Multi-Objective Optimization with Iterated Density Estimation Evolutionary Algorithms using Mixture Models

Transcription

1 Multi-Objective Optimization with Iterated Density Estimation Evolutionary Algorithms using Mixture Models Dirk Thierens Institute of Information and Computing Sciences Utrecht University, The Netherlands Peter Bosman Institute of Information and Computing Sciences Utrecht University, The Netherlands Abstract We propose an algorithm for multi-objective optimization based on iterated density estimation evolutionary algorithms (M IDE A). By making use of the Pareto dominance concept in the selection step, the algorithm searches for the Pareto front. The M IDE A algorithm builds a mixture distribution to model the dependencies in the population at each generation. This not only allows for a powerful representation of complicated dependencies but also for an elegant way to keep the diversity in the population, needed to cover the Pareto front. As a specific example we implement an algorithm with a normal mixture model applied to real valued multi-objective optimization problems. Introduction In classical evolutionary computation search is driven by two interacting processes: selection focuses the search to more promising points in the search space while mutation and crossover try to generate new and better points from these selected solutions. Efficient exploration requires that some information of what makes the parents good solutions needs to be transfered to the offspring solutions. If there were no correlation between the fitness of the parents and the offspring the search process would essentially be an unbiased random walk. Whether or not information is passed between parents and offspring depends on the representation and accompanying exploration operators. For mutation this is usually accomplished by letting it take small randomized steps in the local neighbourhood of the parent solution. Crossover recombines parts of two parent solutions which results in a more globally oriented exploration step. However, this broader exploration requires a careful choice of genotype representation and crossover operator. A common practice in the design of evolutionary search algorithms is to develop a number of representations and operators by using prior domain knowledge, and picking the best after a considerable amount of experimental runs. An alternative to this labour intensive task is to try to learn the structure of the search landscape automatically, an approach often called linkage learning (see for instance [6]). In a similar effort to learn the structure of the problem representation a number of researchers have taken a more probabilistic view of the evolutionary search process ([9,,, 4, 6, 7, 8, 9, 2, 22, 23, 24]). The general idea here is to build a probabilistic model of the current population and learn the structure of the problem representation by inducing the dependence structure of the problem variables. The exploration operators mutation and crossover are now replaced by generating new samples according to this probabilistic model (for a survey see [2]). In [] we have given a general algorithmic framework for this paradigm called iterated density estimation evolutionary algorithm (IDEA). In this paper we will propose an algorithm for multi-objective optimization within the IDEA framework, called M IDE A. The probabilistic model build is a mixture distribution that not only gives us a powerful and computationally tractable representation to model the dependencies in the population, but also provides us with an elegant way to keep the diversity in the population, needed to cover the Pareto front. In the next section we will discuss some concepts from multiobjective evolutionary computation. Thereafter we will introduce the multi-objective mixture-based IDEA: M IDE A. Finally we give an example by implementing an algorithm with a normal mixture model and applying it to real valued multi-objective optimization problems. 2 Multi-objective optimization Optimization is generally considered to be a search process for optimal or near optimal solutions in some search space where it is implicitly assumed that given any arbitrary solution one can always tell which solution is preferred. However in practice there are many problems where such a single preference criterion does not exist. Moreover it is not always possible to rank the solutions when only considering the optimization objectives. The problem is that the objective functions are opposing each other, meaning that an optimal solution to all objectives cannot be found by simply optimizing each objective independently from the others. In multiobjective optimization problems different objective functions have to be optimized simultaneously. A typical example of a multi-objective optimization problem is the design of a product where one objective is the quality of the design and another objective is its cost. Minimizing the cost will in general lead to products of inferior quality, while maximizing the quality will give rise to an increasing cost. A key characteristic of multi-objective optimization problems is the existence of whole sets of solutions that cannot be ordered in terms of preference when only considering the objective function values. To formalize this we define a number of relevant concepts. Suppose we have a problem with k objective functions f i (x);i =:::k which - without loss of generality - should all be minimized.

