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A Hypervolume-Based Optimizer for High-Dimensional Objective Spaces

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Book cover New Developments in Multiple Objective and Goal Programming

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 638))

Abstract

In the field of evolutionary multiobjective optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance. This property is of high interest and relevance for multiobjective search involving a large number of objective functions. However, the high computational effort required for calculating the indicator values has so far prevented to fully exploit the potential of hypervolume-based multiobjective optimization. This paper addresses this issue and proposes a fast search algorithm that uses Monte Carlo sampling to approximate the exact hypervolume values. In detail, we present HypE (Hypervolume Estimation Algorithm for Multiobjective Optimization), by which the accuracy of the estimates and the available computing resources can be traded off; thereby, not only many-objective problems become feasible with hypervolume-based search, but also the runtime can be flexibly adapted. The experimental results indicate that HypE is highly effective for many-objective problems in comparison to existing multiobjective evolutionary algorithms.

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Notes

  1. 1.

    Here, a Pareto set approximation may also contain dominated solutions as well as duplicates, in contrast to the notation in Zitzler et al. (2003).

  2. 2.

    For reasons of simplicity, we will use the term “u weakly dominates v” resp. “u dominates v” independently of whether u and v are elements of X, Z, or Ψ. For instance, A weakly dominates b with A ∈ Ψ and b ∈ X means A ≼ b and a dominates z with a ∈ X and z ∈ Z means f(a) ≤ z ∧ z ≰ f(a).

  3. 3.

    The number of decision variables (first value in parenthesis) and their decomposition into position (second value) and distance variables (third value) as used by the WFG test function for different number of objectives are: 2d (24, 4, 20); 3d (24, 4, 20); 5d (50, 8, 42); 7d (70, 12, 58); 10d (59,9,50); 25d (100, 24, 76); 50d (199,49,150).

References

  • Bader J, Zitzler E (2008) HypE: an algorithm for fast hypervolume-based many-objective optimization. TIK Report 286, Computer Engineering and Networks Laboratory (TIK), ETH Zurich, November 2008

    Google Scholar 

  • Bader J, Deb K, Zitzler E (2008) Faster hypervolume-based search using Monte Carlo sampling. In: Conference on multiple criteria decision making (MCDM 2008). Springer, Berlin

    Google Scholar 

  • Beume N, Rudolph G (2006) Faster S-metric calculation by considering dominated hypervolume as Klee’s measure problem. Technical Report CI-216/06, Sonderforschungsbereich 531 Computational Intelligence, Universität Dortmund. Shorter version published at IASTED international conference on computational intelligence (CI 2006)

    Google Scholar 

  • Beume N, Fonseca CM, Lopez-Ibanez M, Paquete L, Vahrenhold J (2007a) On the complexity of computing the hypervolume indicator. Technical Report CI-235/07, University of Dortmund, December 2007

    Google Scholar 

  • Beume N, Naujoks B, Emmerich M (2007b) SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur J Oper Res 181:1653–1669

    Article  Google Scholar 

  • Bleuler S, Laumanns M, Thiele L, Zitzler E (2003) PISA – a platform and programming language independent interface for search algorithms. In: Fonseca CM et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2003). LNCS, vol 2632. Springer, Berlin, pp 494–508

    Chapter  Google Scholar 

  • Bradstreet L, Barone L, While L (2006) Maximising hypervolume for selection in multi-objective evolutionary algorithms. In: Congress on evolutionary computation (CEC 2006), pp 6208–6215, Vancouver, BC, Canada. IEEE

    Google Scholar 

  • Bringmann K, Friedrich T (2008) Approximating the volume of unions and intersections of high-dimensional geometric objects. In: Hong SH, Nagamochi H, Fukunaga T (eds) International symposium on algorithms and computation (ISAAC 2008), LNCS, vol 5369. Springer, Berlin, pp 436–447

    Google Scholar 

  • Brockhoff D, Zitzler E (2007) Improving hypervolume-based multiobjective evolutionary algorithms by using objective reduction methods. In: Congress on evolutionary computation (CEC 2007), pp 2086–2093. IEEE Press

