Approximating the Knee of an MOP with Stochastic Search Algorithms

  • Oliver Schütze
  • Marco Laumanns
  • Carlos A. Coello Coello
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)


In this paper we address the problem of approximating the ’knee’ of a bi-objective optimization problem with stochastic search algorithms. Knees or entire knee-regions are of particular interest since such solutions are often preferred by the decision makers in many applications. Here we propose and investigate two update strategies which can be used in combination with stochastic multi-objective search algorithms (e.g., evolutionary algorithms) and aim for the computation of the knee and the knee-region, respectively. Finally, we demonstrate the applicability of the approach on two examples.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN VIII 2004. LNCS, vol. 3242, pp. 722–731. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Branke, J., Kaussler, T., Schmeck, H.: Guidance in evolutionary multi-objective optimization. Advances in Engineering Software 32, 499–507 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Das, I.: On characterizing the ”knee” of the Pareto curve based on Normal Boundary Intersection. Structural Optimization 18, 107–115 (1999)CrossRefGoogle Scholar
  4. 4.
    Deb, K.: Multi-objective evolutionary algorithms: introducing bias among pareto-optimal solutions, pp. 263–292 (2003)Google Scholar
  5. 5.
    di Pierro, F., Khu, S.F., Savic, D.A.: An investigation on preference order ranking scheme for multiobjective evolutionary optimization. IEEE Transactions on Evolutionary Computation 11(1), 17–45 (2007)CrossRefGoogle Scholar
  6. 6.
    Handl, J., Knowles, J.: Exploiting the trade-off: the benefits of multiple objectives in data clustering. In: Proceedings of the Third International Conference on Evolutionary Multicriterion Optimization, pp. 547–560Google Scholar
  7. 7.
    Ishibuchi, H., Nojima, Y., Narukawa, K., Doi, T.: Incorporation of decision maker’s preference into evolutionary multiobjective optimization algorithms. In: GECCO, pp. 741–742 (2006)Google Scholar
  8. 8.
    Kukkonen, S., Deb, K.: A fast and effective method for pruning of non-dominated solutions in many-objective problems. In: PPSN, pp. 553–562 (2006)Google Scholar
  9. 9.
    Mattson, C.A., Mullur, A.A., Messac, A.: Smart Pareto filter: Obtaining a minimal representation of multiobjective design space. Engineering Optimization 36, 721–740 (2004)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Mehnen, J., Trautmann, H.: Integration of expert’s preferences in pareto optimization by desirability function techniques. In: Proceedings of the 5th CIRP International Seminar on Intelligent Computation in Manufacturing Engineering (CIRP ICME 2006) (2006)Google Scholar
  11. 11.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Dordrecht (1999)zbMATHGoogle Scholar
  12. 12.
    Rachmawati, L., Srinivasan, D.: A multi-objective evolutionary algorithm with weighted-sum niching for convergence on knee regions. In: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pp. 749–750 (2006)Google Scholar
  13. 13.
    Rachmawati, L., Srinivasan, D.: A Multi-Objective Genetic Algorithm with Controllable Convergence on Knee Regions. In: IEEE Congress on Evolutionary Computation, CEC 2006, pp. 1916–1923 (2006)Google Scholar
  14. 14.
    Witting, K., Hessel-von Molo, M.: Private communication (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Oliver Schütze
    • 1
  • Marco Laumanns
    • 2
  • Carlos A. Coello Coello
    • 1
  1. 1.CINVESTAV-IPN, Computer Science DepartmentMexico CityMexico
  2. 2.ETH Zurich, Institute for Operations ResearchZurichSwitzerland

Personalised recommendations