A Survey on the Hypervolume Indicator in Evolutionary Multiobjective Optimization

Hypervolume is widely used as a performance indicator in the field of evolutionary multiobjective optimization (EMO). It is used not only for performance evaluation of EMO algorithms (EMOAs) but also in indicator-based EMOAs to guide the search. Since its initial proposal in the late 1990s, a wide variety of studies have been done on various topics, including hypervolume calculation, optimal $\mu $ -distribution, subset selection, hypervolume-based EMOAs, and extensions of the hypervolume indicator. However, currently there is no work to systematically survey the hypervolume indicator for these topics whereas it has been frequently used in the EMO field. This article aims to fill this gap and provide a comprehensive survey on the hypervolume indicator. We expect that this survey will help EMO researchers to understand the hypervolume indicator more deeply and thoroughly, and promote further utilization of the hypervolume indicator in the EMO field.

[1]  Tobias Friedrich,et al.  Approximating the least hypervolume contributor: NP-hard in general, but fast in practice , 2008, Theor. Comput. Sci..

[2]  Carlos M. Fonseca,et al.  Greedy Hypervolume Subset Selection in the Three-Objective Case , 2015, GECCO.

[3]  Frank Neumann,et al.  Multiplicative approximations and the hypervolume indicator , 2009, GECCO.

[4]  Hisao Ishibuchi,et al.  Regular Pareto Front Shape is not Realistic , 2019, 2019 IEEE Congress on Evolutionary Computation (CEC).

[5]  Jingda Deng,et al.  Approximating Hypervolume and Hypervolume Contributions Using Polar Coordinate , 2019, IEEE Transactions on Evolutionary Computation.

[6]  Ferrante Neri,et al.  A fast hypervolume driven selection mechanism for many-objective optimisation problems , 2017, Swarm Evol. Comput..

[7]  Qingfu Zhang,et al.  Combining Simple and Adaptive Monte Carlo Methods for Approximating Hypervolume , 2020, IEEE Transactions on Evolutionary Computation.

[8]  Michael T. M. Emmerich,et al.  Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation , 2013, EMO.

[9]  Thomas Bäck,et al.  Efficient computation of expected hypervolume improvement using box decomposition algorithms , 2019, Journal of Global Optimization.

[10]  Luís Paquete,et al.  An optimal algorithm for a special case of Klee's measure problem in three dimensions , 2006 .

[11]  Hisao Ishibuchi,et al.  Reference point specification in hypervolume calculation for fair comparison and efficient search , 2017, GECCO.

[12]  Lucas Bradstreet,et al.  Maximising Hypervolume for Selection in Multi-objective Evolutionary Algorithms , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[13]  Thomas Bäck,et al.  Multi-Objective Bayesian Global Optimization using expected hypervolume improvement gradient , 2019, Swarm Evol. Comput..

[14]  Kalyanmoy Deb,et al.  An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach , 2014, IEEE Transactions on Evolutionary Computation.

[15]  Hisao Ishibuchi,et al.  A Review of Evolutionary Multi-modal Multi-objective Optimization , 2020, arXiv.org.

[16]  Michael T. M. Emmerich,et al.  Maximum Volume Subset Selection for Anchored Boxes , 2018, SoCG.

[17]  Yang Yu,et al.  Subset Selection by Pareto Optimization , 2015, NIPS.

[18]  Carlos M. Fonseca,et al.  A Fast Dimension-Sweep Algorithm for the Hypervolume Indicator in Four Dimensions , 2012, CCCG.

[19]  Jonathan E. Fieldsend,et al.  Efficient real-time hypervolume estimation with monotonically reducing error , 2019, GECCO.

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

[21]  Tobias Friedrich,et al.  Don't be greedy when calculating hypervolume contributions , 2009, FOGA '09.

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

[23]  Carlos M. Fonseca,et al.  A Box Decomposition Algorithm to Compute the Hypervolume Indicator , 2015, Comput. Oper. Res..

[24]  Carlos A. Coello Coello,et al.  A Study of the Parallelization of a Coevolutionary Multi-objective Evolutionary Algorithm , 2004, MICAI.

[25]  Hisao Ishibuchi,et al.  R2-Based Hypervolume Contribution Approximation , 2018, IEEE Transactions on Evolutionary Computation.

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

[27]  Lucas Bradstreet,et al.  Updating exclusive hypervolume contributions cheaply , 2009, 2009 IEEE Congress on Evolutionary Computation.

[28]  Frank Neumann,et al.  Analyzing Hypervolume Indicator Based Algorithms , 2008, PPSN.

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

[30]  Carlos M. Fonseca,et al.  The Hypervolume Indicator , 2020, ACM Comput. Surv..