    Google Scholar 

  • Conover WJ (1999) Practical nonparametric statistics, 3rd edn. Wiley, New York

    Google Scholar 

  • Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester, UK

    Google Scholar 

  • Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Schoenauer M et al (eds) Conference on parallel problem solving from nature (PPSN VI). LNCS, vol 1917. Springer, Berlin, pp. 849–858

    Chapter  Google Scholar 

  • Deb K, Thiele L, Laumanns M, Zitzler E (2005) Scalable test problems for evolutionary multi-objective optimization. In: Abraham A, Jain R, Goldberg R (eds) Evolutionary multiobjective optimization: theoretical advances and applications, chap 6. Springer, Berlin, pp 105–145

    Chapter  Google Scholar 

  • Emmerich M, Beume N, Naujoks B (2005) An EMO algorithm using the hypervolume measure as selection criterion. In: Conference on evolutionary multi-criterion optimization (EMO 2005). LNCS, vol 3410. Springer, Berlin, pp 62–76

    Google Scholar 

  • Emmerich M, Deutz A, Beume N (2007) Gradient-based/evolutionary relay hybrid for computing Pareto front approximations maximizing the S-metric. In: Hybrid metaheuristics. Springer, Berlin, pp 140–156

    Google Scholar 

  • Everson R, Fieldsend J, Singh S (2002) Full elite-sets for multiobjective optimisation. In: Parmee IC (ed) Conference on adaptive computing in design and manufacture (ADCM 2002), pp 343–354. Springer, London

    Google Scholar 

  • Fleischer M (2003) The measure of Pareto optima. Applications to multi-objective metaheuristics. In: Fonseca CM et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2003), Faro, Portugal. LNCS, vol 2632. Springer, Berlin, pp 519–533

    Google Scholar 

  • Fonseca CM, Paquete L, López-Ibáñez M (2006) An improved dimension-sweep algorithm for the hypervolume indicator. In: Congress on evolutionary computation (CEC 2006), pp 1157–1163, Sheraton Vancouver Wall Centre Hotel, Vancouver, BC Canada. IEEE Press

    Google Scholar 

  • Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading, MA

    Google Scholar 

  • Huband S, Hingston P, White L, Barone L (2003) An evolution strategy with probabilistic mutation for multi-objective optimisation. In: Congress on evolutionary computation (CEC 2003), vol 3, pp 2284–2291, Canberra, Australia. IEEE Press.

    Google Scholar 

  • Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506

    Article  Google Scholar 

  • Igel C, Hansen N, Roth S (2007) Covariance matrix adaptation for multi-objective optimization. Evol Comput 15(1):1–28

    Article  Google Scholar 

  • Knowles JD (2002) Local-search and hybrid evolutionary algorithms for Pareto optimization. PhD thesis, University of Reading

    Google Scholar 

  • Knowles J, Corne D (2003) Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Trans Evol Comput 7(2):100–116

    Article  Google Scholar 

  • Laumanns M, Rudolph G, Schwefel H-P (1999) Approximating the Pareto set: concepts, diversity issues, and performance assessment. Technical Report CI-7299, University of Dortmund

    Google Scholar 

  • Mostaghim S, Branke J, Schmeck H (2007) Multi-objective particle swarm optimization on computer grids. In: Proceedings of the 9th annual conference on genetic and evolutionary computation (GECCO 2007), pp 869–875, New York, USA. ACM

    Google Scholar 

  • Nicolini M (2005) A two-level evolutionary approach to multi-criterion optimization of water supply systems. In: Conference on evolutionary multi-criterion optimization (EMO 2005). LNCS, vol 3410. Springer, Berlin, pp 736–751

    Google Scholar 

  • Van Veldhuizen DA (1999) Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. PhD thesis, Graduate School of Engineering, Air Force Institute of Technology, Air University

