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Computational Optimization and Applications

, Volume 64, Issue 2, pp 589–618 | Cite as

Optimal averaged Hausdorff archives for bi-objective problems: theoretical and numerical results

  • Günter Rudolph
  • Oliver Schütze
  • Christian Grimme
  • Christian Domínguez-Medina
  • Heike TrautmannEmail author
Article

Abstract

One main task in evolutionary multiobjective optimization (EMO) is to obtain a suitable finite size approximation of the Pareto front which is the image of the solution set, termed the Pareto set, of a given multiobjective optimization problem. In the technical literature, the characteristic of the desired approximation is commonly expressed by closeness to the Pareto front and a sufficient spread of the solutions obtained. In this paper, we first make an effort to show by theoretical and empirical findings that the recently proposed Averaged Hausdorff (or \(\Delta _p\)-) indicator indeed aims at fulfilling both performance criteria for bi-objective optimization problems. In the second part of this paper, standard EMO algorithms combined with a specialized archiver and a postprocessing step based on the \(\Delta _p\) indicator are introduced which sufficiently approximate the \(\Delta _p\)-optimal archives and generate solutions evenly spread along the Pareto front.

Keywords

Evolutionary computation \(\Delta _p\) indicator Hausdorff distance Evolutionary multiobjective optimization 

Notes

Acknowledgments

HT and CG acknowledge support by the European Center of Information Systems (ERCIS). OS acknowledges support from Conacyt Project No. 128554. CDM acknowledges support by the Consejo Nacional de Ciencia y Tecnología (CONACYT). All authors acknowledge support from CONACYT Project No. 207403, DFG Project No. TR 891/5-1 and DAAD Project No. 57065955.