[31]  Chao Qian,et al.  Distributed Pareto Optimization for Large-Scale Noisy Subset Selection , 2020, IEEE Transactions on Evolutionary Computation.

[32]  Stefan Roth,et al.  Covariance Matrix Adaptation for Multi-objective Optimization , 2007, Evolutionary Computation.

[33]  Liang Feng,et al.  Towards adaptive weight vectors for multiobjective evolutionary algorithm based on decomposition , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[34]  Carlos M. Fonseca,et al.  Computing Hypervolume Contributions in Low Dimensions: Asymptotically Optimal Algorithm and Complexity Results , 2011, EMO.

[35]  Tobias Glasmachers.  Optimized Approximation Sets for Low-Dimensional Benchmark Pareto Fronts , 2014, PPSN.

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

[37]  Anne Auger,et al.  Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications , 2012, Theor. Comput. Sci..

[38]  Ferrante Neri,et al.  Covariance matrix adaptation pareto archived evolution strategy with hypervolume-sorted adaptive grid algorithm , 2016, Integr. Comput. Aided Eng..

[39]  M. Hansen,et al.  Evaluating the quality of approximations to the non-dominated set , 1998 .

[40]  Michael T. M. Emmerich,et al.  Time Complexity and Zeros of the Hypervolume Indicator Gradient Field , 2012, EVOLVE.

[41]  E. Zitzler,et al.  Directed Multiobjective Optimization Based on the Weighted Hypervolume Indicator , 2013 .

[42]  Tobias Friedrich,et al.  Convergence of hypervolume-based archiving algorithms I: effectiveness , 2011, GECCO '11.

[43]  Tobias Friedrich,et al.  The logarithmic hypervolume indicator , 2011, FOGA '11.

[44]  Iryna Yevseyeva,et al.  On Reference Point Free Weighted Hypervolume Indicators based on Desirability Functions and their Probabilistic Interpretation , 2014 .

[45]  Tobias Friedrich,et al.  Convergence of Hypervolume-Based Archiving Algorithms , 2014, IEEE Transactions on Evolutionary Computation.

[46]  Hisao Ishibuchi,et al.  Use of Two Reference Points in Hypervolume-Based Evolutionary Multiobjective Optimization Algorithms , 2018, PPSN.

[47]  Xin Yao,et al.  A Scalable Indicator-Based Evolutionary Algorithm for Large-Scale Multiobjective Optimization , 2019, IEEE Transactions on Evolutionary Computation.

[48]  Enrique Alba,et al.  On the Effect of the Steady-State Selection Scheme in Multi-Objective Genetic Algorithms , 2009, EMO.

[49]  Michael T. M. Emmerich,et al.  Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels , 2006, IEEE Transactions on Evolutionary Computation.

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

[51]  Tobias Friedrich,et al.  Approximating the volume of unions and intersections of high-dimensional geometric objects , 2008, Comput. Geom..

[52]  Johannes Bader,et al.  Hypervolume-based search for multiobjective optimization: Theory and methods , 2010 .

[53]  Kalyanmoy Deb,et al.  R-HV: A Metric for Computing Hyper-volume for Reference Point Based EMOs , 2014, SEMCCO.

[54]  Anne Auger,et al.  Theoretically Investigating Optimal µ-Distributions for the Hypervolume Indicator: First Results for Three Objectives , 2010, PPSN.

[55]  Carlos M. Fonseca,et al.  Computing and Updating Hypervolume Contributions in Up to Four Dimensions , 2018, IEEE Transactions on Evolutionary Computation.

[56]  Frank Neumann,et al.  Maximizing Submodular Functions under Matroid Constraints by Multi-objective Evolutionary Algorithms , 2014, PPSN.

[57]  Michael T. M. Emmerich,et al.  Faster Exact Algorithms for Computing Expected Hypervolume Improvement , 2015, EMO.

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

[59]  Akira Oyama,et al.  On the fast hypervolume calculation method , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[60]  Michael G. Epitropakis,et al.  Progressive preference articulation for decision making in multi-objective optimisation problems , 2017, Integr. Comput. Aided Eng..

[61]  Hisao Ishibuchi,et al.  A New Framework of Evolutionary Multi-Objective Algorithms with an Unbounded External Archive , 2020 .

[62]  Karl Bringmann,et al.  Two-dimensional subset selection for hypervolume and epsilon-indicator , 2014, GECCO.

[63]  Shengxiang Yang,et al.  On the use of hypervolume for diversity measurement of Pareto front approximations , 2016, 2016 IEEE Symposium Series on Computational Intelligence (SSCI).

