A radial space division based evolutionary algorithm for many-objective optimization

Abstract In evolutionary many-objective optimization, diversity maintenance plays an important role in pushing the population towards the Pareto optimal front. Existing many-objective evolutionary algorithms mainly focus on convergence enhancement, but pay less attention to diversity enhancement, which may fail to obtain uniformly distributed solutions or fall into local optima. This paper proposes a radial space division based evolutionary algorithm for many-objective optimization, where the solutions in high-dimensional objective space are projected into the grid divided 2-dimensional radial space for diversity maintenance and convergence enhancement. Specifically, the diversity of the population is emphasized by selecting solutions from different grids, where an adaptive penalty based approach is proposed to select a better converged solution from the grid with multiple solutions for convergence enhancement. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on a variety of benchmark test problems. Experimental results demonstrate the competitiveness of the proposed algorithm in terms of both convergence enhancement and diversity maintenance.

[1]  Georges G. Grinstein,et al.  DNA visual and analytic data mining , 1997 .

[2]  Marco Laumanns,et al.  Combining Convergence and Diversity in Evolutionary Multiobjective Optimization , 2002, Evolutionary Computation.

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

[4]  Jun Zhang,et al.  Particle Swarm Optimization With a Balanceable Fitness Estimation for Many-Objective Optimization Problems , 2018, IEEE Transactions on Evolutionary Computation.

[5]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[6]  Sanaz Mostaghim,et al.  Heatmap Visualization of Population Based Multi Objective Algorithms , 2007, EMO.

[7]  Markus Wagner,et al.  Evolutionary many-objective optimization: A quick-start guide , 2015 .

[8]  Shengxiang Yang,et al.  A Grid-Based Evolutionary Algorithm for Many-Objective Optimization , 2013, IEEE Transactions on Evolutionary Computation.

[9]  Kalyanmoy Deb,et al.  An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints , 2014, IEEE Transactions on Evolutionary Computation.

[10]  Ye Tian,et al.  An improved reference point sampling method on Pareto optimal front , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[11]  Ye Tian,et al.  A Decision Variable Clustering-Based Evolutionary Algorithm for Large-Scale Many-Objective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[12]  Ye Tian,et al.  An Efficient Approach to Nondominated Sorting for Evolutionary Multiobjective Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[13]  Bernhard Sendhoff,et al.  A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[14]  John W. Sammon,et al.  A Nonlinear Mapping for Data Structure Analysis , 1969, IEEE Transactions on Computers.

[15]  M. Ashby.  MULTI-OBJECTIVE OPTIMIZATION IN MATERIAL DESIGN AND SELECTION , 2000 .

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

[17]  Shengxiang Yang,et al.  A Strength Pareto Evolutionary Algorithm Based on Reference Direction for Multiobjective and Many-Objective Optimization , 2017, IEEE Transactions on Evolutionary Computation.

[18]  Peter J. Fleming,et al.  Preference-Inspired Coevolutionary Algorithms for Many-Objective Optimization , 2013, IEEE Transactions on Evolutionary Computation.

[19]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

[20]  Matthew O. Ward,et al.  High Dimensional Brushing for Interactive Exploration of Multivariate Data , 1995, Proceedings Visualization '95.

[21]  Ye Tian,et al.  A multi-objective evolutionary algorithm based on an enhanced inverted generational distance metric , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[22]  Gary G. Yen,et al.  Many-Objective Evolutionary Algorithm: Objective Space Reduction and Diversity Improvement , 2016, IEEE Transactions on Evolutionary Computation.

[23]  Peter J. Fleming,et al.  Many-Objective Optimization: An Engineering Design Perspective , 2005, EMO.

[24]  Patrick M. Reed,et al.  Many‐objective groundwater monitoring network design using bias‐aware ensemble Kalman filtering, evolutionary optimization, and visual analytics , 2011 .

[25]  Xin Yao,et al.  Many-Objective Evolutionary Algorithms , 2015, ACM Comput. Surv..

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

[27]  Yiu-Ming Cheung,et al.  Self-Organizing Map-Based Weight Design for Decomposition-Based Many-Objective Evolutionary Algorithm , 2018, IEEE Transactions on Evolutionary Computation.

[28]  Teuvo Kohonen,et al.  The self-organizing map , 1990 .

[29]  José C. Matos,et al.  MOEA/PC: Multiobjective Evolutionary Algorithm Based on Polar Coordinates , 2015, EMO.

