Patent Grant
7428514
This application is related to U.S. application Ser. No. 11/042,991 filed on Jan. 24, 2005.
The underlying invention generally relates to the field of Estimation of Distribution Algorithms (EDA), especially for optimization problems, including single-objective optimization (SOO) and multi-objective optimization (MOO).
A Hybrid Representation proposed in Okabe, T., Jin, Y. and Sendhoff, B. Evolutionary Multi-Objective Optimisation with a Hybrid Representation, In Proceedings of Congress on Evolutionary Computation (CEC-2003), pages 2262-2269, 2003, which is incorporated by reference in its entirety, is exploiting the different dynamics of Genetic Algorithm (GA) and Evolution Strategy (ES). This difference is caused by the different offspring distribution. The new algorithm shows superior performance compared to the state-of-the-art MOO methods. A natural for the extension of the Hybrid Representation algorithm is to try to model the optimal offspring distribution directly. This new algorithm generates a stochastic model from the offspring distribution in the Parameter Space (PS). Based on this stochastic model, offspring will be generated. Neither crossover nor mutation are used. This type of algorithms is often called Estimation of Distribution Algorithm (EDA).
Genetic Algorithms (GAs) are well known to be powerful tools to obtain optimal solutions for complex optimization problems. As an extension of GAs, recently Estimation of Distribution Algorithms (EDAs) have received considerable attention, see Tsutsui, et al, cited below. The first authors to use the term “EDA” for estimating the distribution of promising points were Mühlenbein and Paass. In the state of the art, several EDAs have been proposed. Since the research for EDAs was triggered by the problem of disruption in GA, a lot of methods are suitable for discrete problems only. However, recently, several papers for the continuous problem have been also published.
According to a recent survey of EDAs, Pelikan, M., Goldberg, D. E. and Lobo, F. A Survey of Optimization by Building and Using Probabilistic Models. Technical Report 99018, University of Illinois, Urbana-Champaign Urbana, Ill. 61801, 1999 which is incorporated by reference herein in its entirety, the proposed methods can be classified into three classes: (1) no interaction, (2) pairwise interaction, and (3) multivariable interaction.
The first class, i.e. no interaction, does not consider any epistasis at all. Thus, each locus is treated independently. The second class, i.e., pairwise interaction, considers pairwise dependency only. Finally, multivariable interaction takes into account any type of dependency between variables.
The classification of the state of the art EDA methods can be seen in the following Table 1-a for discrete domains and in Table 1-b for continuous domains. All of the references cited in these tables are incorporated by reference herein in their entirety.
Since the method proposed by the present invention belongs to the class of multivariable interaction, some of the popular methods in this category will be explained next.
To learn the linkage among parameters and therefore the structure of the problem, Bayesian networks are used. With the Bayesian Networks, the conditional probability is approximated. Each node and connection in the Bayesian networks correspond to the parameters and the conditional probability, respectively. Finally, the factorized probability is used to generate offspring. Recently, this method has been applied to MOO problems and has gathered much attention as per Khan, N., Goldberg, D. E. and Pelikan, M. Multi-objective Bayesian Optimization Algorithm. Technical Report 2002009, Univeristy of Illinois, Urbana-Champaign, Urbana, Ill. 61801, 2002 and Laumanns, M. and Ocenasek, J. Bayesian Optimization Algorithms for Multi-objective Optimization. In Proceedings of Parellel Problem Solving from Nature VII (PPSN-VII), pages 298-307, 2002 which are incorporated by reference herein in their entirety. Iterated Density Estimation Evolutionary Algorithm (IDEA)
Bosman and Thierens have proposed four types of EDAs that all belong to the class of IDEA. The first one is for the discrete domain where the conditional probability is used to build up the stochastic model. The others are for the continuous domain. A normalized Gaussian, a histogram method and a kernel method are used to generate the stochastic model. The kernel based method has been also applied to MOO, called Mixture-based IDEA (MIDEA).
Parzen-Based Estimation of Distribution Algorithm (PEDA)
To generate the stochastic model, a Parzen estimator is used to approximate the probability density of solutions. Based on the stochastic model, new offsprings will be generated. This method has been used for MOO problems.
Marginal Histogram Model (MHM)
For each parameter, the search space is divided into small bins. The ratio of the number of individuals in each bin to the whole number is assigned as the selection probability. With this probability, a bin is selected randomly. In the selected bin, an offspring is generated uniformly.
In view of these references, it is the object of the present invention to propose a more efficient algorithm for optimization.
