. The obetive untion values as obtained rom suessive trials using e present algorim are ompared wi e best, worst and e average o 0 trials reported in Reerene [8]. Table provides a omparison o results or e above eample. Table : Comparison o Results Number o Funtion Value Funtion Evaluations Reported [8] 0, worst -69. avg best Present 8, 8,4 9, Reported [8],400, worst avg best Present 7,4 7,44 74, Eample : The ird eample is a onstrained single obetive problem [8] wi ree variables. The easible region o e searh spae onsists o disoint spheres all o em having a radius o 0.. The global maimum is loated at,, wi e obetive untion value o.00. The obetive untion values as obtained rom suessive trials using e present algorim are ompared wi e best, worst and e average o 0 trials reported in Reerene [8]. It an be seen rom Table, at e algorim arrived at a solution 4.970,.09,.008 wi an obetive value o.00 in 98 untion evaluations using e obetive-onstraint mating sheme. It also reahed 4.988,.000, wi an obetive value o.00 in 90 untion evaluations using e onstraint-onstraint mating sheme. aimize Subet to or at least one set o p, q, r where Table : Comparison o Results Number o Funtion Value Funtion Evaluations Reported [8], worst avg..000 best Present Ob-Con Present Con-Con Eample 4: This is a two-variable onstrained biobetive problem [9]. inimize = = 9 Subet to g 0 g 0 0 where = 00 p p, q, r =,,,7 and 9. 0 i 0, 0, 0 /00 q i =,,. r The initial population is presented in Figure while e inal pareto optimal ront is presented in Figure. The inal pareto ront as presented in Figure was obtained ater untion evaluations and onsists o 9 pareto optimal points. The above problem was solved using a 0.

7 obetive-onstraint mating sheme wi a initial population size o 00. ront Figure. The algorim is urrently being tested on a wide range o single and multiobetive onstrained test problems to establish its suitability as a generi onstrained optimization meodology. Aknowledgment The auors would like to aknowledge e support or is work reeived rom e Institute o High Perormane Computing and e National Siene and Tehnology Board RICURF researh und ET/98/0. Figure : Initial population Figure : Final pareto ront 4 SUARY AND CONCUSIONS This paper presents an evolutionary algorim or onstrained optimization. The meod is problem independent and an handle any omputable onstraint and in addition optimizes e true obetive untion and not some transormed untion. The perormane o e algorim on bo onstrained single and multiobetive problems show a signiiant derease in e number o untion evaluations or omparable obetive untion values. It an be seen rom Eample at a onstraintonstraint mating is eetive or highly onstrained problems where inding even a single easible solution might be diiult. In e same eample an obetiveonstraint mating ails to loate a solution. On e oer hand, or problems where e easible spae is large, an obetive-onstraint mating results in omparable solutions in a signiiantly less number o untion evaluations. The presene o ooperative learning rough obetive-onstraint or onstraint-onstraint mating results is a aster onvergene while e presene o nihing allows e solution to be evenly distributed on e pareto Reerenes. ihalewiz, Z. 99. A Survey o Constraint Handling Tehniques in Evolutionary Computation eods, Proeedings o e 4 Annual Conerene on Evolutionary Programming, IT Press, Cambridge, A, pp. -.. Fonsea, C.. and Flemming, P.J. 99. An Overview o Evolutionary Algorims in ultiobetive Optimization, Evolutionary Computation,, pp Jimenez, F. and Verdegay, J Constrained ultiobetive Optimization by Evolutionary Algorims, Pros. O e International ICSC Symposium on Engineering o Intelligent Systems EIS 98, Spain, pp Srinivas, N. and Deb, K ultiobetive Optimization Using Nondominated Sorting in Geneti Algorims, Evolutionary Computation,, pp Surry, P., Radlie, N.J. and Boyd, I. 99. A multiobetive approah to onstrained optimization o gas supply networks, Pros. o e AISB-9 Workshop on Evolutionary Computing, Springer Verlag, Vol. 99, pp Fonsea, C.. and Flemming, P.J ultiobetive Optimization and ultiple Constraint Handling wi Evolutionary Algorims--Part I: A Uniied Formulation. IEEE Transations on Systems, an, and Cybernetis, Part A: Systems and Humans, 8, pp Hinterding, R. and ihalewiz, Z Your Brains and y Beauty: Parent athing or Constrained Optimisation, Pros. o e International Conerene on Evolutionary Computation, Alaska, pp Koziel, S. and ihalewiz, Z Evolutionary Algorims, Homomorphous appings, and Constrained Parameter Optimization, Evolutionary Computation, Vol.7, No., pp Deb, K Evolutionary Algorims or ulti- Criterion Optimization in Engineering Design, Pros. o Evolutionary Algorims in Engineering and Computer Siene EUROGEN-99,Finland.