An Application example of the breeder genetic algorithm to function optimization

AN APPLICATION EXAMPLE OF THE BREEDER GENETIC ALGORITHM TO FUNCTION OPTIMIZATION Lluis A. Belanche Secció d'/nrel·ligencia Artificial Dep t. de Llenguatges i Sistemes /nformntics Ulliversitat Politecllica de Catalullya [email protected] ABSTRACT Evolutionary Algorithms (EA) have demonstrated their ab iLit y to solve optimiza ti on tasks in a wide range of application s. In this paper, after outlining the basics of such algorithm s, the possibilities of one of the latest to emerge , the Br eeder Genetic Algorithm ( BGA ) are exemp li ed by addressing a c1a ssical nume ri cal optimization problem: the Fletcher-Powell pseudo-random function. 1. INTRODUCTION Since th e ea rl y days of comput er science, function optimjza ti on has been one of th e topics of more active research, because of the enormous potential of applicat ion to solve engineering or math ema ti cal problems. For di screte solution spaces, traditional Articial Intelligence ( Al ) techniques for solving constrained combinatorial optimization tasks, such as th eA * or IDA * al gorithms and de ri vations [1] have proven to be u sef ul for ce rtain appli cat ion s. However, the se methods rely on heuristics to guide th e search process that are not always easy to fi nd or to express. In addi ti on, if tbe search space is big, th e co mput a ti onal cos t of the al go rithm ca n b eco me unacceptably hi gh. An alte rn ative to these methods are Evolutionary Algorithms (EAs) [2], a mong w hi ch Genetic Al go rithms (GAs) [3] have been extensively used for dis-crete as well as continuous optimization task s, although they were not conceived for this [4]. Th eir main advantages over oth er methods are the coarsegrained global search mechanism, th e neat tradeoff between exploration of the search space for n ew solutions and exploita ti on of th e promising ones, the lack of an explicit heuris ti cs, and the easiness of paralle li zat io n. In addition, no kn ow ledge of th e problem has lo be coded apart from telling the algorithm h ow good (h ow fit, in EA jargon) a potential solution i s. Among the latest algorithms we find the Br eeder Genetic Algorithm ( BGA ) [5]. lt is characte ri zed by a truncation selec ti on procedure and direct representa ti on of continuous varia bl es. Trunca tí on selec ti on is a simple u e of rank-based se lec ti on, proven to be very u sef ul in traditional GAs [6] . The direct representa ti on of var iables eliminates th e need for a coding scheme -that usually changes th e search space- and permits to develop new, continuous genetic operator s. The BGA has been shown 56 to be supe ri or to trad it ional GAs for classical continuous optimiza ti on test problems [7]. We sh ow in this paper h ow a rela tí vely simple algorithm like the BGA ís a bl e to cope with th e Fletcher- Powe ll function, an extremely difficult optimization problem, of ten u se d for benchmarkin g. Our intention is n ot to solve this problem to a full, achieving th e best attainable solution, because this would require a more thorough study a nd much hi gh er computa ti onal demands than those used here. Rather, we use it to illustrate th e possibilities of EA s as black-box and robust techniques for optimization. In the next two sections, the ba sics of EAs are outlined, and in partic ul ar th e BGA is briefly introduced. It th en fo ll ows a descríption of the selected problem. Next, a small experimental setup is devi se d and th e results on it are commented on, togeth er with a comparison to oth er EAs. The paper ends wit h so rn e co nclusions and afterthought s. 2. BASICS OF AN EVOLUTIONARY ALGORITHM The term Evolutionary Al gorithms refers to a big family of search methods based on concepts taken from Darwinian evolution of species and natural selec ti on of th e fittes t. Sorne concepts from genetics are also present. Gi ven a problem to be sol ved EAs maintain a population of individuals th at represe nt potential solulions to it. Each individu aJ in th e popula ti on is represented by achromosome consisting of a string of atomic elements ca ll ed genes. Each gene cont ai ns (represents) a varia bl