[64]  Carlos A. Coello Coello,et al.  A GPU-Based Algorithm for a Faster Hypervolume Contribution Computation , 2015, EMO.

[65]  Joshua D. Knowles,et al.  Bounded archiving using the lebesgue measure , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[66]  Luís M. S. Russo,et al.  Extending quick hypervolume , 2016, J. Heuristics.

[67]  Lucas Bradstreet,et al.  Applying the WFG algorithm to calculate incremental hypervolumes , 2012, 2012 IEEE Congress on Evolutionary Computation.

[68]  R. Lyndon While,et al.  A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume , 2005, EMO.

[69]  Eckart Zitzler,et al.  Integrating decision space diversity into hypervolume-based multiobjective search , 2010, GECCO '10.

[70]  Luís Paquete,et al.  Implicit enumeration strategies for the hypervolume subset selection problem , 2018, Comput. Oper. Res..

[71]  H. Ishibuchi,et al.  A New Hypervolume-Based Evolutionary Algorithm for Many-Objective Optimization , 2019, IEEE Transactions on Evolutionary Computation.

[72]  Elizabeth Montero,et al.  An Improved S-Metric Selection Evolutionary Multi-Objective Algorithm With Adaptive Resource Allocation , 2018, IEEE Access.

[73]  Oliver Schütze,et al.  Hypervolume Maximization via Set Based Newton’s Method , 2014 .

[74]  H. Schwefel,et al.  Approximating the Pareto Set: Concepts, Diversity Issues, and Performance Assessment , 1999 .

[75]  Tobias Friedrich,et al.  An Efficient Algorithm for Computing Hypervolume Contributions , 2010, Evolutionary Computation.

[76]  Carlos M. Fonseca,et al.  Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms , 2016, Evolutionary Computation.

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

[78]  Marco Laumanns,et al.  Running time analysis of multiobjective evolutionary algorithms on pseudo-Boolean functions , 2004, IEEE Transactions on Evolutionary Computation.

[79]  Anne Auger,et al.  Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences , 2009, GECCO.

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

[81]  Hisao Ishibuchi,et al.  Hypervolume approximation using achievement scalarizing functions for evolutionary many-objective optimization , 2009, 2009 IEEE Congress on Evolutionary Computation.

[82]  Qingfu Zhang,et al.  On Tchebycheff Decomposition Approaches for Multiobjective Evolutionary Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[83]  R. Lyndon While,et al.  Improving the IWFG algorithm for calculating incremental hypervolume , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[84]  Carlos M. Fonseca,et al.  Greedy Hypervolume Subset Selection in Low Dimensions , 2016, Evolutionary Computation.

[85]  Frank Neumann,et al.  Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms , 2015, Evolutionary Computation.

[86]  Frank Neumann,et al.  Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point , 2013, Evolutionary Computation.

[87]  Wolfgang Ponweiser,et al.  Multiobjective Optimization on a Limited Budget of Evaluations Using Model-Assisted -Metric Selection , 2008, PPSN.

[88]  Luís M. S. Russo,et al.  Quick Hypervolume , 2012, IEEE Transactions on Evolutionary Computation.

[89]  Dimo Brockhoff,et al.  Optimal µ-Distributions for the Hypervolume Indicator for Problems with Linear Bi-objective Fronts: Exact and Exhaustive Results , 2010, SEAL.

[90]  Michael T. M. Emmerich,et al.  Hypervolume-based expected improvement: Monotonicity properties and exact computation , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[91]  Frank Neumann,et al.  Benefits and drawbacks for the use of epsilon-dominance in evolutionary multi-objective optimization , 2008, GECCO '08.

[92]  Nicola Beume,et al.  An EMO Algorithm Using the Hypervolume Measure as Selection Criterion , 2005, EMO.

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

[94]  Nicola Beume,et al.  Effects of 1-Greedy -Metric-Selection on Innumerably Large Pareto Fronts , 2009, EMO.

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

[96]  Lothar Thiele,et al.  The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration , 2007, EMO.

[97]  Lucas Bradstreet,et al.  A Fast Incremental Hypervolume Algorithm , 2008, IEEE Transactions on Evolutionary Computation.

[98]  Hisao Ishibuchi,et al.  Modified Distance Calculation in Generational Distance and Inverted Generational Distance , 2015, EMO.

[99]  Michael T. M. Emmerich,et al.  On the construction of pareto-compliant quality indicators , 2019, GECCO.

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

[101]  Hisao Ishibuchi,et al.  Performance of Decomposition-Based Many-Objective Algorithms Strongly Depends on Pareto Front Shapes , 2017, IEEE Transactions on Evolutionary Computation.

