Skip to main content
SpringerLink
Log in

Generalisierte Hypervolumen-Indikatoren für die Mehrzieloptimierung

  • Chapter
  • First Online:
Multikriterielle Optimierung und Entscheidungsunterstützung

  • 1382 Accesses

Zusammenfassung

Der Hypervolumen-Indikator (HVI) wird häufig für die Qualitätsbewertung von finiten Pareto-Front Approximationsmengen in der Mehrzieloptimierung eingesetzt. Approximationsmengen, die den HVI-Wert maximieren, befinden sich i.d.R. auf der Pareto-Front und die Punkte verteilen sich über die Pareto-Front. Die Verteilung ist besonders an Rändern der Pareto-Front und in Kniepunkten dicht. In neueren Arbeiten wurden Generalisierungen des HVIs vorgestellt. Diese Indikatoren werden in diesem Kapitel behandelt. Insbesondere werden Fragen der Berechnungskomplexität und der Verteilung von Punkten in den bezüglich dieser Indikatoren optimierten Approximationsmengen betrachtet, sowie die Frage, welche Entscheidungspräferenzen durch die Wahl des jeweiligen Indikators ausgedrückt werden.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  • [1] M. T. M. Emmerich, A. H. Deutz, A tutorial on multiobjective optimization: fundamentals and evolutionary methods, Natural Computing (2018) 1–25.

    Google Scholar 

  • [2] O. Schütze, X. Esquivel, A. Lara, C. A. C. Coello, Using the averaged hausdorff distance as a performance measure in evolutionary multiobjective optimization, IEEE Transactions on Evolutionary Computation 16 (4) (2012) 504–522.

    Article  Google Scholar 

  • [3] M. Hansen, A. Jaszkiewicz, Evaluating the quality of approximations of the nondominated set. institute of mathematical modeling, technical university of denmark, Tech. rep., IMM Technical Report IMM-REP-1998-7 (1998).

    Google Scholar 

  • [4] D. Brockhoff, T. Wagner, H. Trautmann, On the properties of the r2 indicator, in: Proceedings of the 14th annual conference on Genetic and evolutionary computation, ACM, 2012, pp. 465–472.

    Google Scholar 

  • [5] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, V. G. Da Fonseca, Performance assessment of multiobjective optimizers: An analysis and review, IEEE Transactions on evolutionary computation 7 (2) (2003) 117–132.

    Article  Google Scholar 

  • [6] E. Zitzler, S. Künzli, Indicator-based selection in multiobjective search, in: International Conference on Parallel Problem Solving from Nature, Springer, 2004, pp. 832–842.

    Google Scholar 

  • [7] J. D. Knowles, D. W. Corne, M. Fleischer, Bounded archiving using the lebesgue measure, in: Evolutionary Computation, 2003. CEC’03. The 2003 Congress on, Vol. 4, IEEE, 2003, pp. 2490–2497.

    Google Scholar 

  • [8] V. A. S. Hernández, O. Schütze, M. Emmerich, Hypervolume maximization via set based newtons method, in: EVOLVE-A bridge between probability, set oriented numerics, and evolutionary computation V, Springer, 2014, pp. 15–28.

    Google Scholar 

  • [9] M. Emmerich, A. Deutz, Time complexity and zeros of the hypervolume indicator gradient field, in: EVOLVE-A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation III, Springer, 2014, pp. 169–193.

    Google Scholar 

  • [10] H. Otten, S. Hille, M. T. M. Emmerich, Global optimization for Lipschitz continuous expensive black box functions, Master thesis (2018).

    Google Scholar 

  • [11] E. Zitzler, L. Thiele, Multiobjective optimization using evolutionary algorithmsa comparative case study, in: International conference on parallel problem solving from nature, Springer, 1998, pp. 292–301.

    Google Scholar 

  • [12] E. Zitzler, D. Brockhoff, L. Thiele, The hypervolume indicator revisited: On the design of pareto-compliant indicators via weighted integration, in: International Conference on Evolutionary Multi-Criterion Optimization, Springer, 2007, pp. 862–876.

    Google Scholar 

  • [13] M. Fleischer, The measure of pareto optima applications to multi-objective metaheuristics, in: International Conference on Evolutionary Multi-Criterion Optimization, Springer, 2003, pp. 519–533.

    Google Scholar 

  • [14] A. Auger, J. Bader, D. Brockhoff, E. Zitzler, Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point, in: Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithms, ACM, 2009, pp. 87–102.

    Google Scholar 

  • [15] P. K. Shukla, N. Doll, H. Schmeck, A theoretical analysis of volume based pareto front approximations, in: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, GECCO ’14, ACM, New York, NY, USA, 2014, pp. 1415–1422. https://doi.org/10.1145/2576768.2598348. URL http://doi.acm.org/10.1145/2576768.2598348

  • [16] K. Bringmann, T. Friedrich, Approximating the volume of unions and intersections of high-dimensional geometric objects, in: International Symposium on Algorithms and Computation, Springer, 2008, pp. 436–447.

    Google Scholar 

  • [17] N. Beume, C. M. Fonseca, M. López-Ibáñez, L. Paquete, J. Vahrenhold, On the complexity of computing the hypervolume indicator, IEEE Transactions on Evolutionary Computation 13 (5) (2009) 1075–1082.

    Article  Google Scholar 

  • [18] T. M. Chan, Klee’s measure problem made easy, in: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science(FOCS), 2014, pp. 410–419.