2 . Pareto dominance: a solution x is said to dominate a solution y (or x B y) iff8i 2 ;:::;kg : f i (x)» f i (y) V 9i 2;:::;kg : f i (x) <f i (y): 2. Pareto optimal: a solution x is said to be Pareto optimal : y B x: 3. Pareto optimal set: is the set PS of all Pareto optimal solutions: PS = fx : y B xg: 4. Pareto front: is the set PF of objective function values of all Pareto optimal solutions: PF = ff(x) = (f (x);:::;f k (x)) j x 2PSg: Multi-objective problems have been tackled with different solution strategies:. The most straightforward approach is to transform the multi-objective problem into a single-objective optimization problem by aggregating the different objectives into one single criterion. This is typically done by taking a weighted sum of the objectives. The great advantage of this approach is that one can simply use any single-objective optimization algorithm to solve it. The result however is very dependent on the choice of weighting factors. In addition it is hard - or even impossible - to find certain points on the Pareto front when this front is not convex. 2. A second solution strategy is to optimize according to a single objective during one cycle and to switch between the objectives during the whole optimization process. Here too one can use any single-objective optimization algorithm. A disadvantage is that the search tends to focus and switch between the extreme positions on the Pareto front where one of the single objectives is optimal. 3. A third strategy to solve multi-objective optimization problems is to search for the Pareto front - or for a representative set of Pareto optimal solutions - by making use of the Pareto dominance concept. The idea is to maintain a population of solutions that cover the entire Pareto front. The notion of searching a search space through maintaining a population of solutions is a key characteristic of evolutionary algorithms, which makes them natural candidates as multi-objective optimization algorithms. The field of evolutionary multi-objective optimization has indeed seen an explosive growth in recent years (for a survey see [8],[]). 3 The multi-objective mixture-based IDE A In this section we describe the algorithmic component of this paper. First, we review the IDEA framework as applied in single-criterion optimization problems. In section 3.2, we formalize the IDEA framework. We then note how we can find a conditional factorization as a structure, given a vector of selected samples in section 3.3, and show how we can use the normal pdf in conditional factorizations in section 3.4. Clustered structures are proposed in section 3. by using a mixture distribution as probabilistic model. Finally, we introduce the mixture-based multi-objective optimization algorithm M IDE A in section The IDEA The IDEA is a framework for Iterated Density Estimation Evolutionary Algorithms that use probabilistic models in evolutionary optimization [, 2]. Before we can formalize it, we need to define probabilistic models. To this end, we take the elementary building block of probabilistic models to be the probability density function (pdf). We define a probabilistic model M to consist of some structure & and a vector of parameters for the pdfs implied by &. We write M =(&; ). We write the probability distribution that is described by M as PM. The pdf to fit over every factor implied by & is chosen on beforehand. The way in which the parameters are fit, is also predefined on beforehand. We denote the parameter vector that is obtained in this manner by ψ fit &. With these assumptions, we denote the resulting probability distribution by P &. We assume that the dimensionality of our problem is l and write L = (; ;:::; l ). We furthermore assume that we have a continuous optimization problem C(y ;y ;:::;y l ) =C(yhLi). Without loss of generality, we assume that we want to minimize C(yhLi). For every problem variable y i, we introduce a continuous random variable Y i and get Y = Y hli. Without any prior information on C(yhLi), we might as well take a uniform distribution over Y (assuming a bounded continuous search space). Therefore, we generate an initial (population) vector of n samples at random. Now we let P (Y) be a probability distribution that is uniform over all vectors yhli with C(yhLi)». Sampling from P (Y) gives more samples that evaluate to a value below. Moreover, if we know Λ = min yhlifc(yhli)g, a single sample gives an optimal solution. To use this in an iterated algorithm, we select bfinc samples in each iteration t and let t be the worst selected sample cost. We then estimate the distribution of the selected samples and thereby find ^P & t (Y) as an approximation to the true distribution P t (Y). t New samples can then be drawn from ^P & (Y) and be used to replace some of the current samples. 3.2 Algorithmic framework The definition of the IDEA framework is given in the pseudocode figure. In the IDEA, we have N fi = (; ;:::;bfinc ), fi 2 [ ; ], sel() is the selection operator, rep() replaces a subset of P with a subset of O, ter() is the termination condi- n tion, sea() is a structure search algorithm, est() estimates the model parameters and sam() generates a single sample using the estimated pdfs.

