Quality Assessment of Pareto Set Approximations

  • Eckart Zitzler
  • Joshua Knowles
  • Lothar Thiele
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5252)

Abstract

This chapter reviews methods for the assessment and comparison of Pareto set approximations. Existing set quality measures from the literature are critically evaluated based on a number of orthogonal criteria, including invariance to scaling, monotonicity and computational effort. Statistical aspects of quality assessment are also considered in the chapter. Three main methods for the statistical treatment of Pareto set approximations deriving from stochastic generating methods are reviewed. The dominance ranking method is a generalization to partially-ordered sets of a standard non-parametric statistical test, allowing collections of Pareto set approximations from two or more stochastic optimizers to be directly compared statistically. The quality indicator method — the dominant method in the literature — maps each Pareto set approximation to a number, and performs statistics on the resulting distribution(s) of numbers. The attainment function method estimates the probability of attaining each goal in the objective space, and looks for significant differences between these probability density functions for different optimizers. All three methods are valid approaches to quality assessment, but give different information. We explain the scope and drawbacks of each approach and also consider some more advanced topics, including multiple testing issues, and using combinations of indicators. The chapter should be of interest to anyone concerned with generating and analysing Pareto set approximations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benjamini, Y., Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society, Series B (Methodological) 57, 125–133 (1995)MathSciNetMATHGoogle Scholar
  2. Berezkin, V.E., Kamenev, G.K., Lotov, A.V.: Hybrid adaptive methods for approximating a nonconvex multidimensional pareto frontier. Computational Mathematics and Mathematical Physics 46(11), 1918–1931 (2006)MathSciNetCrossRefGoogle Scholar
  3. Beume, N., Rudolph, G.: Faster S-Metric Calculation by Considering Dominated Hypervolume as Klee’s Measure Problem. In: Proceedings of the Second IASTED Conference on Computational Intelligence, pp. 231–236. ACTA Press, Anaheim (2006)Google Scholar
  4. Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal on Operational Research 181, 1653–1669 (2007)CrossRefMATHGoogle Scholar
  5. Bland, J.M., Altman, D.G.: Multiple significance tests: the bonferroni method. British Medical Journal 310, 170 (1995)CrossRefGoogle Scholar
  6. Chambers, J., Cleveland, W., Kleiner, B., Tukey, P.: Graphical Methods for Data Analysis. Wadsworth, Belmont (1983)MATHGoogle Scholar
  7. Coello Coello, C.A., Van Veldhuizen, D.A., Lamont, G.B.: Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer Academic Publishers, New York (2002)CrossRefMATHGoogle Scholar
  8. Conover, W.J.: Practical Nonparametric Statistics, 3rd edn. John Wiley and Sons, New York (1999)Google Scholar
  9. Czyzak, P., Jaskiewicz, A.: Pareto simulated annealing—a metaheuristic for multiobjective combinatorial optimization. Multi-Criteria Decision Analysis 7, 34–47 (1998)CrossRefMATHGoogle Scholar
  10. Deb, K.: Multi-objective optimization using evolutionary algorithms. Wiley, Chichester (2001)MATHGoogle Scholar
  11. Deb, K., Pratap, A., Agrawal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 181–197 (2002)CrossRefGoogle Scholar
  12. Efron, B., Tibshirani, R.: An introduction to the bootstrap. Chapman and Hall, London (1993)CrossRefMATHGoogle Scholar
  13. Ehrgott, M., Gandibleux, X.: A Survey and Annotated Bibliography of Multiobjective Combinatorial Optimization. OR Spektrum 22, 425–460 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. Fleischer, M.: The Measure of Pareto Optima. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. Fonseca, C.M., Fleming, P.J.: Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization. In: Forrest, S. (ed.) Proceedings of the Fifth International Conference on Genetic Algorithms, pp. 416–423. Morgan Kaufmann, San Mateo (1993)Google Scholar
  16. Fonseca, C.M., Grunert da Fonseca, V., Paquete, L.: Exploring the performance of stochastic multiobjective optimisers with the second-order attainment function. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 250–264. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. Fonseca, C.M., Paquete, L., López-Ibáñez, M.: An Improved Dimension-Sweep Algorithm for the Hypervolume Indicator. In: Congress on Evolutionary Computation (CEC 2006), Sheraton Vancouver Wall Centre Hotel, Vancouver, BC Canada, pp. 1157–1163. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  18. Grunert da Fonseca, V., Fonseca, C.M., Hall, A.O.: Inferential Performance Assessment of Stochastic Optimisers and the Attainment Function. In: Zitzler, E., Deb, K., Thiele, L., Coello Coello, C.A., Corne, D.W. (eds.) EMO 2001. LNCS, vol. 1993, pp. 213–225. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. Hansen, M.P., Jaszkiewicz, A.: Evaluating the quality of approximations of the non-dominated set. Technical report, Institute of Mathematical Modeling, Technical University of Denmark. IMM Technical Report IMM-REP-1998-7 (1998)Google Scholar