References

  1. 1.
    Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal \(\mu \)-distributions and the choice of the reference point. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 87–102. ACM Press (2009)Google Scholar
  2. 2.
    Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Coello Coello, C.A., Cruz Cortés, N.: Solving multiobjective optimization problems using an Artificial Immune System. Genet. Program. Evolvable Mach. 6(2), 163–190 (2005)CrossRefGoogle Scholar
  4. 4.
    Coello Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems, 2nd edn. Springer, New York (2007)zbMATHGoogle Scholar
  5. 5.
    Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)zbMATHGoogle Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  7. 7.
    Dominguez-Medina, C., Rudolph, G., Schütze, O., Trautmann, H.: Evenly spaced Pareto fronts of quad-objective problems using PSA partitioning technique. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC 2013), pp. 3190–3197. IEEE Press, Piscataway, NJ (2013)Google Scholar
  8. 8.
    Durillo, J.J., Nebro, A.J.: jMetal: a Java framework for multi-objective optimization. Adv. Eng. Softw. 42(10), 760–771 (2011)CrossRefGoogle Scholar
  9. 9.
    Emmerich, M., Deutz, A., Kruisselbrink, J., Shukla, P.: Cone-based hypervolume indicators: construction, properties, and efficient computation. In: Purshouse, R., Fleming, P., Fonseca, C., Greco, S., Shaw, J. (eds.) Proceedings of Evolutionary Multi-Criterion Optimization (EMO 2013), pp. 111–127. Springer, Berlin (2013)Google Scholar
  10. 10.
    Emmerich, M.T., Deutz, A.H., Kruisselbrink, J.W.: On quality indicators for black-box level set approximation. In: EVOLVE-A Bridge Between Probability, Set Oriented Numerics and Evolutionary Computation, pp. 157–185. Springer, Berlin (2013)Google Scholar
  11. 11.
    Gerstl, K., Rudolph, G., Schütze, O., Trautmann, H.: Finding evenly spaced fronts for multiobjective control via averaging Hausdorff-measure. In: Proceedings of 8th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), pp. 1–6. IEEE Press (2011). doi: 10.1109/ICEEE.2011.6106656
  12. 12.
    Hansen, M.P., Jaszkiewicz, A.: Evaluating the quality of approximations of the non-dominated set. IMM Technical Report IMM-REP-1998-7, Institute of Mathematical Modeling, Technical University of Denmark, Lyngby (1998)Google Scholar
  13. 13.
    Hillermeier, C.: Nonlinear Multiobjective Optimization—A Generalized Homotopy Approach. Birkhäuser, Basel (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  15. 15.
    Huang, V., Qin, A., K.Deb, Zitzler, E., Suganthan, P., Liang, J., Preuss, M., Huband, S.: Problem definitions for performance assessment of multi-objective optimization algorithms. Technical Report TR-13, Nanyang Technological University, Singapore (2007). http://www3.ntu.edu.sg/home/epnsugan/index_files/CEC-07/CEC07.htm
  16. 16.
    Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10(5), 477–506 (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    Knowles, J., Corne, D.: On metrics for comparing nondominated sets. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC 2002), vol. 1, pp. 711–716. IEEE Press, Piscataway, NJ (2002)Google Scholar
  18. 18.
    Knowles, J.D., Corne, D.W., Fleischer, M.: Bounded archiving using the Lebesgue measure. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC 2003), vol. 4, pp. 2490–2497. IEEE Press, Piscatawa, NJ (2003)Google Scholar
  19. 19.
    Kukkonen, S., Deb, K.: Improved pruning of non-dominated solutions based on crowding distance for bi-objective optimization problems. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC 2006), pp. 1179–1186. IEEE Press, Piscataway, NJ (2006)Google Scholar
  20. 20.
    Mehnen, J., Wagner, T., Rudolph, G.: Evolutionary optimization of dynamic multi-objective test functions. In: Proceedings of the Second Italian Workshop on Evolutionary Computation (GSICE2). ACM Press (2006). CD-ROM; http://ls11-www.cs.uni-dortmund.de/people/rudolph/publications/papers/MWR06.pdf
  21. 21.
    Pareto, V.: Manual of Political Economy. The MacMillan Press, London (1971)Google Scholar
  22. 22.
    Pottharst, A., Baptist, K., Schütze, O., Böcker, J., Fröhlecke, N., Dellnitz, M.: Operating point assignment of a linear motor driven vehicle using multiobjective optimization methods (2004). In: Proceedings of the 11th International Conference EPE-PEMC 2004. Riga, LatviaGoogle Scholar
  23. 23.
    Powell, M.J.D.: On search directions for minimization algorithms. Math. Program. 4(1), 193–201 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rudolph, G., Trautmann, H., Schütze, O.: Homogene Approximation der Paretofront bei mehrkriteriellen Kontrollproblemen. Automatisierungstechnik (at) 60(10), 612–621 (2012)CrossRefGoogle Scholar
  25. 25.
    Rudolph, G., Trautmann, H., Sengupta, S., Schütze, O.: Evenly spaced Pareto front approximations for tricriteria problems based on triangulation. In: Proceedings of 7th International Conference on Evolutionary Multi-Criterion Optimization (EMO 2013), pp. 443–458. Springer, Berlin (2013)Google Scholar
  26. 26.
    Salomon, S., Avigad, G., Goldvard, A., Schütze, O.: PSA—a new scalable space partition based selection algorithm for MOEAs. In: Schütze, O., et al. (eds.) EVOLVE—A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation II (Proceedings), vol. 175, pp. 137–151. Springer, Berlin (2013)Google Scholar
  27. 27.
    Schütze, O., Esquivel, X., Lara, A., Coello Coello, C.A.: Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE Trans. Evol. Comput. 16(4), 504–522 (2012)CrossRefGoogle Scholar
  28. 28.
    Trautmann, H., Rudolph, G., Dominguez-Medina, C., Schütze, O.: Finding evenly spaced Pareto fronts for three-objective optimization problems. In: Schütze, O., et al. (eds.) EVOLVE—A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation II (Proceedings), pp. 89–105. Springer, Berlin (2013)Google Scholar
  29. 29.
    Veldhuizen, D.A.V.: Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. Ph.D. thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio (1999)Google Scholar
  30. 30.
    Witting, K.: Numerical Algorithms for the Treatment of Parametric Optimization Problems and Applications. PhD thesis, University of Paderborn (2012)Google Scholar
  31. 31.
    Witting, K., Schulz, B., Dellnitz, M., Böcker, J., Fröhleke, N.: A new approach for online multiobjective optimization of mechatronic systems. Int. J. Softw. Tools Technol. Transf. 10(3), 223–231 (2008)CrossRefGoogle Scholar
  32. 32.
    Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)CrossRefGoogle Scholar
  33. 33.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)CrossRefGoogle Scholar
  34. 34.
    Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)CrossRefGoogle Scholar
  35. 35.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Günter Rudolph
    • 1
  • Oliver Schütze
    • 2
  • Christian Grimme
    • 3
  • Christian Domínguez-Medina
    • 4
  • Heike Trautmann
    • 3
    Email author
  1. 1.Department of Computer ScienceTU Dortmund UniversityDortmundGermany
  2. 2.Department of Computer ScienceCINVESTAV IPNMexico CityMexico
  3. 3.Department of Information SystemsUniversity of MünsterMünsterGermany
  4. 4.Computer Research CenterNational Polytechnic InstituteMexico CityMexico

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