[102]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[103]  Hisao Ishibuchi,et al.  Selecting a small number of representative non-dominated solutions by a hypervolume-based solution selection approach , 2009, 2009 IEEE International Conference on Fuzzy Systems.

[104]  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..

[105]  Hao Wang,et al.  Hypervolume Indicator Gradient Ascent Multi-objective Optimization , 2017, EMO.

[106]  Gary B. Lamont,et al.  Multiobjective evolutionary algorithms: classifications, analyses, and new innovations , 1999 .

[107]  Adriana Menchaca-Mendez,et al.  A new selection mechanism based on hypervolume and its locality property , 2013, 2013 IEEE Congress on Evolutionary Computation.

[108]  Lucas Bradstreet,et al.  A fast many-objective hypervolume algorithm using iterated incremental calculations , 2010, IEEE Congress on Evolutionary Computation.

[109]  Lucas Bradstreet,et al.  Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems , 2005, 2005 IEEE Congress on Evolutionary Computation.

[110]  Silviu Maniu,et al.  Hypervolume Subset Selection with Small Subsets , 2019, Evolutionary Computation.

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

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

[113]  Hisao Ishibuchi,et al.  Comparison of Hypervolume, IGD and IGD+ from the Viewpoint of Optimal Distributions of Solutions , 2019, EMO.

[114]  Kay Chen Tan,et al.  Fast hypervolume approximation scheme based on a segmentation strategy , 2020, Inf. Sci..

[115]  Mark Fleischer,et al.  The measure of pareto optima: Applications to multi-objective metaheuristics , 2003 .

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

[117]  Carlos M. Fonseca,et al.  An Improved Dimension-Sweep Algorithm for the Hypervolume Indicator , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[118]  Mark H. Overmars,et al.  New upper bounds in Klee's measure problem , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[119]  Hisao Ishibuchi,et al.  Hypervolume Subset Selection for Triangular and Inverted Triangular Pareto Fronts of Three-Objective Problems , 2017, FOGA '17.

[120]  Jie Zhang,et al.  A Simple and Fast Hypervolume Indicator-Based Multiobjective Evolutionary Algorithm , 2015, IEEE Transactions on Cybernetics.

[121]  Andrzej Jaszkiewicz,et al.  Improved quick hypervolume algorithm , 2016, Comput. Oper. Res..

[122]  Nicola Beume,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Gradient-based / Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric , 2007 .

[123]  Dario Izzo,et al.  Empirical Performance of the Approximation of the Least Hypervolume Contributor , 2014, PPSN.

[124]  Hisao Ishibuchi,et al.  A new R2 indicator for better hypervolume approximation , 2018, GECCO.

[125]  Lucas Bradstreet,et al.  Incrementally maximising hypervolume for selection in multi-objective evolutionary algorithms , 2007, 2007 IEEE Congress on Evolutionary Computation.

[126]  Adriana Menchaca-Mendez,et al.  An alternative hypervolume-based selection mechanism for multi-objective evolutionary algorithms , 2017, Soft Comput..

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

[128]  Xin Yao,et al.  R-Metric: Evaluating the Performance of Preference-Based Evolutionary Multiobjective Optimization Using Reference Points , 2018, IEEE Transactions on Evolutionary Computation.

[129]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers: an analysis and review , 2003, IEEE Trans. Evol. Comput..

[130]  Eckart Zitzler,et al.  Evolutionary algorithms for multiobjective optimization: methods and applications , 1999 .

[131]  Heike Trautmann,et al.  Online convergence detection for evolutionary multi-objective algorithms revisited , 2010, IEEE Congress on Evolutionary Computation.

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

[133]  Michel Minoux,et al.  Accelerated greedy algorithms for maximizing submodular set functions , 1978 .

[134]  Nicola Beume,et al.  S-Metric Calculation by Considering Dominated Hypervolume as Klee's Measure Problem , 2009, Evolutionary Computation.

[135]  Hisao Ishibuchi,et al.  Selecting a small number of non-dominated solutions to be presented to the decision maker , 2014, 2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC).

[136]  Hao Wang,et al.  The Set-Based Hypervolume Newton Method for Bi-Objective Optimization , 2020, IEEE Transactions on Cybernetics.

[137]  R. Filomeno Coelho.  Bi-objective hypervolume-based Pareto optimization , 2015 .

[138]  Nicola Beume,et al.  On the Complexity of Computing the Hypervolume Indicator , 2009, IEEE Transactions on Evolutionary Computation.

[139]  Tobias Friedrich,et al.  Approximation quality of the hypervolume indicator , 2013, Artif. Intell..