Self-Adaptive Parent to Mean-Centri Reombination for Real-Parameter Optimization Kalyanmoy Deb and Himanshu Jain Department of Mehanial Engineering Indian Institute of Tehnology Kanpur Kanpur, PIN 86 {deb,hjain}@iitk.a.in http://www.iitk.a.in/kangal/pub.htm KanGAL Report Number Abstrat Most real-parameter geneti algorithms (RGAs) use a blending of partiipating parent solutions to reate offspring solutions through its reombination operator. The blending operation reates solutions either around one of the parent solutions (having a parent-entri approah) or around the entroid of the parent solutions (having a mean-entri approah). In this paper, we argue that a self-adaptive approah in whih a parent-entri or a mean-entri approah is adopted based on population statistis is a better proedure than either approah alone. We propose a self-adaptive simulated binary rossover (SA-) approah for this purpose. On a suite of eight unimodal and multi-modal test problems, we demonstrate that a RGA with SA- approah performs reliably and onsistently better in loating the global optimum solution than the RGA with the original parent-entri operator and the well-known CMA-ES approah. Introdution One of the hallenges in designing an effiient real-parameter evolutionary optimization algorithm is the reombination operator in whih two or more population members are blended to reate one or more new (offspring) solutions. The existing reombination operators an be lassified into two broad ategories: (i) variable-wise operators and (ii) vetor-wise operators. In the variable-wise ategory, eah variable from partiipating parent solutions are reombined independently with a probability to reate a new value. The resulting offspring solution is then formed by onatenating the reated values from reombinations or hosen from one of the parent solutions. This proedure does not honor the neessary linkage between different variables needed to solve a problem adequately and is most suited for variable-wise separable problems. Some examples of variable-wise reombination operators are blend rossover (BLX) [5], fuzzy reombination (FR) [9], simulated binary rossover () [], and others. On the other hand, in the vetorwise reombination operators, a linear ombination of the omplete variable vetors of partiipating parent solutions is usually made to reate an offspring variable vetor. Suh operators are able to propagate and honor the linkage among variables. Some examples are UNDX operator [8], simplex rossover (SPX) [7], parent-entri rossover (PCX) [], and others. It is worth mentioning here that the searh operation of differential evolution (DE) and partile swarm optimization (PSO) use a vetor-wise reombination operation The reombination operators (variable or vetor-wise) an also be lassified into two funtionally different lasses based on the loation of reating offspring solution vis-a-vis the loation of parent solutions. An earlier study [] lassified most of the above-mentioned real-parameter
reombination operators into two lasses: (i) mean-entri operators and (ii) parent-entri operators. In a mean-entri operator, offspring solutions are reated around the mean of the partiipating parent solutions. BLX, UNDX and SPX operators an be onsidered to be meanentri reombination operators. In a parententri operator, offspring solutions are reated around one of the parent solution. FR,, PCX are examples of suh an operator. Sine parent solutions are tested to be better by the preeding seletion operator, performing a parent-entri operator was argued to be a better operator than a mean-entri operator. This is partiularly true in the beginning of a simulation when parent solutions are expeted to be away from eah other thereby making the mean of parent solutions to lie ompletely in a new region whih may not have been tested for its goodness by the seletion operator. In this paper, we argue and demonstrate that while a parent-entri operator may be judged to be better early on in a simulation, a meanentri operator may be found to be benefiial when population members rowd around the optimum. Thus, instead of using a parententri or a mean-entri reombination operator throughout a simulation, a better strategy