A new algorithm, which can be called Voronoi-based Estimation of Distribution Algorithm (VEDA), is using a Voronoi mesh to represent the stochastic model. The Voronoi mesh is usually used in another context, i.e. finite element methods (FEMs) for generating the mesh. The mesh covers the parameter space. Each cell of the mesh is represented by its center. The rank of the center is determined by the fitness of the individuals belonging to its cell.
One of the characteristics of VEDA is to exploit all available information. In the canonical EAs and EDAs, only selected (promising) individuals contribute to the generation of offspring. However, even solutions of poor quality can serve as “bad examples”, therefore, they are not ignored but exploited by the technique according to the present invention. In particular, in the context of complex optimization problems with computationally expensive evaluations, it is of importance to use essentially all available information. In the context of EDAs poor solutions are helpful as they induce a low probability of the stochastic model in their neighborhood in the parameter space.
To generate offspring efficiently, clustering and Principal Component Analysis (PCA) are used in VEDA. In particular, in higher dimensional cases, it becomes more time consuming to generate the stochastic model. To reduce the computational cost, both techniques are used.
The basic idea behind the Voronoi-based Estimation of Distribution Algorithm (VEDA) stems from the observations on the intrinsic interplay between the distributions in the parameter space and in the fitness space. On the one hand, it is sensible to directly estimate the most appropriate search distribution for multi-objective optimization problems. Therefore, EDAs is a sensible starting point. On the other hand, both in canonical evolutionary algorithms as well as in EDAs, individuals which are not selected are completely discarded, see
One aspect of the invention relates to the offspring distribution in the Parameter Space (PS) with the information given in the Fitness Space (FS). Based on the offspring, a stochastic model is generated from which the next generation will be determined.
Further advantages of the underlying invention result from the subordinate claims as well as from the following description of two preferred embodiments of the invention which are depicted in the following drawings.
Note that all references to prior are documents in brackets are to be understood as explicitly referring to these documents and incorporating the disclosure thereof by reference. According to the invention, a new algorithm for tackling optimization problems, including Single-Objective Optimization (SOO) and Multi-Objective Optimization (MOO), is proposed, namely the Voronoi-based Estimation of Distribution Algorithm (VEDA).
Based on the offspring distribution in the PS with the information given in the FS, a stochastic model is generated using a Voronoi mesh. With the model, the offspring are generated stochastically.
The basic flow of VEDA is shown in
To generate a new individual, a cell is selected (S4h) based on the assigned probability. After choosing a cell, a new individual is generated (S4i) in the selected cell uniformly. Finally, the new individual is projected (S4j) to the real coordinate system. In a further embodiment of the present invention, a system for optimization problems, including Single-Objective Optimization (SOO) and Multi-Objective Optimization (MOO), comprising means for carrying out the steps of the presented optimization method is proposed. The use of this method can be for optimizing aerodynamic or hydrodynamic designs.
The following items distinguish the VEDA concept according to the present invention over a histogram method with a Voronoi mesh:
Since the way to make the model and generate offspring are issues in EDAs, each step is explained in more detail in the following sections.
Clustering
In the VEDA proposed in the present invention, data clusters are determined in the first step. The reason is that it is easier and more efficient to determine several local stochastic models, i.e. one for each cluster, instead of one global model.
Furthermore,
In the context of the present invention, e.g. k-means clustering proposed by MacQueen in Kaufman, L. and Rousseeuw, P. J. Finding Groups in Data—An Introduction to cluster Analysis. Wiley-Interscience, 1990 can be used, which is incorporated by reference herein in its entirety. The calculation of k-means clustering proceeds via the following steps:
As pointed out in Kaufman et al, the drawback of k-means clustering is the dependency on the order of the data.
To use the k-means clustering, one has to determine the value of k, i.e. the number of clusters. In this work, the value of k will be determined at random within a range of [1, 10]. Better performance can be reached when k is set using some a priori knowledge on the Pareto front in the parameter space, e.g., the number of cluster can be determined by the number of disconnected pieces of the Pareto front. This also indicates why k=1, i.e. no clustering, can be viable choice, e.g., for SCH1.
Principal Component Analysis
To reduce the dimensionality and to generate the model efficiently, Principal Component Analysis (PCA) is used.
The PCA can be explained as follows (see Takahashi, M. and Kita, H. A Crossover Operator Using Independent Component Analysis for Real-coded Genetic Algorithms. In Proceedings of Congress on Evolutionary Computation (CEC-2001), pages 643-649, 2001 which is incorporated by reference herein in its entirety for additional details). Suppose there exist m sets of data of n variables, which are represented by an n×m matrix
X={xij}(i=1, . . . , n, j=1, . . . , m).
The variance-covariance matrix S={sij} of X is given by:
where data are normalized in advance so that the mean of each variable becomes zero. Transforming the data X with an n×m matrix A to Y=AX, the variance-covariance matrix
is given by:
S′=ASAT (Eq.2)
The target of the principal component analysis (PCA) is to find a matrix A that makes S′ the identity matrix. That is, to find a matrix that eliminates the correlation among variables of Y. Since S is a real symmetric matrix, there exists an orthogonal matrix P and a diagonal matrix Λ such that PT SP=Λ. A diagonal matrix Λ whose diagonal elements are eigen values λ1, . . . ,λn of S, and a matrix P whose columns are corresponding eigenvectors are a pair of such matrices. Hence, one obtains:
S′=(AP)Λ(AP)T (Eq. 3)
To make S′ the identity matrix, a transformation matrix A can be obtained as follows:
Assuming all the eigenvalues are positive, we can obtain the inverse transformation A−1 as follows:
A−1=P diag(√{square root over (λ1)}, √{square root over (λ2)}, . . . , √{square root over (λn)}) (Eq. 5)
Perturbation of Offspring Distribution
In the research for Real-Coded Genetic Algorithms (RCGAs), the PCA or Gram-Schmidt orthogonalization are also often used. See Deb, K. A Population-based Algorithm-generator for Real-parameter Optimization. Technical Report 2003003, Indian Institute of Technology, Kanpur: Kanpur Genetic Algorithms Laboratory (KanGAL), Kanpur, PIN 208 016, India, 2003, and Deb, K., Anand, A. and Joshi, D. A Computationally Efficient Evolutionary Algorithms for Real-parameter Optimisation. Evolutionary Computation, 10(4):371-395, 2002, and Deb, K. and Agrawal, R. B. Simulated Binary Crossover for Continuous Search Space. Complex Systems, 9:115-148, 1995, and Deb, K. and Beyer, H. -G. Self-adaptation in Real-parameter Genetic Algorithms with Simulated Binary Crossover. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-1999), pages 172-179, 1999, and Eshelman, L. J. and Schaffer, J. D. Real-coded Genetic Algorithms and Interval-schemata. In Proceedings of Foundations of Genetic Algorithms 2 (FOGA-2), pages 187-202, 1993, and Eshelman, L. J., Mathias, K. E. and Schaffer, J. D. Crossover Operator Biases: Exploiting the Population Distribution. In Proceedings of the Seventh International Conference on Genetic Algorithms, pages 354-361, 1997, and Kita, H., Ono, I. and Kobayashi, S. Multi-parental Extension of the Unimodal Normal Distribution Crossover for Real-coded Genetic Algorithms. In Proceedings of Congress on Evolutionary Computation (CEC-1999), volume 2, pages 1581-1587, 1999, and Ono, I. Genetic Algorithms for Optimization Taking Account of Characteristics Preservation (Japanese). PhD thesis, Tokyo Institute of Technology, 1997, and Ono, I. and Kobayashi, S. A Real-coded Genetic Algorithm for Function Optimization Using Unimodal Normal Distribution Crossover. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 246-253, 1997, and Ono, I., Kita, H. and Kobayashi, S. A Robust Real-Coded Genetic Algorithm using Unimodal Normal Distribution Crossover Augmented by Uniform Crossover: Effects of Self-Adaptation of Crossover Probabilities. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-1999), volume 1, pages 496-503, 1999, and Ono, I., Satoh, H. and Kobayashi, S. A Real-coded Genetic Algorithm for Function Optimization Using the Unimodal Normal Distribution Crossover (Japanese). Journal of Aritificial Intelligence, 14(6):1146-1155, 1999, and Takahashi, M. and Kita, H. A Crossover Operator Using Independent Component Analysis for Real-coded Genetic Algorithms. In Proceedings of Congress on Evolutionary Computation (CEC-2001), pages 643-649, 2001, and Tsutsui, S. Sampling Bias and Search Space Boundary Extension in Real Coded Genetic Algorithms. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-2000), pages 211-218, 2000, and Tsutsui, S. and Goldberg, D. E. Search Space Boundary Extension Method in Real-coded Genetic Algorithms. Information Sciences, 133(3-4):229-247, 2001, and Tsutsui, S. and Goldberg, D. E. Simplex Crossover and Linkage Identification: Single stage Evolution vs. Multi-stage Evolution. In Proceedings of Congress on Evolutionary Computation (CEC-2002), pages 974-979, 2002, and Tsutsui, S., Goldberg, D. E. and Sastry, K. Simplex Crossover and Linkage Learning in Real-coded GAs. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-2001), page 785, 2001, and Tsutsui, S., Yamamura, M. and Higuchi, T. Multi-parent Recombination with Simplex Crossover in Real Coded Genetic Algorithms. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-1999), pages 657-664, 1999, and Ueda, T., Koga, N. and Okamoto, M. Efficient Numerical Optimization Technique Based on Real-coded Genetic Algorithm. Genome Informatics, 12:451-453, 2001. All of these references are incorporated by reference herein in their entirety.
Unimodal Normal Distribution Crossover (UNDX), Bimodal Normal Distribution Crossover (BNDX), Parent-Centric Recombination (PCX), Modified PCX (mPCX) are examples of the usage of orthogonalization. See Ono, I. and Kobayashi, S. A Real-coded Genetic Algorithm for Function Optimization Using Unimodal Normal Distribution Crossover. In Thomas Bäck, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 246-253, 1997, and Ono, I., Kita, H. and Kobayashi, S. A Robust Real-Coded Genetic Algorithm using Unimodal Normal Distribution Crossover Augmented by Uniform Crossover: Effects of Self-Adaptation of Crossover Probabilities. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-1999), volume 1, pages 496-503, 1999, and Takahashi, M. and Kita, H. A Crossover Operator Using Independent Component Analysis for Real-coded Genetic Algorithms. In Proceedings of Congress on Evolutionary Computation (CEC-2001), pages 643-649, 2001, and Ueda, T., Koga, N. and Okamoto, M. Efficient Numerical Optimization Technique Based on Real-coded Genetic Algorithm. Genome Informatics, 12:451-453, 2001, and Deb, K. A Population-based Algorithm-generator for Real-parameter Optimization. Technical Report 2003003, Indian Institute of Technology, Kanpur: Kanpur Genetic Algorithms Laboratory (KanGAL), Kanpur, PIN 208 016, India, 2003 which are all incorporated by reference herein in their entirety. However, the usage of orthogonalization in MOO has problems, see
Clearly, the solution sets are not on the Pareto front. However, they seem to be on similar curves. Taking a closer look, we can find out the reason for this problem. If all solutions are on one line in the parameter space, the search power of the orthogonalization will be lost. A rough image is drawn in
To avoid this undesirable convergence, perturbation is introduced.
In
Voronoi-based Stochastic Model
Voronoi mesh is defined as follows:
Given a set S of m data points in , Voronoi mesh is the partition of
into m polyhedral cells, v(p)(p ∈ S). Each cell v(p), namely Voronoi mesh of p, is defined as the set of points in
which are closer to p than to any other points in S, or more precisely,
v(p)={x ∈; dist(x, p)≦dist(x, q)∀q ∈S\p} (Eq. 6)
where dist( ) is the Euclidean distance.
In the generated area in
To simplify the generation of the Voronoi mesh, a discrete Voronoi mesh is used, see
First of all, a space is divided into small cells, see
The number of cells in each direction, Ds, is determined as:
Ds=ε×|NC| (Eq. 7)
wherein ε and |NC| are respectively a predefined parameter and the number of data in a cluster.
Generate Offspring
In the above section describing the Voronoi-based Stochastic Model, all cells in the generative area were assigned a rank. With the assigned rank, the selection probability is calculated. To calculate the selection probability, the geometry distribution is used. Mathematically, the geometry distribution is calculated as follows:
P=PG(1−PG)r−1 (Eq. 8)
wherein PG and r are the given value in the range [0.0, 1.0] and the rank, respectively. For PG=0.2, 0.5, 0.8, the geometry distributions are drawn in
To generate offspring, a cell is selected according to the above selection probability first. In the selected cell, an offspring is uniformly generated. This flow will be repeated until all necessary offspring are generated.
Comparison of VEDA with NSGA-II
The proposed method of the present invention is tested on the test functions SCH1 (n=2, 5), FON2 (n=2, 5) and OKA4 (n=2). The used parameters are shown in Table 2-a, wherein ε corresponds to the accuracy of a model.
Since one of the merits of VEDA is less fitness evaluations, only 1000 fitness evaluations are allowed. To reduce computational cost in n=5, the accuracy is reduced from ε=1.0 to 0.5.