    Google Scholar 

  • [19] A. P. Guerreiro, C. M. Fonseca, M. T. M. Emmerich, A fast dimension-sweep algorithm for the hypervolume indicator in four dimensions., in: CCCG, 2012, pp. 77–82.

    Google Scholar 

  • [20] L. While, L. Bradstreet, L. Barone, A fast way of calculating exact hypervolumes, IEEE Transactions on Evolutionary Computation 16 (1) (2012) 86–95.

    Article  Google Scholar 

  • [21] R. Lacour, K. Klamroth, C. M. Fonseca, A box decomposition algorithm to compute the hypervolume indicator, Computers & Operations Research 79 (2017) 347–360.

    Google Scholar 

  • [22] K. Bringmann, T. Friedrich, P. Klitzke, Two-dimensional subset selection for hypervolume and epsilon-indicator, in: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, ACM, 2014, pp. 589–596.

    Google Scholar 

  • [23] T. Kuhn, C. M. Fonseca, L. Paquete, S. Ruzika, M. M. Duarte, J. R. Figueira, Hypervolume subset selection in two dimensions: Formulations and algorithms, Evolutionary Computation 24 (3) (2016) 411–425.

    Article  Google Scholar 

  • [24] K. Bringmann, S. Cabello, M. T. M. Emmerich, Maximum volume subset selection for anchored boxes, in: B. Aronov, M. J. Katz (Eds.), 33rd International Symposium on Computational Geometry (SoCG 2017), Vol. 77 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017, pp. 22:1–22:15.

    Google Scholar 

  • [25] I. Hupkens, M. Emmerich, Logarithmic-time updates in sms-emoa and hypervolume-based archiving, in: EVOLVE-A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV, Springer, 2013, pp. 155–169.

    Google Scholar 

  • [26] M. T. M. Emmerich, C. M. Fonseca, Computing hypervolume contributions in low dimensions: asymptotically optimal algorithm and complexity results, in: International Conference on Evolutionary Multi-Criterion Optimization, Springer, 2011, pp. 121–135.

    Google Scholar 

  • [27] A. P. Guerreiro, C. M. Fonseca, Computing and updating hypervolume contributions in up to four dimensions, IEEE Transactions on Evolutionary Computation.

    Google Scholar 

  • [28] M. Emmerich, A. Deutz, J. Kruisselbrink, P. K. Shukla, Cone-based hypervolume indicators: construction, properties, and efficient computation, in: EMO 2013, Springer, 2013, pp. 111–127.

    Google Scholar 

  • [29] P. K. Shukla, M. Emmerich, A. Deutz, A theoretical analysis of curvature based preference models, in: International Conference on Evolutionary Multi-Criterion Optimization, Springer, 2013, pp. 367–382.

    Google Scholar 

  • [30] M. Ehrgott, Multicriteria optimization, Vol. 491, Springer Science & Business Media, 2005.

    Google Scholar 

  • [31] M. T. Emmerich, A. H. Deutz, Test problems based on lamé superspheres, in: International Conference on Evolutionary Multi-Criterion Optimization, Springer, 2007, pp. 922–936.

    Google Scholar 

  • [32] T. Friedrich, K. Bringmann, T. Voß, C. Igel, The logarithmic hypervolume indicator, in: Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms, ACM, 2011, pp. 81–92.

    Google Scholar 

  • [33] T. Wagner, H. Trautmann, Integration of preferences in hypervolume-based multiobjective evolutionary algorithms by means of desirability functions, IEEE Transactions on Evolutionary Computation 14 (5) (2010) 688–701.

    Article  Google Scholar 

  • [34] M. T. M. Emmerich, A. H. Deutz, I. Yevseyeva, On reference point free weighted hypervolume indicators based on desirability functions and their probabilistic interpretation, Procedia Technology 16 (Complete) (2014) 532–541. https://doi.org/10.1016/j.protcy.2014.10.001.

    Article  Google Scholar 

  • [35] A. Custódio, M. Emmerich, J. Madeira, Recent developments in derivative-free multiobjective optimization, Computational Technology Reviews 5 (1) (2012) 1–31.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Leiden Institute of Advanced Computer Science, Leiden University, Leiden, Niederlande

    Michael T. M. Emmerich & André H. Deutz

Authors
  1. Michael T. M. Emmerich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. André H. Deutz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael T. M. Emmerich .

Editor information

Editors and Affiliations

  1. Kaiserslautern, Germany

    Karl-Heinz Küfer

  2. Kaiserslautern, Germany

    Stefan Ruzika

  3. Kaiserslautern, Germany

    Pascal Halffmann

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Emmerich, M.T.M., Deutz, A.H. (2019). Generalisierte Hypervolumen-Indikatoren für die Mehrzieloptimierung. In: Küfer, KH., Ruzika, S., Halffmann, P. (eds) Multikriterielle Optimierung und Entscheidungsunterstützung. Springer Gabler, Wiesbaden. https://doi.org/10.1007/978-3-658-27041-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-658-27041-4_1

  • Published:

  • Publisher Name: Springer Gabler, Wiesbaden

  • Print ISBN: 978-3-658-27040-7

  • Online ISBN: 978-3-658-27041-4

  • eBook Packages: Business and Economics (German Language)

Publish with us

Policies and ethics

Search

Navigation