would be to adaptively hoose a parent-entri or a mean-entri operator depending on whether the population is approahing or rowded around the optimum. In the remainder of the paper, we overview the differene between parent and mean-entri operators and indiate some popular reombination operators. We then onsider a speifi parent-entri operator () and modify the operator so that at every generation it behaves like a parent-entri or a mean-entri operator depending on the population statistis. Thereafter, we demonstrate the effiay of the proposed modified algorithm on a number of unimodal and multi-modal test problems borrowed from the literature. The results are ompared with the original parent-entri operator, a purely mean-entri operator, and the wellknown CMA-ES algorithm [6]. The proposed approah is found to perform onsistently better in most problems than other methods inluding CMA-ES. Moreover, the methodology is also found to be salable in to variable versions of the test problems. With a mention of immediate future studies, onlusions of this paper is finally made. Real-Parameter Reombination Operators The main distinguishing aspet of a reombination operator ompared to other searh operations in an evolutionary algorithm (EA) is that more than one evolving solutions are utilized to reate one or more new offspring solutions. Sine an EA population evolves with the generation ounter, the reation of offspring solutions based on multiple evolving parents naturally provide a self-adaptive feature to the algorithm [3]. Importantly, suh an operation is missing in most other optimization algorithms and is one of the unique features of an EA. When it omes to solving real-parameter optimization problems, reombination between two real-valued vetors beomes more of a blending operation. Most existing realparameter reombination operations an be lassified aording to whether the offspring solutions are reated lose to one of the parent solutions or lose to the entroid of the partiipating parent solutions. We disuss these two lassifiations below.. Parent-Centri Crossover Operators As the name suggests, in these reombination operators, the offsprings are reated around the parents. The motivation behind these operators is that sine parent solutions have been tested to be better in the preeding seletion operation, reating offspring solutions near to parents may lead to better solutions. We disuss a ouple of suh operators... Fuzzy Reombination (FR) Operator This is a variable-wise reombination operator and was proposed by Voigt et al. in 995 [9]. For two parent values of a partiular variable, solutions lose to eah parent is hosen with a triangular probability distribution, as shown in
Figure. A parameter d is introdued to ontrol d(p p )p p Figure : Fuzzy reombination (FR-d) operator. d is a user-defined parameter. the diversity of the reated offspring values visa-vis parent values... Simulated Binary Crossover () is also a variable-wise reombination operator and was proposed by Deb et al. in 995 []. This operator is similar to FR operator, but has the ergodi property suh that any real value in the searh spae an be reated from any two parent values, but with differing probabilities, as shown in Figure. The parameter η ontrols the diversity in the offspring values ompared to that in the parent values. operation, and is given below: (u) η+, if u.5, β(u) = ) η+, otherwise. ( ( u) (3) The resulting weights to p and p are biased in suh a way that values lose to the parent values are more likely than values away from them, thereby providing its parent-entri property...3 PCX Operator The above variable-wise parent-entri operator an also be extended to vetors. One suh operator is the parent-entri rossover or PCX [], in whih three parent solutions (vetors) are hosen and a biased linear ombination of them is proposed to reate an offspring solution ( y) around one of the parent vetors (x (p) ), as follows: n y = x (p) + w ζ d (p) e (p) + w η D e (i), (4) i=, i p where e (i) are the n orthonormal bases that span the subspae. The diretion e (p) is along d (p). The parameters w ζ and w η are zero-mean normally distributed variables with variane σζ and ση, respetively. The operator