e, eith er for th e problem or for th e algorithm itse lf. The possible values of a gene are ca ll ed aUe/es and th e gene' s position in th e chromosome is ca ll ed lo cus ( pI. loci ). Th ere is aJso a di stin ct ion between th e genoryp e, the gene ti c mate ri al of an individual, and th e pheno ry pe, the individual result of genotype development ( th at i s, th e born li v in g thing). In EAs th e genotype coincides wi th th e chromosome, and the phenotype is si mulated via a fitness function, a scalar va lu e -similar to a re in Jorcementex pr ess in g h ow we ll and individual has come out of a given ge notyp e. An EA can be fo rma lJ y described by th e con ce ptual algorithm in g. 1, parameterized by a tupl e: < EA-Setup >=< D o, ( ¡.t ,A), 1,0 , 'f' , e, <1> , :=: > B URAN 014 DI CIEMBRE 19 99 where n, = Ci' " ¡'2' ... ¡'I') is th e population at time t and thus no is the, usua ll y random, initial population, /J. th e pop ul ation size, A the offspring size (out of /1 ), Y the selec ti on operator, n the recombina ti on operator, \fl th e mutation operator, e the termina ti on crite ri on, :=: th e replacement criterion a nd <1> th e fitness function. In this algorithm, operator se quencing on the popula-tion is as follows: n, represe nts the popula ti on at time (i.e., genera ti on) t, nT, th e population af t er sel ec ti on, n·, af ter recombination and n", after muta ti on, to end in a n ew population n '+I . Th e search process usua ll y starts with a randornly ge nera ted population no and evo lves ove r time in a que st for be tter and better individuals where , from ge ner ation to ge neration , n ew popul a ti ons are formed by appl i ca ti on of thr ee fundamental kinds of opera tors to the indi v idual s of a population , forming a ch arac t er is ti c thr ee-s tep p roce dur e: l. Seleetion of the fittest individuals, yielding th e soca ll ed gene pool; 2. Reeombina ti on of (some of) the pr ev iously se-lected individuals forming the ge ne pool, giving ri se 10 an o spring of new indi vi duals; 3. Mutation of (so me of) th e n ew ly cr ea ted individual s. Procedure Evolut io na.ry - Algoritbm ( t : =0; create n ~ ; evaluate <I> (i), V'i E n t ; while not (e (n t» do ( / - Create the gene pool n\ 'Ir / select: nT t := T (n t ); /* Apply genetic operators "' / recombine: n t := n(nT l ); mutate: n' l := o/ In t ); / ,. Evaluate their effect fr / evaluate <I> (i) , 'Vi E n t ; ; '" Forro the new generatian "" replace: n ,O¡ := .:: (n t v nTt ); t : = t+l Fi g ur e J: E vo hai o na ry AL gorilhm. By iterating this three-step mechanism, it is hoped that increasingly better individuals will be found (that is , will appear in the population ). This reasoning is based on the fo ll owing ideas: l. The sel ec ti on of the fittest individuals ensures that o nl y th e best ones 1 will be a ll owe d to have o spring, dri ving the search towards good sol utions, mi micking the natu- 1 Or the luckiesl in so rne EA instances, like m os t GA s. • RAM AS DE EsTUDIANTES DEL IEEE ral process of se lec ti on, in w hi ch o nl y the more adapted sp ec ies are to survive. 2. By r eco mbining the gene ti c material of th ese selected individuals, the possibility of obtaining an offspring where al least one chi ld is better than any of its parents is high. 3. Mutation is meant 10 introduce n ew traits, not prese nt in any of th e pare nt s. It is usua ll y p erforme d on freshJy obtained individuals by s li ghtly altering some of their gene ti c mate ri al. Hence, an EA may be seen as a non-empty sequence of ordered op era tor appli ca ti ons: fitness eva lua ti on, selection, recombination, muta ti on and re pl acement. Th ee ntire process it erates until one oft he following crite ri a is fullfiled: l. Convergenee: it ha pp ens because the individuals are too similar. Fr esh and new ideas are n ee ded, but r eco mbination i incapa bl e of providing them because th e individuals are very close to one another, and mutation alone is not pow erf ul enough to introdu ce the desired variability. Conv ergence can be monitored by on-line (avera ge of the best individuals) and off-line (average of ave ra ge individuals) throug hout the generations; 2. Problem