Dimensionality Reduction in Multiobjective Optimization: The Minimum Objective Subset Problem

  • Dimo Brockhoff
  • Eckart Zitzler
Part of the Operations Research Proceedings book series (ORP, volume 2006)

Abstract

The number of objectives in a multiobjective optimization problem strongly influences both the performance of generating methods and the decision making process in general. On the one hand, with more objectives, more incomparable solutions can arise, the number of which affects the generating method’s performance. On the other hand, the more objectives are involved the more complex is the choice of an appropriate solution for a (human) decision maker. In this context, the question arises whether all objectives are actually necessary and whether some of the objectives may be omitted; this question in turn is closely linked to the fundamental issue of conflicting and non-conflicting optimization criteria. Besides a general definition of conflicts between objective sets, we here introduce the \( \mathcal{N}\mathcal{P} \)-hard problem of computing a minimum subset of objectives without losing information (MOSS). Furthermore, we present for MOSS both an approximation algorithm with optimum approximation ratio and an exact algorithm which works well for small input instances. We conclude with experimental results for a random problem and the multiobjective 0/1-knapsack problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. J. Agrell. On redundancy in multi criteria decision making. European Journal of Operational Research, 98(3):571–586, 1997.CrossRefGoogle Scholar
  2. 2.
    D. Brockhoff and E. Zitzler. On Objective Conflicts and Objective Reduction in Multiple Criteria Optimization. TIK Report 243, February 2006.Google Scholar
  3. 3.
    K. Deb. Multi-objective optimization using evolutionary algorithms. Wiley, Chichester, UK, 2001.Google Scholar
  4. 4.
    K. Deb and D. K. Saxena. On finding pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems. Kangal report no. 2005011, December 2005.Google Scholar
  5. 5.
    M. Ehrgott. Multicriteria Optimization. Springer Berlin Heidelberg, 2005.Google Scholar
  6. 6.
    U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, 1998.CrossRefGoogle Scholar
  7. 7.
    C. M. Fonseca and P. J. Fleming. An overview of evolutionary algorithms in multiobjective optimization. Evolutionary Computation, 3(1):1–16, 1995.Google Scholar
  8. 8.
    T. Gal and H. Leberling. Redundant objective functions in linear vector maximum problems and their determination. European Journal of Operational Research, 1(3):176–184, 1977.CrossRefGoogle Scholar
  9. 9.
    I. T. Jolliffe. Principal component analysis. Springer, 2002.Google Scholar
  10. 10.
    K. M. Miettinen. Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, 1999.Google Scholar
  11. 11.
    R. C. Purshouse and P. J. Fleming. Conflict, harmony, and independence: Relationships in evolutionary multi-criterion optimisation. In EMO 2003 Proceedings, pages 16–30. Springer, Berlin, 2003.Google Scholar
  12. 12.
    P. Slavík. A tight analysis of the greedy algorithm for set cover. In STOC’ 96: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 435–441, New York, NY, USA, 1996. ACM Press.Google Scholar
  13. 13.
    K. C. Tan, E. F. Khor, and T. H. Lee. Multiobjective Evolutionary Algorithms and Applications. Springer, London, 2005.Google Scholar
  14. 14.
    W. T. Trotter. Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore and London, 1992.Google Scholar
  15. 15.
    L. While. A new analysis of the lebmeasure algorithm for calculating hypervolume. In EMO 2005 Proceedings, pages 326–340. Springer, 2005.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Dimo Brockhoff
    • 1
  • Eckart Zitzler
    • 1
  1. 1.Computer Engineering and Networks LaboratoryETH ZurichSwitzerland

Personalised recommendations