is desribed pitorially in Figure 3(a) by reating a number of offspring solutions from three parent solutions. p p Figure : -η operator. η is a user-defined parameter. Here we hoose η =. Sine we have modified this operator in our study here, we desribe this operator in somewhat more details. For two partiipating parent values (p and p )ofthei-th variable, two offspring values ( and ) are reated as a linear ombination of parent values, as follows: =.5( β(u))p +.5( + β(u))p, () =.5( + β(u))p +.5( β(u))p. () The parameter β(u) depends on a random number u reated within [,] for eah reombination (a) PCX operator. (b) UNDX operator. Figure 3: PCX is a parent-entri and UNDX is a mean-entri operator.. Mean-Centri Crossover Operators In these operators, offsprings are reated around the entroid of the partiipating parents. The 3
motivation for these operators ome from the realization that sine all parents were judged to be better by the seletion operator, the intermediate region surrounded by parents may also be onsidered good. Some suh operators are desribed below. 8 7 6 5 4 3 d best d avg Centroid.. Blend Crossover (BLX) BLX-α is a variable-wise reombination operator proposed by Eshelman and Shaffer [5]. Offsprings are reated uniformly around the two parent values. The user-defined parameter α allows offsprings to be reated within or outside the range of parents... Vetor-wise Mean-Centri Reombination Operators In the unimodal normally distributed rossover (UNDX) operator (Figure 3(b)), three or more parents are hosen and points around the entroid of the parents are reated with a multinomial normal distribution. Simplex rossover (SPX) [7] is another example of a mean-entri reombination operator. 3 Parent or Mean-Centri Reombination Although both types of reombination operators exist, it is important to understand under what senarios eah type of reombination operator will be useful. Let us investigate two senarios for this purpose. Figure 4 depits a senario in whih an EA population is approahing the optimum. In suh a senario, the population-best solution is likely to lie at the boundary of the urrent population, partiularly when the population lies within the optimal basin. Ideally, in suh a senario, a parent-entri reombination is likely to yield better offspring solutions, as solutions beyond the population-best solution are expeted to be loser to the optimum. Thus, while approahing an optimum, it is better to employ a parententri reombination operator for a faster approah towards the optimum. On the other hand, Figure 5 shows a senario in whih the EA population surrounds the op- Best Optimum Point 5 5 5 5 35 45 55 65 758 Figure 4: Senario : Population is approahing the optimum. Here λ>. timum and the population best is likely to be an intermediate population member. In suh a Best Point Centroid d best d avg Optimum Figure 5: Senario : Population surrounds the optimum. Here, λ<. ase, a mean-entri reombination operator is likely to produe better solutions. Thus, while surrounding an optimum, it is better to employ a mean-entri reombination operator. While the above two senarios and orresponding nature of reombination operations are somewhat lear, it is not lear how to know whih senario exist at any partiular generation so that a suitable type of reombination operator an be applied. In this study, we ompute a parameter λ for this purpose based on the loation of population-best and population-average points at every generation: λ = d best d avg, (5) 4
where d best is distane of the population-best solution from the entroid of population in the variable spae and d avg is the average distane of population members from the entroid of population. A little thought from Figures 4 and 5 will reveal that in the first senario, λ is likely to be greater than one and in the seond senario it is likely to be smaller than one. Sine the parameter λ aptures the extent of approah or rowding near the optimum in the urrent population, we use this parameter to reate an offspring based on parent-entri or mean-entri approah. 