A diversity indicator based on reference vectors for many-objective optimization

Abstract Diversity estimation of Pareto front (PF) approximations is a critical issue in the field of evolutionary multiobjective optimization. However, the existing diversity indicators are usually inappropriate for PF approximations with more than three objectives. Many of them can be utilized only when compared with approximations obtained by multiple multiobjective optimizers, which makes them difficult to use online. In this paper, we propose a unary diversity indicator based on reference vectors (DIR) to estimate the diversity of PF approximations for many-objective optimization. In DIR, a set of uniform and widespread reference vectors are generated. The coverage of each solution in the objective space is evaluated by the number of representative reference vectors it is associated with. The diversity (both spread and uniformity) is determined by the standard deviation of the coverage for all the solutions. The smaller value of DIR, the better the diversity of a PF approximation is. DIR can be applied to a unary approximation without any compared approximations needed. Thus, DIR is easy to use as either an offline indicator to estimate the performance of an optimizer or an online indicator for the selection of solutions in a MOEA. In the experimental studies, both the artificial and the real PF approximations generated by seven different many-objective algorithms are used to verify DIR as an offline indicator. The effects of the number of reference vectors on DIR are also investigated. In addition, as an online indicator, DIR is integrated into a Pareto-dominance-based evolutionary multiobjective optimizer, NSGA-II. The experimental studies show it has the significant performance enhancements over the original NSGA-II on many-objective optimization problems.

[1]  Lothar Thiele,et al.  On Set-Based Multiobjective Optimization , 2010, IEEE Transactions on Evolutionary Computation.

[2]  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.

[3]  Dirk Thierens,et al.  The balance between proximity and diversity in multiobjective evolutionary algorithms , 2003, IEEE Trans. Evol. Comput..

[4]  Xin Yao,et al.  Performance Scaling of Multi-objective Evolutionary Algorithms , 2003, EMO.

[5]  Nicola Beume,et al.  Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization , 2007, EMO.

[6]  Kalyanmoy Deb,et al.  Reliability-Based Optimization Using Evolutionary Algorithms , 2009, IEEE Transactions on Evolutionary Computation.

[7]  Carlos A. Coello Coello,et al.  Study of preference relations in many-objective optimization , 2009, GECCO.

[8]  Jinhua Zheng,et al.  Spread Assessment for Evolutionary Multi-Objective Optimization , 2009, EMO.

[9]  Sanghamitra Bandyopadhyay,et al.  Multiobjective GAs, quantitative indices, and pattern classification , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[10]  Zhong Zhi-hua,et al.  MULTI-OBJECTIVE OPTIMIZATION FOR CRASH SAFETY DESIGN OF VEHICLES USING STEPWISE REGRESSION MODEL , 2007 .

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

[12]  Hisao Ishibuchi,et al.  A many-objective test problem for visually examining diversity maintenance behavior in a decision space , 2011, GECCO '11.

[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]  Yaochu Jin,et al.  A Critical Survey of Performance Indices for Multi-Objective Optimisation , 2003 .

[15]  Tong Heng Lee,et al.  Evolutionary Algorithms for Multi-Objective Optimization: Performance Assessments and Comparisons , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

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

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

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

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

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

[21]  Qingfu Zhang,et al.  A Constrained Decomposition Approach With Grids for Evolutionary Multiobjective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

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

[23]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[24]  Jürgen Teich,et al.  A New Approach on Many Objective Diversity Measurement , 2005, Practical Approaches to Multi-Objective Optimization.

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

[26]  Shengxiang Yang,et al.  Diversity Comparison of Pareto Front Approximations in Many-Objective Optimization , 2014, IEEE Transactions on Cybernetics.

[27]  Kalyanmoy Deb,et al.  Running performance metrics for evolutionary multi-objective optimizations , 2002 .

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

[29]  Kalyanmoy Deb,et al.  Multiobjective optimization , 1997 .

[30]  Carlos A. Coello Coello,et al.  Multi-Objective Ant Colony Optimization: A Taxonomy and Review of Approaches , 2011, Integration of Swarm Intelligence and Artificial Neural Network.

[31]  David W. Corne,et al.  Techniques for highly multiobjective optimisation: some nondominated points are better than others , 2007, GECCO '07.

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

[33]  Shengxiang Yang,et al.  Shift-Based Density Estimation for Pareto-Based Algorithms in Many-Objective Optimization , 2014, IEEE Transactions on Evolutionary Computation.

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

[35]  Lothar Thiele,et al.  Quality Assessment of Pareto Set Approximations , 2008, Multiobjective Optimization.

[36]  Kay Chen Tan,et al.  Online Diversity Assessment in Evolutionary Multiobjective Optimization: A Geometrical Perspective , 2015, IEEE Transactions on Evolutionary Computation.

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

[38]  Qingfu Zhang,et al.  An External Archive Guided Multiobjective Evolutionary Algorithm Based on Decomposition for Combinatorial Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[39]  Anne Auger,et al.  Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point , 2009, FOGA '09.

[40]  David Corne,et al.  The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[41]  Shapour Azarm,et al.  Metrics for Quality Assessment of a Multiobjective Design Optimization Solution Set , 2001 .

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

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

[44]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[45]  Jinhua Zheng,et al.  Uniformity assessment for evolutionary multi-objective optimization , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[46]  Jouni Lampinen,et al.  GDE3: the third evolution step of generalized differential evolution , 2005, 2005 IEEE Congress on Evolutionary Computation.

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

[48]  Alfred Inselberg,et al.  Parallel Coordinates: Visual Multidimensional Geometry and Its Applications , 2003, KDIR.

[49]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.