solved: the global optimum is found up to a sa ti sfactory accuracy (if optimum kn ow n); 3. End of resourees: the maximum number of function evalua ti ons has b ee n reached. Evolutionary Algorithms are effec ti ve mai nl y because their search m ec hanism keeps a well-balanced tradeoff between exploration (trying to a lw ays drive the search to the discovery of new, more u sef ul , gene ti c material) and exploitation (trying to fine-tune good alreadyfo und solutions). Exploration is m ai nl y d ea lt with by th e mutation ope ralor. Exploitation is carr i ed o ul by th e sel ec ti on process and Ih e use of recombina ti on operators, although mutation may a1so pl aya role in th e neruning of solution s. Th e tn ess function is built out of th e function lo be optimized (ca ll ed th e objeetive funetion). A lI EAs represent th e decision varia bl es in the chr omosome in one way or another, eilh er directly as r ea l values or resorting to a discrete coding, usually binary (Iike most GAs). The particular coding sch eme is the classical knowledge representa ti on problem in Al, and comp letely co ndilions the result s. In add it ion, some algorithms (like the Evolution Strategies, ES) append their ow n variables lO the representation in the form of auxiliary information that evolves with time li ke the oth er variables. An exce ll ent state-of-the-art and r ev i ew ofEAs , and a u sef ul departure point because of its ri ch set of references 57 is [8]. There is also a very complete F AQ with pointers to papers, books, software and the main groups working on EAs all over the world [9]. 3. THE BREEDER GENETIC ALGORITHM In traditional GAs, selection is stochastic and meant to rnimic -to sorne degree- Darwinian evolution; instead, BGA selection (named truncation selection) is a deterministic and articial procedure driven by a breeding mechanism (as used in livestock), where only the best individuals -usually afixed percentage of'r total population size ¡.1- are selected and enter the gene pool to be recombined and mutated, as the basis to form a new generation. Genetic (recombination and mutation) operators are applied by randomly and uniforml y selecting two parents until the number of offspring equals ¡.1-q. Then, the former q best elements are re-inserted into the population, forming a new generation of ¡.1 indi viduals that replaces the previous one. This guaranteed survival of sorne of the best individuals is called q-elitism. For the BGA, the typical value is q = 1. The BGA selection mechanism is then deterministic (there are no probabilities), extinctive (the best elements are guaranteed to be selected and the worst are guaranteed not to be selected) and 1- elitist( the bestelementis always to survive fromgeneration to generation). The BGA chromosomes are potential solution vectors 'it of n components, where n is the problem size, the number of free variables of the function to be op-tirnized. The common aspect of BGAs with ordinary GAs is the fact that both are mainly driven by recombination, with mutation regarded as an important but background operator, in the double role of solution fine-tuner (for very small mutations) and as the main discovery force (for moderate ones). We will now briefly describe the different possibilities for the genetic operators. The reader is referred to [10] for a detailed description. 3.1 Recombination Any operator Q combining the genetic material of the parents is called a recombination operator. In BGAs, recombination is applied unconditionally, Pr(Q) = 1. Let ~ ~ x = (x , ... , x ), Y = (y , ... , y ) be two selected gene-pool ¡ ~ ~ ¡ -"4' ~ ~ individuals x , Y such that x :;t: y . Let z = (z¡, ... , z.) be the result of recombination and 1 ::;; i ::;; n. The following are some of the more common possibilities to obtain an o . ~ spnng z: 1. Discrete Recombination (DR). Z¡E {x¡,y) (chosenwithequalprobability) 2. Line Recombination (LR). with a fixed a E [O, 1]. Typically, a= 0.5. 