4 Self-Adaptive (SA- ) Operator We onsider the operator [] and modify it using the λ value. First, we reate two virtual parents v and v from two partiipating parents p and p as follows: v = p + p v = p + p λ p p, (6) + λ p p. (7) Next, the virtual parents (instead of p and p ) are then used to reate two offspring values using equations and. Unlike in another study [4] in whih the parameter η was updated from one generation to the other, here we keep it same from the start to the end of a simulation. The above two equations yields: (v v )=λ(p p ). Thus, if λ = (population-best solution is at the entroid of the population), two virtual parents are idential and lie at the mean value of the two parent values. This auses a mean-entri operation. On the other hand, if λ = (populationbest solution is at the average distane from entroid of the population), virtual parents are idential to the original parents. This makes the SA- to be idential to the original operator. For λ>, virtual parents move outside the original parent values and the reated offspring values are likely to have a greater diversity than that in the original operation. Figure 6 illustrates three different senarios. Sine the parameter λ is not pre-fixed and depends on the loation of population-best solution from the entroid of the population and a measure of an average diversity of the population, λ value will hange from one generation to another. If the population-best solution is at the periphery of the population (as in senario in Figure 4), SA- reates a diverse set of offspring solutions and the searh tends to explore the searh spae beyond the region represented by the population. Figure 7 shows the distribution of offspring values for a partiular variable. However, when the population-best solution is lose to the entroid of the population at any generation (as in senario in Figure 5), the above SA- behaves like a mean-entri operator and diversity of the reated offsprings is expeted to redue, thereby fousing the searh. Figure 8 shows the orresponding distribution of offsprings. Interestingly, all these happen in a self-adaptive manner and without any user intervention. We now present simulation results using the SA- operator. 5 Results We hoose five unimodal and three multi-modal test problems from the EA literature. They are tabulated in Table. To ompare the performane of the proposed SA-, we use the wellknown CMA-ES [6] and two variations of the reombination operator: RGA with the original operator (a purely parent-entri operator) and RGA with a purely mean-entri operator, in whih two offspring points ( and ) are reated from two parents (p and p )using the following probability distribution: P (β) = (η+)/(β+) η+,whereβ = ( )/(p p ). In eah ase, runs are made from different initial populations until a speified number of funtion evaluations are elapsed. Thereafter, best, median and worst objetive value attained in runs are tabulated and plotted. In all ases, a population of size 5n (where n is the number of variables in the problem), a rossover probability of.9, and η = are used. To evaluate the effet of reombination operator alone, we have not used any mutation operator in this study. For all test problems, we have hosen two initialization proesses: (i) symmetri initialization around optimum: x i [, ] for all variables, 5
Different loations of best individual λ=.5 λ =.5 Centroid SA Atual Parents Virtual Parents Atual Parents Virtual Parents d λ< avg λ= λ> SA p v p v v p p v Figure 6: Parameter λ for different senarios. Figure 7: Distribution of offspring values for Senario. Figure 8: Distribution of offspring values for Senario. and (ii) one-sided initialization away from optimum: x i [, 6] for all variables. Eah problem is onsidered with different sizes, n =, 5, and. To investigate the working of the proposed adaptive reombination approah, we plot the variation of the parameter λ with generation for a typial run on problem P in Figure. Sine 5. Problem P Figure 9 shows the hange in objetive funtion value of the population-best member with generation for four different algorithms applied to the -variable version of P. In this ase, popula- 4 4 var ellipsoidal pop= 6 SA 8 3 4 5 6 7 Figure 9: Redution in objetive value for - variable problem P with symmetri initialization. tions are initialized around the optimum. It is