3 If we set m to (n bfinc), sel() to selection by taking the best bfinc vectors and rep() to replacing the worst (n bfinc) vectors by the new samples, we have that t+ = t " with ". This assures that the search for Λ is conveyed through a monotonically decreasing series ::: t. We call an IDEA with m, sel() and rep() so chosen, end a monotonic IDEA. IDEA(n,fi,m,sel(),rep(),ter(),sea(),est(),sam()) Initialize an empty vector of samples P ψ () 2 Add and evaluate n random samples for i ψ to n do 2. P ψ P t NEWRANDOMVECTOR() 2.2 c[p i ] ψ C(P i ) 3 Initialize the iteration counter t ψ 4 Iterate until termination while :ter() do 4. Select bfinc samples (y hli;y hli;:::;y bfinc hli) ψ sel() 4.2 Set t to the worst selected cost t ψ c[y k hli] such that 8 i2nfi hc[y i hli]» c[y k hli]i 4.3 Search for a structure & & ψ sea() 4.4 Estimate the parameters ψ fit & ψ est() 4. Create an empty vector of new samples O ψ () 4.6 Draw m new samples from ^P & (Y) for i ψ to m do 4.6. O ψ O t sam() 4.7 Replace a part of P with a part of O rep() 4.8 Evaluate the new samples in P f or each unevaluated P i do 4.8. c[p i ] ψ C(P i ) 4.9 Update the iteration counter t ψ t + Denote the required iterations by t end t end ψ t 3.3 Learning conditional factorizations In step 4:3 of the IDEA, we search for a structure. In this section, we only focus on the specific structure of a conditional factorization. A factorization implies a product of pdfs. If such a product is a function of variables yhai, it is a valid structure over random variables Y hai. We can state that variable Y i is either conditionally dependent on variables Y hai with i 62 a, or it is not. By identifying a vertex with each variable Y i and and arc (Y i ;Y j ) if and only if Y j is conditionally dependent on Y i, we get the conditional factorization graph. A conditional factorization is valid if and only if its factorization graph is acyclic. The resulting probability distribution is the product of l conditional pdfs in which each variable is conditioned on at most l parents. We have to learn such a conditional factorization from the vector of selected samples. To this end, a variety of approaches can be taken [2]. We use an incremental algorithm that starts from the empty graph with no arcs. Each iteration, the arc to add is selected as the arc that increases some metric the most. If no addition of any arc further increases the metric, the final factorization graph has been found. The metric that we use in this paper, is commonly known as the Bayesian Information Criterion (BIC). For a derivation of this metric in the IDEA context, we refer the reader to a more detailed report [3]. Let S = (y ; y ;:::;y jsj ) be the selected vector of l dimensional samples. The BIC metric is parameterized by a regularization parameter that determines the amount of penalization of more complex models in order to favour more simple models. The BIC metric is defined by: ln(l(sj ^PM(Y))) z } z } + ln(jsj)j j Error( ^PM(Y)jS) Complexity( ^PM(Y)jS) In equation, we have used the likelihood of M: 3.4 Gaussian pdfs L(Sj ^PM(Y)) = jsj Y i= () ^PM(Y)(y i ) (2) A widely used parametric pdf is the normal pdf. sample average in dimension j is Y j = jsj The P jsj i= (y i ) j. The sample covariance matrix over variables Y hai is S = P jsj i= (y i hai Y hai)(y i hai Y hai) T. Let s ij = jsj S (i; j). To compute the BIC measure, we require to evaluate the likelihood. However, the average negative logarithm of the likelihood can be seen to be equal to the entropy of the normal pdf [4]. For this pdf, the entropy can be evaluated significantly faster than the likelihood. The required conditional pdf and the entropy can be stated as follows [2]: fn (y a jyha a i)= where p (ya μ) 2 ff 2ß e ( ff = p s μ = Ya s P jaj i= (ya i Ya i )s i s 2ff 2 (3) h(y hai) = 2 (jaj + ln((2ß)jaj det(s))) (4) 3. Factorization mixtures by clustering The structure of the sample vector may be highly non linear. This non linearity can force us to use probabilistic models of a high complexity to retain some of this non linearity. However, especially using relatively simple pdfs such as the normal pdf, the non linear interactions cannot be captured even