58 3. Extended Intermediate Recombination (EIR). with a [-b, l+b] chosen with uniform probabilify. The param~ter b expresses to what degree an offspring can be generated out of the parents' s scope, the imaginary line that joins them in 9\ . More precisely, it works by controlling the maximum fraction a = b[y¡ - x¡ ]of the distance between parents where the offspring can be placed, either left to the leftmost parent or right to the rightmost parent. A typical value for b= 0.25, although any non-negativereal number not exceeding 0.5 is a potential value. The bigger the , the more the effect of the parents is dirninished in creating offspring. A method for dynarnically setting its value called range, was introduced in [10] and shown to have a remarkable effect in performance. It works as follows: such that a¡ E [-b¡-, 1 + bt] with uniform probability and, Y¡ -r¡ 1);-=-+-- r¡ - r¡ + r. -x. 1)+=_1 __ 1 1 + r¡ -r¡ This procedure assigns different values for the left (b-) andright(b+) lirnits ofthe interval from which a¡ is to b~ selected, anct does never generate a value outside the range [r.- , r+] forthe variable i, an aspectnotfullled by the other ~ethods that otherwise has to be dealt with a posteriori. 4. Fuzzy Recombination (FR). This operator, introduced in [11], basically replaces the uniformpdj(probability distribution function) by a bimodal one, where the two modes are located at x¡ and y¡. The label "fuzzy" comes from the fact that the two parts Pr .(t); Pr(t) of the XI yl probability distribution resemble triangular fuzzy numbers. 3.2 Mutation A mutation operator 'P is applied to each gene with some probability Pr('P) = l/n so that, on average, one gene is mutated for each individual. Let -:¿ = (z¡, ... , z) denote the result of mutation of an individual 'it. The elements of -:¿ are formed as follows: 1. Discrete Mutation (DM). z. = x. + signo range .. b l l l with signE {-1, + 1} chosen with equal probability, range¡=p(r¡+ - r), PE [0.1, 0.5] and BURAN N°14 DICIEMBRE 1999 k-l . D= L <p.T l i=O l where C{J¡E {O,l} from a Bernouilli probability distribution where Pr( C{J¡ = 1) = l/k.In this setting k E N+ is a parameter originally related to the precision with which the optimum was to be located, a machinedependent constant. In practice, however, the value of k is related to the expected value of mutation steps: the higher k is, the more fine-grained is the resultant mutation operator. The factor p is the range ratio, related to the maximum step that mutation is allowed to produce as a ratio of variable range. 2. Continuous Mutation (CM). Same as DM but with where PE [O, 1] with uniform probability. 4 THE FLETCHER-POWELL FUNCTION This highly multimodal function was introduced in 1963 as a regression problem [12], where a collection of parameters have to be estimated such thata quadratic error term, depending on a non-linear expression, has to be minimized. Let us denote the n parameters as a ~ vector x . Let A = (a . .), B = (b .. ) be two n x n real --? lJ lJ matrices and a a fixed real vector of dimension n. Define the two terms A and B as: 1 1 n A¡(a) = l.Ja¡jsina j +b¡j cosaj) H n B¡ (x) = ¿ (a¡jsinxj + b¡j COSX j) j=1 Then, the function is defined as: n FP(i) = 2: (A¡(a) - B¡ (i))2 j=1 The solution is FP( t') = O, wheret *= ~ . The function is c1early non-syrnmetric (which would make it easier) and non-separable (it cannot be optimized on each x. separately). The sinusoidal factors lead to a high J number of hills and valleys, making a gradient-based method unfeasible. There are up to 2n extrema located in the subspace IX}$;1t, the one usually selected to constrain the solution space. The vector ~ (the position of the absolute minimum) is chosen atrandom from this subspace ajE [-1t, 1t]. Moreover, the matrices A, B are also chosen at random with a .. , b.. E [-100, 100]. This makes the lJ lJ • RAMAS PE ESTUDIANTES DEL IEEE suboptimal extrema to be randomIy distributed over the search space. In our case, a value of n = 30 is chosen, so that the function inc1udes a total ofl ,830 random numbers. 