lear that CMA-ES has outperformed all RGAs with any of the variations. However, the enhanement in performane of SA- ompared to the original is lear from the figure. The mean-entri version of the operator did not perform so well. Interestingly, in all ases, the algorithms do not seem to be sensitive on runs started from different initial populations. λ d best d avg d opt best d opt.5.5 5 5 5 3 4 5 5 5 3 4 5 5 5 3 Generations Figure : Variation of λ for -variable problem P with symmetri initialization. the initial population already surrounds the optimum in a symmetri initialization, λ is expeted to be less than one. Importantly, the RGA with SA- redues λ with generations, as evident from the figure. It is interesting that original operator would have kept λ = in all generations, but how the proposed SA- operator uses a varying λ to help onverge to the optimum in a omputationally fast manner. After suffiient number of generations when population is lose to optimum there are large flutuations in λ value, this is attributed to that fat that near to optimum the entire population shrinks to a very small region hene both d avg and d best beome very small to give flutuating values of λ. The middle sub-figure shows that the differ- 6
Table : Unimodal and Multi-modal test problems used in the study. Prob. Objetive funtion f Unimodal Problems P f ellipsoidal = n i= ix i P f disus = 4 x + n i= x i P3 f igar = x +4 n i= x i P4 f ridge = x + n ( i= x i P5 f stepellip =.max s / 4, ) n i i= n s i {.5+x i, if x i >.5, s i =.5+x i /, otherwise. Multi-modal Problems P6 f rastrigin = n [ i= x i + ( os(πx i )) ] n P7 f griewangk = 4 i= x i ( ) n i= os xi ( P8 f shaffer = where s i = i + n n i= si + s i sin (5s /5 i ) x i + x i+ ) ene between the average population-distane from entroid and population-best point from entroid is positive, thereby indiating that population-best solution is loser to the entroid of the population throughout the simulation run than a random population member. Eventually with generations, the differene omes lose to zero, indiating the onverging property of the population with generation. The bottom-most sub-figure shows the differene between distane of the entroid from the known optimum and the distane of the population-best solution from optimum. The negative values indiate that the population-best solution is somewhat away from the true optimum than the population-entroid in initial generations, but eventually both distanes beome equal, thereby demonstrating the onverging property of the algorithm. We now study the effet of one-sided initialization, in whih the optimum is not surrounded by the initial population members. Figure shows the performane of all four algorithms. Again, CMA-ES performs the best. The proposed SA- performs better than the original by a few orders of magnitude. Interestingly, the mean-entri version of does not do as well as the original, for the reason given earlier. Sine P is a unimodal problem, in early generations, one-sided initialization makes one 4 4 var ellipsoidal pop= 6 SA 8 3 4 5 6 7 8 Figure : Redution in objetive value for - variable problem P with one-sided initialization. of the boundary points as the population-best point. Hene, λ is expeted to be greater than one early on. Figure verifies this fat. Interestingly, this trend is opposite to that observed in the symmetri initialization ase (Figure ). An inreased λ enables a larger diversity in offspring solutions thereby ausing a faster approah towards the optimum. When the population omes around the optimum, a situation similar to the symmetri initialization happens and the algorithm finds it to be better to redue λ to enable onvergene to the optimum. 7
