Skip to main content

An Expedition to Multimodal Multi-objective Optimization Landscapes

  • Conference paper
  • First Online:
Evolutionary Multi-Criterion Optimization (EMO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10173))

Included in the following conference series:


The research in evolutionary multi-objective optimization is largely missing a notion of functional landscapes, which could enable a visual understanding of multimodal multi-objective landscapes and their characteristics by connecting decision and objective space. This consequently leads to the negligence of decision space in most algorithmic approaches and an almost complete lack of Exploratory Landscape Analysis (ELA) tools. This paper dares a first step into this unexplored field based on gradient properties of the multi-objective landscape. For a first time, basins of attraction and superpositions of local optima are visualized and thereby made intuitively accessible. With this work, we hope to highlight the importance of detailed decision space analysis in multi-objective optimization and to stimulate further research in that direction.

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

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Similar content being viewed by others


  1. 1.

    Note that we restricted ourselves to bi-objective mixed-sphere problems as we build upon the definitions and problems from [8], but in general our proposed visualization technique can also be applied to problems with more than two objectives.

  2. 2.

    A vector \(\mathbf {a} = (a_1, a_2, \ldots , a_n)^T\) dominates another vector \(\mathbf {b} = (b_1, b_2, \ldots , b_n)^T\), i.e., \(\mathbf {a} \prec \mathbf {b}\), if and only if \(\forall i \in \{1, \ldots , n\}: \; a_i \le b_i\) and \(\exists j \in \{1, \ldots , n\}: \; a_j < b_j\).

  3. 3.

    One can also use more coarse configurations without losing a lot of accuracy. For instance, we tried smaller grids (consisting of 1 000 by 1 000 points) and a smaller threshold (\(\delta = 10^{-3}\)) while detecting only minor differences in the resulting figures.


  1. Beyer, H.G.: The Theory of Evolution Strategies. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  2. Bossek, J.: smoof: Single and Multi-Objective Optimization Test Functions (2016)., r package version 1.4

  3. Fonseca, C.M.M.: Multiobjective genetic algorithms with application to control engineering problems. Ph.D. thesis, Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK (1995)

    Google Scholar 

  4. Garrett, D., Dasgupta, D.: Multiobjective landscape analysis and the generalized assignment problem. In: Maniezzo, V., Battiti, R., Watson, J.-P. (eds.) LION 2007. LNCS, vol. 5313, pp. 110–124. Springer, Heidelberg (2008). doi:10.1007/978-3-540-92695-5_9

    Chapter  Google Scholar 

  5. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays. Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)

    Google Scholar 

  6. Kerschke, P., Preuss, M., Wessing, S., Trautmann, H.: Detecting funnel structures by means of exploratory landscape analysis. In: Proceedings of the 17th Annual Conference on Genetic and Evolutionary Computation, pp. 265–272. ACM (2015)

    Google Scholar 

  7. Kerschke, P., Preuss, M., Wessing, S., Trautmann, H.: Low-budget exploratory landscape analysis on multiple peaks models. In: Proceedings of the 18th Annual Conference on Genetic and Evolutionary Computation. ACM (2016, accepted)

    Google Scholar 

  8. Kerschke, P., Wang, H., Preuss, M., Grimme, C., Deutz, A., Trautmann, H., Emmerich, M.: Towards analyzing multimodality of continuous multiobjective landscapes. In: Handl, J., Hart, E., Lewis, P.R., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds.) PPSN 2016. LNCS, vol. 9921, pp. 962–972. Springer, Heidelberg (2016). doi:10.1007/978-3-319-45823-6_90

    Chapter  Google Scholar 

  9. Knowles, J.D., Corne, D.W.: Towards landscape analyses to inform the design of hybrid local search for the multiobjective quadratic assignment problem. In: HIS, Second International Conference on Hybrid Intelligent Systems (2002)

    Google Scholar 

  10. Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings on the Second Berkeley Symposium on Mathematical Statististics and Probability, pp. 481–492. University of California Press (1951)

    Google Scholar 

  11. Mersmann, O., Bischl, B., Trautmann, H., Preuss, M., Weihs, C., Rudolph, G.: Exploratory landscape analysis. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, GECCO 2011, NY, USA, pp. 829–836. ACM, New York (2011)

    Google Scholar 

  12. Rosenthal, S., Borschbach, M.: A concept for real-valued multi-objective landscape analysis characterizing two biochemical optimization problems. In: Mora, A.M., Squillero, G. (eds.) EvoApplications 2015. LNCS, vol. 9028, pp. 897–909. Springer, Cham (2015). doi:10.1007/978-3-319-16549-3_72

    Google Scholar 

  13. Schwefel, H.P.: Evolution and Optimum Seeking. Wiley, New York (1995)

    MATH  Google Scholar 

  14. Tušar, T.: Visualizing Solution Sets in Multiobjective Optimization. Ph.D. thesis, Jožef Stefan International Postgraduate School (2014)

    Google Scholar 

  15. Tušar, T., Filipič, B.: Visualization of Pareto front approximations in evolutionary multiobjective optimization: a critical review and the prosection method. IEEE Trans. Evol. Comput. 19(2), 225–245 (2015)

    Article  Google Scholar 

  16. Wessing, S.: Two-stage methods for multimodal optimization. Ph.D. thesis, Technische Universität Dortmund (2015).

  17. Wessing, S.: optproblems: Infrastructure to Define Optimization Problems and Some Test Problems for Black-Box Optimization (2016)., python package version 0.9

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pascal Kerschke .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Kerschke, P., Grimme, C. (2017). An Expedition to Multimodal Multi-objective Optimization Landscapes. In: Trautmann, H., et al. Evolutionary Multi-Criterion Optimization. EMO 2017. Lecture Notes in Computer Science(), vol 10173. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-54156-3

  • Online ISBN: 978-3-319-54157-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics