An investigation of clustering strategies in many-objective optimization: the I-Multi algorithm as a case study

A variety of general strategies have been applied to enhance the performance of multi-objective optimization algorithms for many-objective optimization problems (those with more than three objectives). One of these strategies is to split the solutions to cover different regions of the search space (clusters) and apply an optimizer to each region with the aim of producing more diverse solutions and achieving a better distributed approximation of the Pareto front. However, the effectiveness of clustering in this context depends on a number of issues, including the characteristics of the objective functions. In this paper we show how the choice of the clustering strategy can greatly influence the behavior of an optimizer. We investigate the relation between the characteristics of a multi-objective optimization problem and the efficiency of the use of a clustering combination (clustering space, metric) in the resolution of this problem. Using as a case study the Iterated Multi-swarm (I-Multi) algorithm, a recently introduced multi-objective particle swarm optimization algorithm, we scrutinize the impact that clustering in different spaces (of variables, objectives and a combination of both) can have on the approximations of the Pareto front. Furthermore, employing two difficult multi-objective benchmarks of problems with up to 20 objectives, we evaluate the effect of using different metrics for determining the similarity between the solutions during the clustering process. Our results confirm the important effect of the clustering strategy on the behavior of multi-objective optimizers. Moreover, we present evidence that some problem characteristics can be used to select the most effective clustering strategy, significantly improving the quality of the Pareto front approximations produced by I-Multi.

[1]  H. Kita,et al.  Failure of Pareto-based MOEAs: does non-dominated really mean near to optimal? , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[2]  Lamia Benameur,et al.  A New Hybrid Particle Swarm Optimization Algorithm for Handling Multiobjective Problem Using Fuzzy Clustering Technique , 2009, 2009 International Conference on Computational Intelligence, Modelling and Simulation.

[3]  Qingfu Zhang,et al.  An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition , 2015, IEEE Transactions on Evolutionary Computation.

[4]  Aurora Trinidad Ramirez Pozo,et al.  C-Multi: A competent multi-swarm approach for many-objective problems , 2016, Neurocomputing.

[5]  Martin Pelikan,et al.  Hierarchical Bayesian optimization algorithm: toward a new generation of evolutionary algorithms , 2010, SICE 2003 Annual Conference (IEEE Cat. No.03TH8734).

[6]  Maurice K. Wong,et al.  Algorithm AS136: A k-means clustering algorithm. , 1979 .

[7]  Tobias Friedrich,et al.  Speeding up many-objective optimization by Monte Carlo approximations , 2013, Artif. Intell..

[8]  Kalyanmoy Deb,et al.  Faster Hypervolume-Based Search Using Monte Carlo Sampling , 2008, MCDM.

[9]  Bernhard Sendhoff,et al.  Voronoi-based estimation of distribution algorithm for multi-objective optimization , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[10]  Charu C. Aggarwal,et al.  On the Surprising Behavior of Distance Metrics in High Dimensional Spaces , 2001, ICDT.

[11]  Samir W. Mahfoud.  Niching methods for genetic algorithms , 1996 .

[12]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation) , 2006 .

[13]  Aurora Trinidad Ramirez Pozo,et al.  Effective Linkage Learning Using Low-Order Statistics and Clustering , 2009, IEEE Transactions on Evolutionary Computation.

[14]  Lucas Bradstreet,et al.  A Fast Way of Calculating Exact Hypervolumes , 2012, IEEE Transactions on Evolutionary Computation.

[15]  Enrique Alba,et al.  SMPSO: A new PSO-based metaheuristic for multi-objective optimization , 2009, 2009 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making(MCDM).

[16]  Ching-Shih Tsou,et al.  An Improved Particle Swarm Pareto Optimizer with Local Search and Clustering , 2006, SEAL.

[17]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..

[18]  Kalyanmoy Deb,et al.  A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II , 2000, PPSN.

[19]  Stefan M. Rüger,et al.  Fractional Distance Measures for Content-Based Image Retrieval , 2005, ECIR.

[20]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[21]  Kalyanmoy Deb,et al.  Improved Pruning of Non-Dominated Solutions Based on Crowding Distance for Bi-Objective Optimization Problems , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[22]  Gian Mauricio Fritsche,et al.  Capturing Relationships in Multi-objective Optimization , 2015, 2015 Brazilian Conference on Intelligent Systems (BRACIS).

[23]  Donald W. Bouldin,et al.  A Cluster Separation Measure , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[24]  Concha Bielza,et al.  Multiobjective Estimation of Distribution Algorithm Based on Joint Modeling of Objectives and Variables , 2014, IEEE Transactions on Evolutionary Computation.

[25]  Jonathan A. Wright,et al.  SOLUTION ANALYSIS IN MULTI-OBJECTIVE OPTIMIZATION , 2012 .