λ d best d avg d opt best d opt.5.5 5 5 5 3 35 4 5 5 5 5 5 3 35 4 6 4 5 5 5 3 35 4 Generations Figure : Variation of λ for -variable problem P with one-sided initialization. 5-variable and -variable versions of P are also attempted using all four methodologies using symmetri initializations. Table tabulate the results. For the 5-variable problem with 6,47 funtion evaluations, the median objetive value of runs is found to be around 5( 7 ), whereas using CMA-ES a median objetive value of 3( 4 ) is ahieved with an idential number of funtion evaluations. Clearly, CMA-ES has performed better in this problem. Similar results are obtained for -variable version of the problem and also for one-sided initialization. For brevity, they are not disussed here. 5. ProblemsPtoP5 Similar experiments are exeuted on the remaining unimodal test problems. The variation of population-best objetive funtion value for symmetri and one-sided initializations with generation are plotted in Figures 3 and 6, respetively. In both ases, RGA with the proposed SA- outperforms other algorithms. Results on to -variable problems are tabulated in Table. Here, CMA-ES does not perform well partiularly for 5 and -variable problems, for both symmetri and one-sided initializations, however the RGA with proposed SA- performs better than other three algorithms. CMA-ES performs well on P3 (igar problem), however the proposed RGA with SA- outperformed other three methods on P4 and P5, as they are lear from the table. In general, the original fairs better than the mean-entri version of the operator. Based on these extensive results, it an onluded that the RGA with the proposed SA- performs onsistently well on all unimodal test problems used in this study. 5.3 Multi-modal Problems P6 to P8 Figures 4 and 7 show the redution in objetive funtion value with generations for symmetri and one-sided initializations on -variable version of P6. The Rastrigin funtion (P6) has many loal optimum and the proposed SA- operator through its parent-entri to meanentri adaptations helps to avoid all loal optimum even when initialized around x i [, 6] (far away from global basin) and onverge to the global optimum. The variation of λ in a typial run for the one-sided initialization is shown in Figure 9. In all runs, the performane is d avg d best d opt best d opt λ 5 5 5 3 35 4 5 5 5 5 5 3 35 4 5 5 5 5 5 3 35 4 Generations Figure 9: Variation of λ for -variable problem P6 with one-sided initialization. quite similar, thereby demonstrating the robustness of the proposed approah. Based on the urrent loation of population-best and population diversity, a foused or a broader searh takes plae. While onverging to a loal optimum, as soon as a better solution on a better loal optimal basin is found, a large value of λ is obtained and a broader searh takes plae, thereby taking the population out of the urrent loal basin. The flutuating λ value explains this phenomenon. All other algorithms inluding CMA- ES seem to get stuk to a loal optimum. RGAs with the original operator ould never find 8
Table : Objetive funtion value (median of runs) obtained after a fixed alloated number of funtion evaluations for all four algorithms are shown. Funtion Initiali- n Fun. SA- parent-entri mean-entri CMA-ES zation evals. f ellipsoidal [, ] n 5,893.65 6.9 8.39 3.48 5 5 6,47 5.9 7.6 3 8.4.69 4 4,.56 7.5 4 5.39.7 4 [, 6] n 5,989.4 3. 3.9 3.35 4 5 7,5 8.69 5 4.6 3 7.78 4 3.45 5 39,78.34 4 5.96 4.76 5.5 4 f disus [, ] n 8,. 9.6 7.5 5.34 4 5 7,5.6 7 8.99 9.44 3. 37,5.48 7 5.6.6 7.53 [, 6] n,. 8.78.9 4 8.56 9 5,5.74 7.8. 4.5 47,5.54 7.6 3.9 4. f igar [, ] n 6,85.8 4 4.6 3.8 3 8.7 9 5 5,46.77 3 6.8 5 4.54 4 8.95 9 3,6 5.9 3 5. 6.63 5 9.46 9 [, 6] n 6,685.736 5.36 4.48 7 8.65 9 5 6,7.44 3.63 6 4.79 7 9.49 9 3,7 4.33.38 7 6.95 7 9.7 9 f sharpridge [, ] n 9,5 5.94 7..85.3 5 33,75 5.85 7..35 8.99 7, 6.74 7 6.9.55.45 [, 6] n,5 5.7 7.43.74 3 4.85 5 37,5 5.68 7.4.6 3 5.3 8, 6. 7. 3.9 3 3.93 f stepellisoidal [, ] n 3,5. 8. 3.8.63 5 8,75 4.48 7 6.8 6.64.83,. 9.9.6 5.3 [, 6] n 5,5. 8.77. 3 4. 5 3,75 3.7 8.7 3.7 3.45 3,. 9.9 3 4.9 3 8.98 f rastrigin [, ] n,5.54 7 5.7.93.59 5,. 7 3.98 9.97.4 4,5.44 7.3 3 3.6.67 [, 6] n 5,665. 7 6. 8..88 5 6,5.4 7 4.8.6 3 9.95 5,5.6 7.49.57 3.56 f shaffer [, ] n, 3.43 7.49 4.3 8. 5,5 4. 7.8.86 4. 45, 3.77 7 8.83.97.4 [, 6] n 4,5 3.94 7 3.4 3.99 9.65 5 6,5 3.66 7 3.93 4.66.3 5,5 3.59 7.33 3.48 3.4 f griewangk [, ] n 5,59.7 5 6.6 5.6 4.44 5 5 4,46.3 7 9.96.65 6. 5 3,.38 7.7.46 3.6 4 [, 6] n 5,665.4.3.88.66 5 5 4,596.79 5.3.3 9. 5 3,465 9.3 6.59.66 3.7 4 9