To compare the results, NSGA-II (non-dominated Sorting Genetic Algorithm II), see Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182-197, 2002, is also tested with only 1000 fitness evaluations. The parameters for NSGA-II are shown in Table 2-b.
The simulation results of SCH1, FON2 and OKA4 for VEDA and NSGA-II are shown in
On the SCH1 and FON2 with 2 dimensional cases, the differences are small. However, VEDA is better than NSGA-II. On the SCH1 and FON2 with 5 dimensional cases, the performance of both is not sufficient. However, clearly VEDA shows a better result than NSGA-II. On the OKA4, the difference is clear. VEDA nearly converges to the Pareto front but NSGA-II does not.
From these results, VEDA shows better performance than NSGA-II in the early generation. This indicates that a real-world optimization problem may be solved without many fitness evaluation by the proposed VEDA. However, VEDA has also drawbacks, e.g. the increase of the computational complexity in higher dimensional cases. Like other EDAs, the current capacity of VEDA is about 10 dimension. Since many real-world optimization problems have a large number of design parameters, the current capacity of EDAs is not sufficient.
Comparison of VEDA with Other EDAs for MOO
Although EDAs are mainly studied on single objective optimization (SOO) problems, recently some methods have applied EDAs to MOO. Thierens and Bosman have applied Mixture-based Iterated Density Estimation Evolutionary Algorithm (MIDEA) to MOO and Costa and Minisci have applied Parzen-based EDA (PEDA) to MOO. Since both of them have used the same test functions, we use these test functions for the comparison. Since MIDEA did not show a sufficient quality on test function ZDT4 (n=10), ZDT4 is not used in this comparison. The maximum number of iterations is set as Table 3. The numbers of data and offspring are 100 and the value of ε is 1.0 for test functions FON2 and KUR1 and 0.1 for test function DEB4. See Thierens, D. and Bosmann, P. A. N. Multi-Objective Mixture-based Iterated Density Estimation Evolutionary Algorithms. In Proceedings of Genetic and Evolutionary Computation Conference (GECCO-2001), pages 663-670, 2001 for MIDEA, which is incorporated by reference herein in its entirety and Costa et al. for PEDA.
The results and a comparison of VEDA with state of the art methods are shown in
Piece-wise Linear Approximation of Pareto Front with VEDA
In the state of the art, nearly all MOO methods output a set of solutions to represent the Pareto front. There is no MOO method to output a mathematical description of the Pareto front in the parameter space. Jin and Sendhoff have proposed a way to represent the Pareto front by piece-wise linear functions in the PS. Since many test functions have a linear Pareto front in the PS, they have exploited this characteristic to get more accurate solutions. See Jin, Y. and Sendhoff, B. Connectedness, Regularity and the Success of Local Search in Evolutionary Multi-Objective Optimization. In Proceedings of Congress on Evolutionary Computation (CEC-2003), volume 3, pages 1910-1917, 2003 which is incorporated by reference herein in its entirety.
One of the implicit characteristics of VEDA is the availability of a mathematical description instead of just a solution set. Since VEDA uses the PCA and the maximum and minimum values in each axis, VEDA can output mathematical description. As an example, the output of VEDA on SCH1 (n=2) is shown in Table 4.
The graphical output is shown in
Summary of the Advantages of the Present Invention
The invention proposes the Voronoi-based Estimation of Distribution Algorithm (VEDA) for tackling MOO problems. In VEDA, the offspring distribution with the ranking information is directly used to generate promising solutions. Using the concept of Voronoi mesh, the stochastic model is made and the promising solutions are generated according to the generated stochastic model.
On several test functions, the performance of VEDA has been compared with a state-of-the-art MOO algorithm called NSGA-II proposed in Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182-197, 2002 which is incorporated by reference herein in its entirety. The performance of VEDA is better than NSGA-II. With a small number of fitness evaluations, VEDA can identify the Pareto front in most cases. This indicates the possibility to reduce the computational cost in real-world problems using VEDA. Although VEDA shows good performance on several test functions, the remaining problem is the increase of the computational complexity in higher dimensional cases. Additionally, VEDA should be modified to consider additional constraints.
In the VEDA, several parameters are introduced. Since the performance of the VEDA seems to be stable against the parameters, the fixed values are used in this invention. However, the parameter dependency should be taken into account. Although most of MOO methods in the state of the art output only solution set, VEDA can output a mathematical description.
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