Skip to main content
SpringerLink
Log in

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

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. We note that \(\Delta _p(\cdot ,\cdot )\) is not a metric but an inframetric (a metric with relaxed triangle inequality) [27]. Keeping this fact in mind, we take the liberty to use the term distance in this context.

  2. A tridiagonal matrix is irreducibly diagonally dominant if it is weakly diagonal dominant, both off-diagonals have nonzero entries, and there exists at least one row that is strictly diagonal dominant.

  3. This will not lead to exactly uniformly distributed Pareto-optimal solutions. However, the number of reference solutions is by far larger than the approximated set’s size.

References

  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)

  2. Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  5. Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  8. Durillo, J.J., Nebro, A.J.: jMetal: a Java framework for multi-objective optimization. Adv. Eng. Softw. 42(10), 760–771 (2011)

    Article  Google Scholar 

  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)

  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)

  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. 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)

  13. Hillermeier, C.: Nonlinear Multiobjective Optimization—A Generalized Homotopy Approach. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  14. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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)

  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)

  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)

  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. Pareto, V.: Manual of Political Economy. The MacMillan Press, London (1971)

    Google Scholar 

  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, Latvia

  23. Powell, M.J.D.: On search directions for minimization algorithms. Math. Program. 4(1), 193–201 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rudolph, G., Trautmann, H., Schütze, O.: Homogene Approximation der Paretofront bei mehrkriteriellen Kontrollproblemen. Automatisierungstechnik (at) 60(10), 612–621 (2012)

    Article  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  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)

  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)

  30. Witting, K.: Numerical Algorithms for the Treatment of Parametric Optimization Problems and Applications. PhD thesis, University of Paderborn (2012)

  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)

    Article  Google Scholar 

  32. Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)

    Article  Google Scholar 

  33. Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Computer Science, TU Dortmund University, Dortmund, Germany

    Günter Rudolph

  2. Department of Computer Science, CINVESTAV IPN, Mexico City, Mexico

    Oliver Schütze

  3. Department of Information Systems, University of Münster, Münster, Germany

    Christian Grimme & Heike Trautmann

  4. Computer Research Center, National Polytechnic Institute, Mexico City, Mexico

    Christian Domínguez-Medina

Authors
  1. Günter Rudolph
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver Schütze
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Grimme
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christian Domínguez-Medina
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Heike Trautmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heike Trautmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rudolph, G., Schütze, O., Grimme, C. et al. Optimal averaged Hausdorff archives for bi-objective problems: theoretical and numerical results. Comput Optim Appl 64, 589–618 (2016). https://doi.org/10.1007/s10589-015-9815-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-015-9815-8

Keywords

Search

Navigation