    Google Scholar 

  • Wagner T, Beume N, Naujoks B (2007) Pareto-[4], aggregation-, and indicator-based methods in many-objective optimization. In: Obayashi S et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 742–756. Extended version published as internal report of Sonderforschungsbereich 531 Computational Intelligence CI-217/06, Universität Dortmund, September 2006

    Google Scholar 

  • While L (2005) A new analysis of the LebMeasure algorithm for calculating hypervolume. In: Conference on evolutionary multi-criterion optimization (EMO 2005), Guanajuato, México. LNCS, vol 3410. Springer, Berlin, pp 326–340

    Google Scholar 

  • While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evol Comput 10(1):29–38

    Article  Google Scholar 

  • Yang Q, Ding S (2007) Novel algorithm to calculate hypervolume indicator of Pareto approximation set. In: Advanced intelligent computing theories and applications. With aspects of theoretical and methodological issues, Third international conference on intelligent computing (ICIC 2007), vol 2, pp 235–244

    Google Scholar 

  • Zamora LP, Burguete STG (2008) Second-order preferences in group decision making. Oper Res Lett 36:99–102

    Article  Google Scholar 

  • Zitzler E (1999) Evolutionary algorithms for multiobjective optimization: methods and applications. PhD thesis, ETH Zurich, Switzerland

    Google Scholar 

  • Zitzler E (2001) Hypervolume metric calculation. ftp://ftp.tik.ee.ethz.ch/pub/people/zitzler/hypervol.c

  • Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: Yao X et al (eds) Conference on parallel problem solving from nature (PPSN VIII). LNCS, vol 3242. Springer, Berlin, pp 832–842

    Chapter  Google Scholar 

  • Zitzler E, Thiele L (1998a) An evolutionary approach for multiobjective optimization: the strength Pareto approach. TIK Report 43, Computer Engineering and Networks Laboratory (TIK), ETH Zurich

    Google Scholar 

  • Zitzler E, Thiele L (1998b) Multiobjective optimization using evolutionary algorithms – a comparative case study. In: Conference on parallel problem solving from nature (PPSN V), pp 292–301, Amsterdam

    Google Scholar 

  • Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271

    Article  Google Scholar 

  • Zitzler E, Laumanns M, Thiele L (2002) SPEA2: improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou KC et al (eds) Evolutionary methods for design, optimisation and control with application to industrial problems (EUROGEN 2001), pp 95–100. International Center for Numerical Methods in Engineering (CIMNE)

    Google Scholar 

  • Zitzler E, Thiele L, Laumanns M, Fonseca CM, Grunert da Fonseca V (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132

    Article  Google Scholar 

  • Zitzler E, Brockhoff D, Thiele L (2007) The hypervolume indicator revisited: on the design of Pareto-compliant indicators via weighted integration. In: Obayashi S et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 862–876

    Chapter  Google Scholar 

  • Zitzler E, Thiele L, Bader J (2008) On set-based multiobjective optimization. TIK Report 300, Computer Engineering and Networks Laboratory (TIK), ETH Zurich

    Google Scholar 

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Acknowledgements

Johannes Bader has been supported by the Indo-Swiss Joint Research Program IT14.

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Authors and Affiliations

  1. Computer Engineering and Networks Lab, ETH Zurich, 8092, Zurich, Switzerland

    Johannes Bader

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  1. Johannes Bader
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  2. Eckart Zitzler
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Correspondence to Johannes Bader .

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Editors and Affiliations

  1. University of Portsmouth, Lion Terrace, Portsmouth, PO1 3HE, United Kingdom

    Dylan Jones

  2. College of Business & Administration, Kuwait University, Safat, Kuwait

    Mehrdad Tamiz

  3. Dept. Mathematics, University of Portsmouth, Lion Terrace, Portsmouth, PO1 3HE, United Kingdom

    Jana Ries

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Bader, J., Zitzler, E. (2010). A Hypervolume-Based Optimizer for High-Dimensional Objective Spaces. In: Jones, D., Tamiz, M., Ries, J. (eds) New Developments in Multiple Objective and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol 638. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10354-4_3

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