5. EXPERIMENTAL RESULTS An execution with the BGA involves the choice of a mutation operator \f and its parameters p and k and of a recombination operator Q and its parameter o (only for EIR). The truncation threshold 1" and population size ¡.t are also to be set. These last two values are of great importance because they are strongly interelated. For the sake of c1arity and simplicity we do not discuss them here. Instead, an educated guess is made just in order to obtain reasonable results within low computing resources. To this end, a value of 1" = 15 is set, to ensure a high selective pressure so as to enforce the quick discovery of fairly good solutions. The low value of entails a moderate-to-high population size in correspondence, to account for enough diversity. We set then Ji = 100. The choice of .genetic operators is more elaborated. To begin with, p is set to 0.5, a value that stands for high average mutation steps, needed to broad-tune rather than to rene solutions. In addition, a low value of k (lower than 16) is in favour of this scheme, and known to have a much deeper impact on performance [10] than , so we carried out sorne preliminary runs with k E {6, 8, 10, 12, 16} that lead to the selection of k = 10. In addition, the continuous operator showed to be better. All this tuning runs take only a few seconds (always less than a minute) in a shared SUNtm Ultra- 60 System. The stopping criterion is based on the number of function evaluations permitted (given by the vari-able FFEvals). In particular, given a nite number of FFEvals, the algorithm will stop each run whenever [FFEvals / J..l] generations are reached. This stopping criterion allows to compare different general settings in a fair way, since, for example, a smaller population would be allotted more generations, but always keeping the number of evaluations in similar values. For each conguration, a number of independent runs are performed - denoted by NRunskeeping track of the mean and best solutions found. In these initial experiments, FFEvals=50,000 and NRuns=5. Regarding the recombination operator, ten pos sibilities are tested from within the set {DR, LR with a = 0.5, EIR with 0= Oto 0.30, EIR with rangeoandFR}. Among them, the EIR operator quickly stand s out over the rest, with varying performance that depends on its parameter D. The results for this last operator are 59 presented in a single table for ease of reading -Table 1- summarizing the information as a function of the sample values for tested. For each conguration, average and best solutions found are shown. EIRO Average Best 0.05 51,541 18,296 0.10 28,837 23,151 0.15 25,368 2,265 0.20 32,459 11,798 0.25 41,262 14,017 0.30 47,356 20,561 ranged 25,953 9,030 Table 1: Resultsfor the E1R (o) recombination operator. Each entry shows the average and best results across five runs. As it can be seen, EIR (0.15) shows to be the best setting, both on average and in terms of the best solution found. Note that the curve of performance across is concave, with EIR (0.15) at the bottom. The operator with the modication range shows to have a performance almost equal than the best one achieved with a fixed o. This is remarkable because it means that a comparable performance can be obtained without the need of a search along . Both settings (O = 0.15 and range~ are selected for further experiments. It has to be said that the already obtained results are quite good. The initial function evaluations are in the order of millions (between one and five, depending on the run). Once the algorithm has been roughly tuned to the problem landscape, a series of runs are performed to assess its potential to a deeper degree. Operators are then set to continuous mutation with p= 0.5, k = 10 for mutation, and the two choÍCes EIR ( 0= 0.15), EIR (range ~ for recombination, using the knowledge gained so far; this time, however, NRuns=100 runs of FFEvals=100, 000 are carried out. The results are presented collectively in Table 2, along with sorne other results present in the literature. Biick [2] (p. 157), for example, reports results on this function with a standard genetic algorithm (SGA), an evolutionary prograrnming algorithrn (EP) and the powerful evolution strategies (ES), reproduced in the table for Algorithm Average Best SGA 4.581.10 4 1.032 .10 4 EP 1.107 . 10 5 2.997.10 4 BGA-1 1.987. 