KDT-MOEA: A multiobjective optimization framework based on K-D trees

Abstract This paper presents KDT-MOEA, a framework that takes advantage of a special kind of binary search tree, named K-D tree, to solve multiobjective optimization problems (MOPs). Our main goal is to explore the capabilities of this data structure to define neighborhood structures either in decision variables space or in objective space, as well as by switching between them at any time. The KDT-MOEA framework performance is compared with five state-of-the-art algorithms on the DTLZ, WFG and LZ09 benchmarking problems with up to 15 objectives. Statistical tests demonstrate that KDT-MOEA was able to outperform the compared methods on most problems. In addition, in order to evaluate the flexibility and the potential of the proposed operators, extended versions of the compared algorithms are also presented. Empirical results pointed out that the new versions were superior to all five original MOEAs, indicating that the proposed operators can also be easily incorporated into existing MOEAs with different strategies to achieve better results.

[1]  Qingfu Zhang,et al.  Decomposition-Based Algorithms Using Pareto Adaptive Scalarizing Methods , 2016, IEEE Transactions on Evolutionary Computation.

[2]  Peter J. Fleming,et al.  Genetic Algorithms for Multiobjective Optimization: FormulationDiscussion and Generalization , 1993, ICGA.

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

[4]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[5]  Peter J. Fleming,et al.  Evolutionary many-objective optimisation: an exploratory analysis , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[6]  Dipti Srinivasan,et al.  A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition , 2017, IEEE Transactions on Evolutionary Computation.

[7]  Charles Gide,et al.  Cours d'économie politique , 1911 .

[8]  Ricardo H. C. Takahashi,et al.  On the Performance Degradation of Dominance-Based Evolutionary Algorithms in Many-Objective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[9]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

[10]  Peter J. Fleming,et al.  Generalized decomposition and cross entropy methods for many-objective optimization , 2014, Inf. Sci..

[11]  H. Scheffé.  Experiments with Mixtures , 1958 .

[12]  Marco Laumanns,et al.  SPEA2: Improving the Strength Pareto Evolutionary Algorithm For Multiobjective Optimization , 2002 .

[13]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

[14]  Eckart Zitzler,et al.  HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization , 2011, Evolutionary Computation.

[15]  Kiyoshi Tanaka,et al.  Ieee Transactions on Evolutionary Computation Computational Cost Reduction of Non-dominated Sorting Using the M-front , 2022 .

[16]  Antonio J. Nebro,et al.  A Study of the Parallelization of the Multi-Objective Metaheuristic MOEA/D , 2010, LION.

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

[18]  Qingfu Zhang,et al.  Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II , 2009, IEEE Transactions on Evolutionary Computation.

[19]  Eckart Zitzler,et al.  Indicator-Based Selection in Multiobjective Search , 2004, PPSN.

[20]  Ye Tian,et al.  PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum] , 2017, IEEE Computational Intelligence Magazine.

[21]  Jon Louis Bentley,et al.  Multidimensional binary search trees used for associative searching , 1975, CACM.

[22]  Tome Eftimov,et al.  A Novel Approach to statistical comparison of meta-heuristic stochastic optimization algorithms using deep statistics , 2017, Inf. Sci..

[23]  Hisao Ishibuchi,et al.  How to Specify a Reference Point in Hypervolume Calculation for Fair Performance Comparison , 2018, Evolutionary Computation.

[24]  Qingfu Zhang,et al.  Are All the Subproblems Equally Important? Resource Allocation in Decomposition-Based Multiobjective Evolutionary Algorithms , 2016, IEEE Transactions on Evolutionary Computation.

[25]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[26]  Yanxia Zhang,et al.  The Application of kd-tree in Astronomy , 2008 .

[27]  Mikkel T. Jensen,et al.  Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms , 2003, IEEE Trans. Evol. Comput..

[28]  Kalyanmoy Deb,et al.  Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms , 1994, Evolutionary Computation.

[29]  Tsung-Che Chiang,et al.  MOEA/D-AMS: Improving MOEA/D by an adaptive mating selection mechanism , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[30]  Gary B. Lamont,et al.  Multiobjective Evolutionary Algorithms: Analyzing the State-of-the-Art , 2000, Evolutionary Computation.

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

[32]  Andrzej Jaszkiewicz,et al.  ND-Tree-Based Update: A Fast Algorithm for the Dynamic Nondominance Problem , 2016, IEEE Transactions on Evolutionary Computation.

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

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

[35]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems , 2002, Genetic Algorithms and Evolutionary Computation.

[36]  Carlos A. Brizuela,et al.  A survey on multi-objective evolutionary algorithms for many-objective problems , 2014, Computational Optimization and Applications.

[37]  Qingfu Zhang,et al.  Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition , 2009 .

[38]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[39]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[40]  Peter J. Fleming,et al.  On the Evolutionary Optimization of Many Conflicting Objectives , 2007, IEEE Transactions on Evolutionary Computation.

[41]  Andreas Nüchter,et al.  Collision detection between point clouds using an efficient k-d tree implementation , 2015, Adv. Eng. Informatics.

[42]  Qingfu Zhang,et al.  MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition , 2007, IEEE Transactions on Evolutionary Computation.

[43]  R. Storn,et al.  Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces , 2004 .

[44]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[45]  Maria Luisa Sapino,et al.  Data Management for Multimedia Retrieval , 2010 .