4 with higher order models. The key issue is the use of clusters. The use of clusters allows us to efficiently break up non linear interactions so that we can use simple models to get an adequate representation of the sample vector. Furthermore, computationally efficient clustering algorithms exist that provide useful results. A short survey [3] has shown that the randomized euclidic leader algorithm is both fast as well as useful for IDEAs. Each cluster is processed separately in order to have a probability distribution fit over it. We let k be the amount of clusters and let K =(; ;:::;k ). In general, we write f for a factorization. For a mixture of factorizations, we write fhki. The resulting probability distribution is a weighted sum of the individual probability distributions over each cluster: ^P fhki(y) = jkj X i= fi i ^P i f i (Y) () An effective way to set the mixture coefficients fi i,isto proportionally assign larger coefficients to the clusters with a better average cost. By taking the absolute value of the difference of the average cluster cost and the average initial sample vector cost, we allow for both maximization as well as minimization problems. In the randomized euclidic leader algorithm, we use the normalized Euclidean distance measure. The leader algorithm is one of the fastest clustering algorithms. The first sample to make a new cluster is appointed to be its leader. The leader algorithm goes over the sample vector exactly once. For each sample it encounters, it finds the first cluster that has a leader being closer to the sample than a given threshold T d. If no such cluster can be found, a new cluster is created containing only this single sample. To prevent the first clusters from becoming quite a lot larger than the later ones, we randomize the order in which the clusters are inspected. To avoid problems with two clusters to which some samples are equally close, the order in which the clusters are scanned is randomized as well. 3.6 Multi-objective mixture-based IDE A The algorithm discussed so far is still a single-objective optimization algorithm. To change it into a multi-objective Pareto covering optimization algorithm we need to make the following changes:. First, we have to search for the Pareto-front: in step 4: of the IDEA framework selection picks out the best bfinc samples. Making this selection on the basis of Pareto dominance allows us to search for the Pareto front. First all non-dominated solutions are selected and removed from the population. Next the following non-dominated solutions are picked out, and this process continues until bfinc samples are taken. 2. Second, we have to cover the Pareto-front: maintaining diversity or niches is needed to prevent the population to converge to a single Pareto optimal point instead of to form a representative set of the entire front. Since the mixture-based IDEA already constructs a set of clusters we can simply use this to maintain the diversity. This can be done in the parameter space or in the objective space. In the following section we will investigate the validity of the proposed strategy by implementing and testing a multiobjective mixture-based IDEA. The mixture distribution considered here is a mixture of gaussian pdfs and is tested on continuous multi-objective optimization problems. Note that the choice of a mixture of gaussians is just a specific instantiation of the algorithm. Other problem domains might suggest the use of other mixture models. For instance for the discrete multi-objective knapsack problem we have proposed and tested a mixture of discrete univariate factorizations and a mixture of trees ([], [2]). Another possibility would be the use of mixtures of Bayesian networks ([26]) but it can be questioned whether this is still computationally feasible. 4 Experimental results The continuous multi objective optimization problems we used for testing are the following (minimization of both objectives in all cases): Name Objectives Domain f = e P l y i= i 2 p l MOP 2 MOP 3 f = e P l i= f = +(A B ) 2 + (A B ) 2 f = (y +3) 2 +(y +) 2 y i+ p l 2 [ 4; 4] 3 A = g( ; ; 2; ; ; 2; 2 ; 2 2) A = g( 2 ; ; ; ; 2; 2; 2 ; 2) B = g( 2 ;y; 2;y; ;y; 2 ;y) = g( 2 ;y; ;y; 2;y; 2 ;y) B g(x) =x sin(x ) x 2cos(x 3)+ x 4sin(x ) x 6cos(x 7) f P q l 2 = i= e :2 y 2 i +y2 i+ MOP 4 f P l = i= jy ij :8 +sin(y 3 i ) f = y EC 4 f = fl q y fl EC 6 fl P l = 9 + i= f = e 4y sin 6 (6ßy i) = fl ( f f ) fl Pl fl = +9 i= y 2 i cos(4ßy i) :2 y i 9 [ ß; ß] 2 [ ; ] 3 [ ; ] [ ; ] 9 [; ] These functions have been taken from the multi-objective evolutionary optimization literature ([7],[2]). In all our testing, we used monotonic IDEAs. We used the rule of thumb by Mühlenbein and Mahnig [6] for FDA and set fi to :3. We fix the selection size to bfinc =2. Selection is performed by truncation on the domination count. Clustering is applied using the leader algorithm in both the objective space as well as the parameter space. In both cases, we computed the fi i