10 4 8.165.10 2 BGA-2 9.238 .10 3 8.649 .10 2 ES 1.749 .10 3 3.190 ·10·\ Table 2: Comparative results found by some evolu-tionary algorithms. BGA-1 is with EIR (0.15), and BGA-2 with EIR (range) 2 Crossover is a form of recombination used in discrete GAs. 60 convenience. The frrst notable point is the inferior performance oftheSGAandEP compared to the otheralgorithms. Eiben and van Kemenade [14] report also sorne results for a GA, product of their study on diagonal and n-point crossover2 operators. They show these operators to be generally superior to traditional 2-parent, 1- or 2-point crossover. Although they do not provide numerical results, this improvement seems to make the GA vary between 30,000 and 10,000 on average, occasionally going below this mark. Altogether then, the binary GA is not likely to go beyond this order of magnitude, unless heavily modied or tailored to the problem. With respect to the BGAs, both have improved their average performance, due to the double number of evaluations they have been allowed. Also noteworthy is the superior average performance ofEIR (range) over EIR (0= 0.15). We believe that, in the long run, the firstmethod is likely to widen the gap because of its adaptive nature. 4.1943e+06 ,----,-___r----.---.---.--,--,--~-___r-__, Frtn+s -e-- ----...¡..----: --- . ..J----+--- ..... --+- .. -----+.-.-----1.----+----+--- I _·_·_--····-i,··----_·_·_-t···-----i---····_-t, -------l----···---l¡:--. ···_----¡--------··-r--------·""i------ ! I ! !; I ---+----~---i-·---~---l-----J----·---L.--f---_j---- . I I i! 262144 . ! i ¡ i ¡ ¡ ! ¡ 16384 ----·--1--- --~·-_·_·-+-·_-_····_1--·_·-···_·r·--··---t-·_---1-·----~ --- .. ---t----- ! i; i! i . i I I i ! i 4096 ····-··--····I·-···_·_--··--·r-····-···--····I--·····- .... i I i 1024 -.-.-.+--.---+-.- .. --+---- .. -+-. --"-.+-'-'---+---'-'--+-"-'--!--'.~"""...d. ~L-~ __ ~ __ -L __ ~ __ ~ __ ~ __ L-~ __ ~ __ ~ o 10 20 30 40 50 60 70 80 90 100 Figure 2: Performance curvefor BGA-2 (EIR with range). The x-axis shows the number of generations (in multiples of ten). The y-axis shows the tness in a log2 scale. Note the quasilinear slope and the fact that further progress was still being achieved. The error decrease from the initial random guess to the final solution is in four orders of magnitude. Note that, very roughly, both departing from similar average values, doubling the number of function evaluations has re ected in an average result for EIR (range~ that is twice as good as that ofEIR (0= 0.15). However, the best absolute result is very similar for both, possibly indicating a limitin whatcan be achieved with the allowed resources and the selected ~, 't setting. A plot of the run nding the best solution for BGA-2 is shown in Figure 2. The results reported by Biick for the ES are by far the best, both on average and, especially, in the best solution found, which is excellent. This can be explained in several ways. First, ES [13] is a far more complicated algorithrn that takes into account the interactions between variables more explicitly by selfoptimizing the amount of mutation BURAN N"14 DICIEMBRE 1999 necessary. Itis nonetheless very sensitive to its parameters so that tuning it is a more dicult task. However, once these are correctly set, ES performance is often superior to other algorithms. Second, itis interesting to note that the data for Back' s experiments were obtained in sim-ilar conditions (~= 100, 't = 15 and 100 runs) but resorting to 200,000 evaluations. The random initialization for the function is also a different one, which could well result in an overall easier or more complicated landscape. Back also reports the great variability found: of the 100 runs, only 44 were "success-fuI runs", defined as those reaching anal value under 400.0 (compare it with the best result obtained: 0.319). In our case, for example, the n deviation was computed to be 16792.8 for BGA-l, and 6563.7 for BGA-2. To see whetherthe BGA would be able to better its performance, we carried out a last experiment, a single run of BGA-2 with 200,000 evaluations. The result was 358.379, a value qualifying as successfu!. 