[26]  Aurora Trinidad Ramirez Pozo,et al.  Using archiving methods to control convergence and diversity for Many-Objective Problems in Particle Swarm Optimization , 2012, 2012 IEEE Congress on Evolutionary Computation.

[27]  David E. Goldberg,et al.  Genetic Algorithms, Clustering, and the Breaking of Symmetry , 2000, PPSN.

[28]  Gary G. Yen,et al.  Diversity-based Information Exchange among Multiple Swarms in Particle Swarm Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[29]  Dirk Thierens,et al.  Multi-objective optimization with diversity preserving mixture-based iterated density estimation evolutionary algorithms , 2002, Int. J. Approx. Reason..

[30]  Carlos A. Coello Coello,et al.  Solving Multiobjective Optimization Problems Using an Artificial Immune System , 2005, Genetic Programming and Evolvable Machines.

[31]  Hisao Ishibuchi,et al.  Behavior of EMO algorithms on many-objective optimization problems with correlated objectives , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[32]  Marco Laumanns,et al.  Scalable multi-objective optimization test problems , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[33]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[34]  Carlos A. Coello Coello,et al.  Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization , 2012, IEEE Transactions on Evolutionary Computation.

[35]  Marco Laumanns,et al.  Stochastic convergence of random search methods to fixed size Pareto front approximations , 2011, Eur. J. Oper. Res..

[36]  MengChu Zhou,et al.  An adaptive particle swarm optimization method based on clustering , 2015, Soft Comput..

[37]  Marco Laumanns,et al.  Scalable Test Problems for Evolutionary Multiobjective Optimization , 2005, Evolutionary Multiobjective Optimization.

[38]  Aurora Trinidad Ramirez Pozo,et al.  Iterated multi-swarm: a multi-swarm algorithm based on archiving methods , 2013, GECCO '13.

[39]  R. E. Lee,et al.  Distribution-free multiple comparisons between successive treatments , 1995 .

[40]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[41]  J. A. Hartigan,et al.  A k-means clustering algorithm , 1979 .

[42]  Salem Fawaz Adra,et al.  Improving Convergence, Diversity and Pertinency in Multiobjective Optimisation , 2007 .

[43]  Carlos A. Coello Coello,et al.  Using Clustering Techniques to Improve the Performance of a Multi-objective Particle Swarm Optimizer , 2004, GECCO.

[44]  Bernhard Sendhoff,et al.  Reducing Fitness Evaluations Using Clustering Techniques and Neural Network Ensembles , 2004, GECCO.

[45]  Janez Demsar,et al.  Statistical Comparisons of Classifiers over Multiple Data Sets , 2006, J. Mach. Learn. Res..

[46]  Fang Liu,et al.  A Multiobjective Evolutionary Algorithm Based on Decision Variable Analyses for Multiobjective Optimization Problems With Large-Scale Variables , 2016, IEEE Transactions on Evolutionary Computation.

[47]  Pavel Berkhin,et al.  A Survey of Clustering Data Mining Techniques , 2006, Grouping Multidimensional Data.

[48]  Pedro Larrañaga,et al.  Learning Factorizations in Estimation of Distribution Algorithms Using Affinity Propagation , 2010, Evolutionary Computation.

[49]  David E. Goldberg,et al.  Multiobjective hBOA, clustering, and scalability , 2005, GECCO '05.

[50]  M. Friedman.  The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1937 .

[51]  Elena Deza,et al.  Encyclopedia of Distances , 2014 .

[52]  W. Kruskal,et al.  Use of Ranks in One-Criterion Variance Analysis , 1952 .

[53]  Jürgen Teich,et al.  Covering Pareto-optimal fronts by subswarms in multi-objective particle swarm optimization , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[54]  Kalyanmoy Deb,et al.  A Hybrid Framework for Evolutionary Multi-Objective Optimization , 2013, IEEE Transactions on Evolutionary Computation.

[55]  Noël Richard,et al.  A Comprehensive Evaluation of Spectral Distance Functions and Metrics for Hyperspectral Image Processing , 2015, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

[56]  Peter J. Rousseeuw,et al.  Clustering by means of medoids , 1987 .

[57]  Masaharu Munetomo,et al.  Linkage Identification by Fitness Difference Clustering , 2006, Evolutionary Computation.

[58]  Hisao Ishibuchi,et al.  Evolutionary many-objective optimization: A short review , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[59]  R. Lyndon While,et al.  A review of multiobjective test problems and a scalable test problem toolkit , 2006, IEEE Transactions on Evolutionary Computation.

[60]  Dun-Wei Gong,et al.  Handling multi-objective optimization problems with a multi-swarm cooperative particle swarm optimizer , 2011, Expert Syst. Appl..

[61]  Qiuming Zhang,et al.  An Improved Multi-Objective Particle Swarm Optimization Algorithm , 2007, ISICA.

[62]  Pedro Larrañaga,et al.  Applying genetic algorithms to search for the best hierarchical clustering of a dataset , 1999, Pattern Recognit. Lett..