[30]  Gary G. Yen,et al.  Visualization and Performance Metric in Many-Objective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[31]  Hao Hu,et al.  Risk design optimization using many-objective evolutionary algorithm with application to performance-based wind engineering of tall buildings , 2014 .

[32]  Slim Bechikh,et al.  A New Decomposition-Based NSGA-II for Many-Objective Optimization , 2018, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[33]  Jonathan E. Fieldsend,et al.  Visualizing Mutually Nondominating Solution Sets in Many-Objective Optimization , 2013, IEEE Transactions on Evolutionary Computation.

[34]  Kalyanmoy Deb,et al.  A combined genetic adaptive search (GeneAS) for engineering design , 1996 .

[35]  Gaoping Wang,et al.  Fuzzy-Dominance and Its Application in Evolutionary Many Objective Optimization , 2007, 2007 International Conference on Computational Intelligence and Security Workshops (CISW 2007).

[36]  R. K. Ursem.  Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[37]  Shengxiang Yang,et al.  The Supplement of the Paper “ Multi-Line Distance Minimization : A Visualized Many-Objective Test Problem Suite ” , 2017 .

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

[39]  Nicola Beume,et al.  SMS-EMOA: Multiobjective selection based on dominated hypervolume , 2007, Eur. J. Oper. Res..

[40]  Martin J. Oates,et al.  PESA-II: region-based selection in evolutionary multiobjective optimization , 2001 .

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

[42]  Shahryar Rahnamayan,et al.  3D-RadVis: Visualization of Pareto front in many-objective optimization , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[43]  Qingfu Zhang,et al.  Combining Model-based and Genetics-based Offspring Generation for Multi-objective Optimization Using a Convergence Criterion , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[44]  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..

[45]  Tapabrata Ray,et al.  A Decomposition-Based Evolutionary Algorithm for Many Objective Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[46]  R. Lyndon While,et al.  A faster algorithm for calculating hypervolume , 2006, IEEE Transactions on Evolutionary Computation.

[47]  Carlos A. Coello Coello,et al.  Evolutionary multi-objective optimization: a historical view of the field , 2006, IEEE Comput. Intell. Mag..

[48]  Xin Yao,et al.  Two_Arch2: An Improved Two-Archive Algorithm for Many-Objective Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[49]  Qingfu Zhang,et al.  A Self-Organizing Multiobjective Evolutionary Algorithm , 2016, IEEE Transactions on Evolutionary Computation.

[50]  Soon-Thiam Khu,et al.  An Investigation on Preference Order Ranking Scheme for Multiobjective Evolutionary Optimization , 2007, IEEE Transactions on Evolutionary Computation.

[51]  Miqing Li,et al.  Benchmark Functions for the CEC'2017 Competition on Many-Objective Optimization , 2017 .

[52]  Xin Yao,et al.  Diversity Assessment in Many-Objective Optimization , 2017, IEEE Transactions on Cybernetics.

[53]  Mario Köppen,et al.  Substitute Distance Assignments in NSGA-II for Handling Many-objective Optimization Problems , 2007, EMO.

[54]  Peter J. Fleming,et al.  A Real-World Application of a Many-Objective Optimisation Complexity Reduction Process , 2013, EMO.

[55]  Tea Tusar,et al.  Visualization of Pareto Front Approximations in Evolutionary Multiobjective Optimization: A Critical Review and the Prosection Method , 2015, IEEE Transactions on Evolutionary Computation.

[56]  M. Farina,et al.  On the optimal solution definition for many-criteria optimization problems , 2002, 2002 Annual Meeting of the North American Fuzzy Information Processing Society Proceedings. NAFIPS-FLINT 2002 (Cat. No. 02TH8622).

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

[58]  Geoffrey E. Hinton,et al.  Visualizing Data using t-SNE , 2008 .

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

[60]  Carlos A. Coello Coello,et al.  Improved Metaheuristic Based on the R2 Indicator for Many-Objective Optimization , 2015, GECCO.

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

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

[63]  Ye Tian,et al.  A region division based diversity maintaining approach for many-objective optimization , 2017, Integr. Comput. Aided Eng..

[64]  Ye Tian,et al.  A Knee Point-Driven Evolutionary Algorithm for Many-Objective Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[65]  Bo Zhang,et al.  Balancing Convergence and Diversity in Decomposition-Based Many-Objective Optimizers , 2016, IEEE Transactions on Evolutionary Computation.

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