5 MOP2 MOP3 MOP4 EC4 EC6 Obj Par None Figure : Average amount of evaluations f Figure 4: Result for MOP 4 (objective clustering) f Figure 2: Result for MOP 2 (objective clustering). 2 run run run 2 run 3 run 4 run run 6 run 7 run 8 run 9 based upon the proportional domination count. For the objective space, we set T d =2, giving approximately 3 clusters. For the parameter space, we set T d to such a value that 2 we again get approximately 3 clusters. For comparison, we also tested an approach using no clustering. We searched for conditional factorizations using the BIC metric with =. 2 In this preliminary study, termination is enforced when the domination count of all of the selected samples equals. At such a point, the selected sample vector contains only non dominated solutions. Note that this does not have to imply at all that full convergence has been obtained since the front itself may not be optimal. To prevent the alternative of allowing an arbitrary amount of generations or evaluations, a good termination criterion might be when none of the selected samples is dominated by any of the selected samples in the previous generation. For now, we restrict ourselves to the simple termination criterion, keeping in mind that premature convergence is possible. No single run was allowed more than evaluations. We have performed independent runs and determined the final front from the combined results. The average amount of required evaluations for each type of clustering is stated in figure. In figures 2, 3 and 4, the results using objective clustering on MOP 2, MOP 3 and MOP 4 are shown respectively Figure : All runs for EC 4 (objective clustering). For each of these three problems, none of the individual runs differ significantly from the combined result. Moreover, the results of parameter clustering as well as no clustering at all are also similar to these results, so we omit further graphs for these three problems. The table in figure indicates that using the IDEA approach requires only a few evaluations to adequately solve the three MOP problems. Compared to EC 4, the three MOP problems are relatively simple. Converging to the optimal front is very difficult in EC 4. In figure, we can see that we indeed require a vastly larger amount of evaluations. Only when we cluster in the objective space, do we on average require less than the maximum of evaluations. However, closely observing the results points out that premature convergence has often taken place in the algorithm with objective clustering. For all of the tested algorithms, the results differ quite largely amongst the different runs. Therefore, we have plotted the f run run run 2 run 3 run 4 run run 6 run 7 run 8 run f Figure 3: Result for MOP 3 (objective clustering) f Figure 6: All runs for EC 4 (parameter clustering).

6 2 run run run 2 run 3 run 4 run run 6 run 7 run 8 run f Figure 7: All runs for EC 4 (no clustering) Objective Clustering Parameter Clustering f Figure 9: Result for EC 6 (objective clustering) f Figure 8: Additional results for EC combined results for objective clustering, parameter clustering and no clustering in figures, 6 and 7 respectively. It becomes clear that clustering in the objective space is by far the most effective in evaluation requirements as well as optimization performance. From the results on EC 4 given so far, it seems that parameter clustering is not very effective either. We note however that the amount of clusters can have an influence on the optimization performance. Taking more clusters in combination with a larger population size to effectively fill up and use these additional clusters, can lead to a good estimation of the promising regions of the multi objective space. To illustrate this, we have plotted the resulting fronts after runs for objective clustering with bfinc = and T d =: and for parameter clustering with bfinc =2T d =4:743. This leads to approximately clusters for objective clustering and 7 clusters for parameter clustering. The results in figure 8 show that very good results are obtained with these settings. For objective clustering the average amount evaluations is 94746, whereas for parameter clustering we find an average of 8242 evaluations. Note that we again have premature convergence because of the termination criterion. Given that the results of no clustering do not really improve with an increasing population size and the maximum of evaluations, we conclude that clustering of some sort is crucial in order to be able to effectively tackle difficult problems such as EC 4. The main difficulty with problem EC 6 is that the optimal front is not uniformly distributed in its solutions. Without clustering, we are therefore very likely to find only a part of Figure : Result for EC 6 (parameter clustering) f Figure : Additional result for EC 6 using a larger population (parameter clustering). f