5 var disus pop= 4 var rastrigin pop=5 var shaffer pop=5 4 4 5 SA 3 4 5 6 7 8 6 SA 8.5.5.5 3 3.5 4 x 4 6 SA 8 4 6 8 Figure 3: Population-best objetive value for -variable problem P with symmetri initialization. Figure 4: Population-best objetive value for -variable problem P6 with symmetri initialization. Figure 5: Population-best objetive value for -variable problem P8 with symmetri initialization. var disus pop= var rastrigin pop=5 var shaffer pop=5 5 4 4 4 5 SA 4 6 8 6 SA 8 3 4 5 x 4 6 SA 8 4 6 8 4 Figure 6: Population-best objetive value for -variable problem P with one-sided initialization. Figure 7: Population-best objetive value for -variable problem P6 with one-sided initialization. Figure 8: Population-best objetive value for -variable problem P8 with one-sided initialization. the global optimum for this problem so far. Similar results of better performane of the proposed approah are obtained for problem P8 as well (Figures 5 and 8). Table tabulates the median objetive values obtained in runs for both symmetri and one-sided initializations for all three multimodal problems. In all three problems, RGA with adaptive performs onsistently well, whereas CMA-ES performs better only in problem P7 (Griewangk). 6 Conlusions In this paper, we have argued and demonstrated that instead of using a purely parent-entri or a purely mean-entri reombination operator, an adaptive use of them in a real-parameter optimization algorithm is a better overall approah. By modifying the existing simulated binary rossover () operator with a selfadaptive parameter that depends on the loation of population-best solution ompared to the diversity assoiated with the population, a selfadaptive transition from parent-entri to meanentri versions of the reombination operator has been ahieved. On a number of unimodal and multi-modal test problems having, 5, and variables, the real-parameter geneti algorithm (RGA) with SA- has onsistently and reliably found the global minimum in most problems, whereas other algorithms inluding the well-known CMA-ES ould not solve all problems for the alloated number of funtion evaluations. This proof-of-priniple study shows promise
for a more detailed study in whih (i) a number of other more omplex problems an be taken up, (ii) the priniple of self-adaptive transition from parent-entri to mean-entri operations an be extended to multi-objetive optimization problems, (iii) instead of basing the λ parameter on the population-best solution alone, the top 5% or % population members may be onsidered for this purpose. We are urrently pursuing some of these extensions, Nevertheless, this study has shown a lear indiation of the importane of an adaptive approah ompared to either purely parent-entri or purely mean-entri reombination approahes. the Seventh International Conferene on Geneti Algorithms (ICGA-7), pages 46 53, 997. [9] H.-M. Voigt, H. Mühlenbein, and D. Cvetković. Fuzzy reombination for the Breeder Geneti Algorithm. In Proeedings of the Sixth International Conferene on Geneti Algorithms, pages 4, 995. Referenes [] K. Deb and R. B. Agrawal. Simulated binary rossover for ontinuous searh spae. Complex Systems, 9():5 48, 995. [] K. Deb, A. Anand, and D. Joshi. A omputationally effiient evolutionary algorithm for realparameter optimization. Evolutionary Computation Journal, (4):37 395,. [3] K. Deb and H.-G. Beyer. Self-adaptation in realparameter geneti algorithms with simulated binary rossover. In Proeedings of the Geneti and Evolutionary Computation Conferene (GECCO- 99), pages 7 79, 999. [4] K. Deb, K. Sindhya, and T. Okabe. Self-adaptive simulated binary rossover for real-parameter optimization. In Proeedings of the Geneti and Evolutionary Computation Conferene (GECCO- 7), pages 87 94. New York: The Assoiation of Computing Mahinery (ACM), 7. [5] L. J. Eshelman and J. D. Shaffer. Real-oded geneti algorithms and interval-shemata. In Foundations of Geneti Algorithms (FOGA-), pages 87, 993. [6] N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation Journal, 9():59 95,. [7] T. Higuhi, S. Tsutsui, and M. Yamamura. Theoretial analysis of simplex rossover for real-oded geneti algorithms. In Parallel Problem Solving from Nature (PPSN-VI), pages 365 374,. [8] I. Ono and S. Kobayashi. A real-oded geneti algorithm for funtion optimization using unimodal normal distribution rossover. In Proeedings of