  20. Helbig, S., Pateva, D.: On several concepts for ε-efficiency. OR Spektrum 16(3), 179–186 (1994)MathSciNetCrossRefMATHGoogle Scholar
  21. Kamenev, G., Kondtratíev, D.: Method for the exploration of non-closed nonlinear models (in Russian). Matematicheskoe Modelirovanie 4(3), 105–118 (1992)MathSciNetGoogle Scholar
  22. Kamenev, G.K.: Approximation of completely bounded sets by the deep holes method. Computational Mathematics And Mathematical Physics 41, 1667–1676 (2001)MathSciNetMATHGoogle Scholar
  23. Knowles, J.: A summary-attainment-surface plotting method for visualizing the performance of stochastic multiobjective optimizers. In: Computational Intelligence and Applications, Proceedings of the Fifth International Workshop on Intelligent Systems Design and Applications: ISDA’05 (2005)Google Scholar
  24. Knowles, J., Corne, D.: On Metrics for Comparing Non-Dominated Sets. In: Congress on Evolutionary Computation (CEC 2002), pp. 711–716. IEEE Press, Piscataway (2002)Google Scholar
  25. Knowles, J., Thiele, L., Zitzler, E.: A Tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers. TIK Report 214, Computer Engineering and Networks Laboratory (TIK), ETH Zurich (2006)Google Scholar
  26. Knowles, J.D.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. Ph.D. thesis, University of Reading (2002)Google Scholar
  27. López-Ibáñez, M., Paquete, L., Stützle, T.: Hybrid population-based algorithms for the bi-objective quadratic assignment problem. Journal of Mathematical Modelling and Algorithms 5(1), 111–137 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. Lotov, A.V., Kamenev, G.K., Berezkin, V.E.: Approximation and Visualization of Pareto-Efficient Frontier for Nonconvex Multiobjective Problems. Doklady Mathematics 66(2), 260–262 (2002)MATHGoogle Scholar
  29. Lotov, A.V., Bushenkov, V.A., Kamenev, G.K.: Interactive Decision Maps. Approximation and Visualization of Pareto Frontier. Kluwer Academic Publishers, Boston (2004)CrossRefMATHGoogle Scholar
  30. Miller, R.G.: Simultaneous Statistical Inference, 2nd edn. Springer, New York (1981)CrossRefMATHGoogle Scholar
  31. Perneger, T.V.: What’s wrong with Bonferroni adjustments. British Medical Journal 316, 1236–1238 (1998)CrossRefGoogle Scholar
  32. Schott, J.: Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization. Master’s thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (1995)Google Scholar
  33. Shaw, K.J., Nortcliffe, A.L., Thompson, M., Love, J., Fonseca, C.M., Fleming, P.J.: Assessing the Performance of Multiobjective Genetic Algorithms for Optimization of a Batch Process Scheduling Problem. In: 1999 Congress on Evolutionary Computation, Washington, D.C., pp. 37–45. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  34. Smith, K.I., Everson, R.M., Fieldsend, J.E., Murphy, C., Misra, R.: Dominance-based multiobjective simulated annealing. IEEE Transactions on Evolutionary Computation. In press (2008)Google Scholar
  35. Ulungu, E.L., Teghem, J., Fortemps, P.H., Tuyttens, D.: Mosa method: A tool for solving multiobjective combinatorial optimization problems. Journal of Multi-Criteria Decision Analysis 8(4), 221–236 (1999)CrossRefMATHGoogle Scholar
  36. Van Veldhuizen, D.A.: Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations. Ph.D. thesis, Graduate School of Engineering, Air Force Institute of Technology, Air University (1999)Google Scholar
  37. Wagner, T., Beume, N., Naujoks, B.: Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 742–756. Springer, Heidelberg (2007), extended version published as internal report of Sonderforschungsbereich 531 Computational Intelligence CI-217/06, Universität Dortmund (September 2006).CrossRefGoogle Scholar
  38. Westfall, P.H., Young, S.S.: Resampling-based multiple testing. Wiley, New York (1993)MATHGoogle Scholar
  39. While, L.: A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 326–340. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  40. While, L., Bradstreet, L., Barone, L., Hingston, P.: Heuristics for Optimising the Calculation of Hypervolume for Multi-objective Optimisation Problems. In: Congress on Evolutionary Computation (CEC 2005), IEEE Service Center, Edinburgh, Scotland, pp. 2225–2232. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  41. While, L., Hingston, P., Barone, L., Huband, S.: A Faster Algorithm for Calculating Hypervolume. IEEE Transactions on Evolutionary Computation 10(1), 29–38 (2006)CrossRefGoogle Scholar
  42. Zitzler, E., Künzli, S.: Indicator-Based Selection in Multiobjective Search. 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 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  43. Zitzler, E., Thiele, L.: Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  44. Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE Transactions on Evolutionary Computation 3(4), 257–271 (1999)CrossRefGoogle Scholar
  45. Zitzler, E., Thiele, L., Laumanns, M., Foneseca, C.M., Grunert da Fonseca, V.: Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)CrossRefGoogle Scholar
  46. Zitzler, E., Brockhoff, D., Thiele, L.: The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Eckart Zitzler
    • 1
  • Joshua Knowles
    • 2
  • Lothar Thiele
    • 1
  1. 1.ETH ZurichSwitzerland
  2. 2.University of ManchesterUK

Personalised recommendations