6 CONCLUDING REMARKS The quest for general-purpose search mechanisms is still an open and very active field. Since the 70's, new and powerful heuristic methods have emerged that are particularly well suited forfunction optirnization -although this was not exactly their original purpose-mainly because of their generality, robustness, and conceptual (though not necessarily analytical) simplicity. Three of these methods are Simulated Annealing (SA), Tabu Search (TS) and Evolutionary Algorithms (EA). In the last decade, these properties and their remarkable successes have boosted their widespread use. The EA family is the biggest and subject of continuous improvement. In addition, many c1assical and modern problems are being reinstantiated and explored under the light of these methods. Articial neural networks are a good representative of this. In particular, the training process of a supervised feed-forward network can be easily cast as a function optirnization problem. In this paper we have given an impression of how different EAs can cope with a well-known task, one for which methods of non-linear optirnization are prone to end up in local rninima of the function. Among the former, the Breeder Genetic Algorithm (BGA) is relatively simple and yields reasonably good solutions in a very lirnited time. Note that although it has not been our intention to solve the task at hand, and only a small experiment setup has been devised for it, the quality of the solutions and the promise of better ones show the BGA (and EAs in general) as fe asible alternatives for a great variety of tasks, ranging from the mentioned neural network training problem [15] to aerofoil design in Aerodynamics [16]. W ork is in progress toward a thorough application of these methods in a principled way. REFERENCES [1] Pearl, J. Heuristics. Intelligent search strategies for computer problem solving. Addison-Wesley, 1984. • RAMAs DE ESTUDIANTES DEL IEEE [2] Back, Th. Evolutionary Algorithms in Theory and Practice. Oxford University Press, New York, 1996. [3] Goldberg, D.E. Genetic Algorithms for Search, Optirnization & Machine Learning. Addison-Wesley, 1989. [4] Holland, J .H. Adaptation in natural and articial systems. The University of Michigan Press. Ann Arbor, MI, 1975. [5] Mühlenbein, H., Schlierkamp-Voosen, D. Predictive Models for the Breeder Genetic AIgorithm. Evolutionary Computation, 1 (1): 25-49, 1993. [6] Whitley, D., The GENITOR AIgorithm and Selection Pressure: Why Rank-BasedAllocationofReproductive Trials is Best. In Procs. of the 3rd Int!. Conf. on Genetic AIgorithms, (ed.) Scha er, J.D., Morgan Kaufmann: San Mateo, 116-121, 1989. [7] De Falco, I., Del Balio, R., Della Cioppa, A., Tarantino, E. A Comparative Analysis ofEvolutionary Algorithms for Function Optirnisation. In Procs. of the Second Workshop on Evolutionary Computing (WEC2), Nagoya, JAPAN. [8] Back, Th., Schwefel, H.P. An Overview of Evolutionary AIgorithms for Parameter Optimization. Evolutionary Computation, 1 (1): 1-23, 1993. [9] Heitkotter, 1., Beasley, D. (Eds.). The Hitch-Hiker's Guide to Evolutionary Computation: A list of Frequently Asked Questions (FAQ). USENET: comp.ai.genetic. A vailable from anonymous FTP at rtfm.mit.edu, on /pub/usenet/news.answers/ai-faq/ genetic. [10] Belanche, Ll. A Study in Function Optirnization with the Breeder Genetic Algorithm. LSI Research Report LSI -99-36-R. Dept. de Llenguatges i Sistemes Inform atics. Univ. Politecnica de Catalunya, 1999. [11] Voigt, R.M., Mühlenbein, H., Cvetkovic, D. Fuzzy recombination for the continuous Breeder Genetic AIgorithm. In Procs. of the 6th Int!. Conf. on Genetic Algorithms, (ed.) L. Eshelman, Morgan Kaufmann: San Mateo, 104-113, 1995. [12] Fletcher, R., Powell, M.J.D. A rapidly conver-gent descent method for rninirnization. Computer Journal, 6:163-168,1963. [13] Back, Th. Evolution Strategies: An alternative Evolutionary Algorithm. Technical Report of the Informatik Centrum, Dortmund. [14] Eiben, A. E. , van Kemenade, C.H.M. Diagonal Crossover in Genetic AIgorithms for Numerical Optirnization. Journal of Control and Cybernetics, 26(3):447-465, 1997. [15] Yao, X. A Review of Evolutionary Articial Networks. Int!. J. ofIntelligent Systems, 8(4): 539-567, 1993. [16] De Falco, I. An introduction to Evolutionary Algorithms and their application to the Aerofoil Design Problem -Part 1: the AIgorithms. Part 11: the Results. Invited papers at the von Karman Lecture Series 1997, Bruxelles. Both pa-pers can be retrieved from the IRSIP Web Pages http:// www.irsip.na.cnr.it/-hotg. 61