The Measure of Pareto Optima Applications to Multi-objective Metaheuristics

  • M. Fleischer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2632)


This article describes a set function that maps a set of Pareto optimal points to a scalar. A theorem is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function’s arguments to be maximally diverse Pareto optimal solutions of a discrete, multi-objective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multiobjective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient, polynomial-time algorithm for calculating this scalar and an analysis of its complexity is also presented.


Objective Function Simulated Annealing Multiobjective Optimization Pareto Optimal Solution Pareto Optimum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fleischer, M.A.: The measure of pareto optima: Applications to multiobjective metaheuristics. Technical Report 2002-32, Institute for Systems Research, University of Maryland, College Park, MD. (2002) This is an unabridged version of the instant article.Google Scholar
  2. 2.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms—a comparative case study. In A.E. Eiben, T. Bäck, M.S.H.S., ed.: Proceedings of the Fifth International Conference on Parallel Problem Solving from Nature—PPSN V, Berlin, Germany (1998)Google Scholar
  3. 3.
    M. Laumanns, G.R., Schwefel, H.: Approximating the pareto set: Diversity issues and performance assessment. (1999)Google Scholar
  4. 4.
    Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. Ph.d. dissertation, Swiss Federal Institute fo Technology (ETH), Zurich, Switzerland (1999) The relevant part pertaining to our Lebesgue measure is discussed in Ch. 3.Google Scholar
  5. 5.
    Wu, J., Azarm, S.: Metrics for quality assessment of a multiobjective design optimization solution set. Transactions of the ASME 123 (2001) 18–25CrossRefGoogle Scholar
  6. 6.
    Fonseca, C., Fleming, P.: On the performance assessment and comparison of stochastic multiobjective optimizers. In G. Goos, J.H., van Leeuwen, J., eds.: Proceedings of the Fourth International Conference on Parallel Problem Solving from Nature—PPSN IV, Berlin, Germany (1998) 584–593Google Scholar
  7. 7.
    Mitra, D., Romeo, F., Sangiovanni-Vincentelli, A.: Convergence and finite-time behavior of simulated annealing. Advances in Applied Probability 18 (1986) 747–771zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Laumanns, L. Thiele, K.D., Zitzler, E.: Archiving with guaranteed convergence and diversity in multi-objective optimization. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO). (2002) 439–447Google Scholar
  9. 9.
    Coello, C.: Bibliograph on evolutionary multiobjective optimization. (2001)
  10. 10.
    Veldhuizen, D.V., Lamont, G.B.: Multiobjective evolutionary algorithms: Analyzing the state-of-the-art. Evolutionary Computation 8 (2000) 125–147CrossRefGoogle Scholar
  11. 11.
    Czyzak, P., Jaskiewicz, A.: Pareto simulated annealing—a metaheuristic technique for multiple-objective combinatorial optimization. Journal of Multi-criteria Decision Analysis 7 (1998) 34–47zbMATHCrossRefGoogle Scholar
  12. 12.
    Reeves, C.: Modern Heuristic Techniques for Combinatorial Problems. John Wiley & Sons, Inc., New York, NY (1993)zbMATHGoogle Scholar
  13. 13.
    Knowles, J.D., Corne, D.W.: Approximating the nondominated front using the pareto archived evolution strategy. Evolutionary Computation 8 (2000) 149–172CrossRefGoogle Scholar
  14. 14.
    Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge, England (1991)zbMATHGoogle Scholar
  15. 15.
    Wu, J.: Quality Assisted Multiobjective and Multi-disciplinary Genetic Algorithms. Department of mechanical engineering, University of Maryland, College Park, College Park, Maryland (2001) S. Azarm, Ph.D. advisor.Google Scholar
  16. 16.
    Knowles, J.: Local Search and Hybrid Evolutionary Algorithms for Pareto Optimization. PhD thesis, The University of Reading, Reading, UK (2002) Department of Computer Science.Google Scholar
  17. 17.
    Kapoor, S.: Dynamic maintenance of maxima of 2-d point sets. SIAM Journal on Computing 29 (2000) 1858–1877zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Frederickson, G., Rodger, S.: A new approach to the dynamic maintenance of maximal points in a plane. Discrete and Computational Geometry 5 (1990) 365–374zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hildebrand, F.B.: Introduction to Numerical Analysis. 2nd edn. Dover Publications, Inc., Mineola, NY (1987)zbMATHGoogle Scholar
  20. 20.
    Fleischer, M.A.: 28: Generalized Cybernetic Optimization: Solving Continuous Variable Problems. In: Metaheuristics: Advances and Trends in Local Search Paradigms for Optimization. Kluwer Academic Publishers (1999) 403–418Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • M. Fleischer
    • 1
  1. 1.Institute for Systems ResearchUniversity of MarylandCollege Park

Personalised recommendations