7 the front. Furthermore, by clustering in parameter space, we also have no guarantee to find a good representation of the front since the parameter space is directly related to the density of the points along the front. On the other hand, clustering this space does give a means of capturing more regions than a single cluster can. If we are to cluster in the objective space, we should have no problem finding a larger part of the front unless the problem itself is very difficult as is the case for instance with EC 4. In figures 9 and, the results for objective clustering and parameter clustering are shown respectively. Using parameter clustering is clearly not effective. Using no clustering, we found results identical to those of parameter clustering, no matter how large the population size. However, by increasing the population size to get bfinc = and setting T d = 4:473, wefind the front in figure with an average of 2839 evaluations. Even though the results are not as good as those of objective clustering, it does show the notable effect of clustering on multi objective optimization. It should also be noted that the Pareto front found in figure 9 (and to a lesser degree in figure ) seems to coincide with the optimal Pareto front, which is not trivial to achieve since the fast elitist non-dominated sorting GA (NSGA-II [2]), the strength Pareto Evolutionary Algorithm (SPEA [27]), and the Pareto-archived evolution strategy (PAES [3]) are all reported to converge to a sub-optimal front ([2]). In the experiments so far, the structure learned at each generation is a conditionally factorized gaussian probability density function. This structure has the advantage of being capable to learn conditional dependences between variables, while at the same time being computationally efficient enough to be applied at each generation. Without detailed knowledge about the fitness function it is not possible to tell whether this structure is optimal. For instance it might well be possible that the fitness function can be optimized without the need to learn the conditional dependences between variables. In this case it would be computationally more efficient to use a probability density structure that ignores the interactions between the variables. To get a feeling of the impact of this choice we have optimized the functions EC 4 and EC 6 with a mixture of univariate factorizations. A population size of bfinc = 2, resp. bfinc =(fi =:3) was used, with a cluster threshold =., resulting in 3 to clusters. Clustering was done in the objective space, and a total of runs were performed. Figures 2 and 3 show that the Pareto front found is of similar quality than in the previous experiment using conditionally factorized gaussian pdfs with objective clustering. The average amount of function evaluations for EC 4 was 2963, while for EC 6 the number was These figures are substantially lower than those found before (see figure ), indicating that for these functions the computational effort spent by learning a more powerful and complicated model seems to be unnecessary. It should be noted that this gives only a rough impression about convergence speed and quality of the algorithms. Future studies will have to look at the influence of population size, selection threshold, and cluster size f Figure 2: Result for EC 4 with univariate factorization (objective clustering) f Figure 3: Result for EC 6 with univariate factorization (objective clustering). Conclusion We have proposed a multi-objective iterated density estimation evolutionary algorithm. The algorithm builds a mixture distribution as probabilistic model which allows to maintain the necessary diversity to cover the Pareto front in a sound statistical way. As a specific instantiation of the proposed algorithm we have implemented an example with gaussian mixture models. Preliminary experimental results show the validity of the proposed method on a set of continuous multi objective optimization problems taken from the multiobjective evolutionary optimization literature. Bibliography [] P.A.N. Bosman and D. Thierens. An algorithmic framework for density estimation based evolutionary algorithms. Utrecht Univ. Tech. Rep. UU CS ftp://ftp.cs.uu.nl/pub/ruu/cs/techreps/cs-999/ ps.gz, 999. [2] P.A.N. Bosman and D. Thierens. Expanding from discrete to continuous estimation of distribution algorithms: The IDEA. In M. Schoenauer, K. Deb, G. Rudolph, X. Yao, E. Lutton, J.J. Merelo, and H.-P. Schwefel, editors, Parallel Problem Solving from Nature PPSN VI, pages Springer, 2. [3] P.A.N. Bosman and D. Thierens. Mixed IDEAs. Utr. Univ. Tech. Rep. UU CS 2 4. ftp://ftp.cs.uu.nl/ pub/ruu/cs/techreps/cs-2/2-4.ps.gz, 2. [4] P.A.N. Bosman and D. Thierens. Negative log likelihood and statistical hypothesis testing as the basis of model selec-

8 tion in IDEAs. Utrecht University Tech. Rep. UU CS ftp://ftp.cs.uu.nl/pub/ruu/ CS/techreps/CS-2/2-36.ps.gz, 2. [] C.A. Coello Coello. A comprehensive survey of evolutionarybased multiobjective optimization techniques. Knowledge and Information Systems, (3):269 38, 999. [6] D.E. Goldberg, K. Deb, H. Kargupta, and G. Harik. Rapid, accurate optimization of difficult problems using fast messy genetic algorithm. In S. Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms ICGA- 93, pages Morgan Kaufmann, 993. [7] K. Deb. Multi-objective genetic algorithms: Problem difficulties and construction of test problems. Evolutionary Computation, 7(3):2 23, 999. [8] C.M. Fonseca and P.J. Fleming. An overview of evolutionary algorithms in multiobjective optimization. Evolutionary Computation, 3(): 6, 99. [9] M. Gallagher, M. Fream, and T. Downs. Real valued evolutionary optimization using a flexible probability density estimator. In W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, and R.E. Smith, editors, Proc. of the GECCO 999 Genetic and Evolutionary Computation Conference, pages Morgan Kaufmann Pub., 999. [] G. Harik. Linkage learning via probabilistic modeling in the ECGA. IlliGAL Tech. Report 99. ftp://ftpilligal.ge.uiuc.edu/pub/papers/illigals/99.ps.z, 999. [] G. Harik, F. Lobo, and D.E. Goldberg. The compact genetic algorithm. In Proc. of the 998 IEEE Int. Conf. on Evolutionary Computation, pages IEEE Press, 998. [2] K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan. A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: Nsga-ii. In M. Schoenauer et al., editor, Parallel Problem Solving from Nature PPSN VI, pages Springer, 2. [3] J. Knowles and D. Corne. The pareto archived evolution strategy: a new baseline algorithm for multi-objective optimisation. In A. Zalzala et al., editor, Proceedings of the 999 Congress on Evolutionary Computation (CEC-99), pages 98. IEEE Press, 999. [4] P. Larranaga, R. Etxeberria, J. Lozano, and J. Pena. Optimization by learning and simulation of bayesian and gaussian networks. TR EHU-KZAA-IK-4-99, 999. [] M. Meila and M.I. Jordan. Estimating dependency structure as a hidden variable. In M.I. Jordan et al., editor, Proceedings of Neural Information Processing Systems, pages MIT Press, 998. [6] H. Mühlenbein and T. Mahnig. FDA a scalable evolutionary algorithm for the optimization of additively decomposed functions. Evol. Comp., 7:33 376, 999. [7] H. Mühlenbein, T. Mahnig, and O. Rodriguez. Schemata, distributions and graphical models in evolutionary optimization. Journal of Heuristics, :2 247, 999. [8] H. Mühlenbein and G. Paaß. From recombination of genes to the estimation of distributions i. binary parameters. In A.E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel, editors, Parallel Problem Solving from Nature PPSN V, pages Springer, 998. [9] M. Pelikan and D.E. Goldberg. Genetic algorithms, clustering, and the breaking of symmetry. In M. Schoenauer, K. Deb, G. Rudolph, X. Yao, E. Lutton, J.J. Merelo, and H.-P. Schwefel, editors, Parallel Problem Solving from Nature PPSN VI, pages Springer, 2. [2] M. Pelikan, D.E. Goldberg, and E. Cantú-Paz. BOA: The bayesian optimization algorithm. In W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, and R.E. Smith, editors, Proc. of the GECCO 999 Genetic and Evolutionary Computation Conference, pages Morgan Kaufmann Pub., 999. [2] M. Pelikan, D.E. Goldberg, and F. Lobo. A survey of optimization by building and using probabilistic models. IlliGAL Tech. Rep ftp://ftp-illigal. ge.uiuc.edu/pub/papers/illigals/998.ps.z, 999. [22] M. Pelikan and H. Mühlenbein. The bivariate marginal distribution algorithm. In R. Roy, T. Furuhashi, K. Chawdry, and K. Pravir, editors, Advances in Soft Computing Engineering Design and Manufacturing. Springer Verlag, 999. [23] S. Baluja and S. Davies. Using optimal dependency trees for combinatorial optimization: Learning the structure of the search space. In D.H. Fisher, editor, Proc. of the 997 Int. Conf. on Machine Learning. Morgan Kauffman Pub., 997. [24] M. Sebag and A. Ducoulombier. Extending population based incremental learning to continuous search spaces. In A.E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel, editors, Parallel Problem Solving from Nature PPSN V, pages Springer, 998. [2] D. Thierens and P.A.N. Bosman. Multi-objective mixturebased iterated density estimation evolutionary algorithms. submitted, 2. [26] B. Thiesson. Learning mixtures of bayesian networks. Microsoft Tech. Report MSR-TR-97-3, 998. [27] E. Zitzler and L. Thiele. Multiobjective optimization using evolutionary algorithms a comparative case study. In A.E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel, editors, Parallel Problem Solving from Nature PPSN V, pages Springer, 998.