Skip to main content
Log in

The hypervolume based directed search method for multi-objective optimization problems

Journal of Heuristics Aims and scope Submit manuscript

Abstract

We present a new hybrid evolutionary algorithm for the effective hypervolume approximation of the Pareto front of a given differentiable multi-objective optimization problem. Starting point for the local search (LS) mechanism is a new division of the decision space as we will argue that in each of these regions a different LS strategy seems to be most promising. For the LS in two out of the three regions we will utilize and adapt the Directed Search method which is capable of steering the search into any direction given in objective space and which is thus well suited for the problem at hand. We further on integrate the resulting LS mechanism into SMS-EMOA, a state-of-the-art evolutionary algorithm for hypervolume approximations. Finally, we will present some numerical results on several benchmark problems with two and three objectives indicating the strength and competitiveness of the novel hybrid.

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

Access this article

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Notes

  1. An overview about recent results on the complexity of HV computation is provided in http://simco.gforge.inria.fr/doku.php?id=openproblems.

References

  • 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 ’09, pp. 87–102. ACM, New York, NY, USA (2009)

  • Beume, N.: S-Metric calculation by considering dominated hypervolume as Klee’s measure problem. Evolut. Comput. 17(4), 477–492 (2009)

    Article  Google Scholar 

  • Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2006)

    Article  MATH  Google Scholar 

  • Bosman, P.A.N.: On gradients and hybrid evolutionary algorithms for real-valued multiobjective optimization. IEEE Trans. Evolut. Comput. 16(1), 51–69 (2012)

    Article  Google Scholar 

  • Bringmann, K., Friedrich, T.: The maximum hypervolume set yields near-optimal approximation. In: Proceedings of the ACM Genetic and Evolutionary Computation Conference (GECCO), pp. 511–518. ACM Press (2010)

  • Bringmann, K., Friedrich, T.: Tight bounds for the approximation ratio of the hypervolume indicator. In: Proceedings of the 11th International Conference Parallel Problem Solving From Nature (PPSN). Lecture Notes in Computer Science, vol. 6238, pp. 607–616, Krakow, Poland, September. Springer (2010)

  • Brown, M., Smith, R.E.: Directed multi-objective optimisation. Int. J. Comput. Syst. Signals 6(1), 3–17 (2005)

    Google Scholar 

  • 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 

  • Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester (2001)

    MATH  Google Scholar 

  • Emmerich, M., Deutz, A.: Time complexity and zeros of the hypervolume indicator gradient field. In: Schütze, O., et al. (eds.) EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation III. Studies in Computational Intelligence, vol. 500, pp. 169–193. Springer, Berlin (2014)

    Google Scholar 

  • Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20, 602–626 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Fliege, J., Fux Svaiter, B.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Hernández, V.A.S., Schütze, O., Emmerich, M.: Hypervolume maximization via set based newtons method. In: Tantar, A., et al. (eds.) EVOLVE—A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation V. Advances in Intelligent Systems and Computing, vol. 288, pp. 15–28. springer, Berlin (2014)

    Google Scholar 

  • Hernández, V.A.S., Schütze, O., Rudolph, G., Trautmann, H.: The directed search method for pareto front approximations with maximum dominated hypervolume. In: Emmerich, M., et al. (eds.) EVOLVE—A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation IV. Advances in Intelligent Systems and Computing, vol. 227, pp. 189–205. Springer, Berlin (2013)

    Google Scholar 

  • Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. Springer, Basel (2001)

    Book  MATH  Google Scholar 

  • Ishibuchi, H., Yoshida, T., Murata, T.: Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling. IEEE Trans. Evolut. Comput. 7(2), 204–223 (2003)

    Article  Google Scholar 

  • Jaszkiewicz, A., Ishibuchi, H., Zhang, Q.: Multiobjective memetic algorithms. Handbook of Memetic Algorithms. Studies in Computational Intelligence, vol. 379, pp. 201–217. Springer, Berlin (2012)

    Chapter  Google Scholar 

  • Knowles, J., Corne, D.: Memetic algorithms for multiobjective optimization: issues, methods and prospects. In: Hart, W.E., Krasnogor, N., Smith, J.E. (eds.) Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166, pp. 313–352. Springer, Berlin (2005)

    Chapter  Google Scholar 

  • Knowles, J.D.: Local-search and hybrid evolutionary algorithms for pareto optimization. PhD thesis, University of Reading, UK (2002). (Section 4.3.4 “S metric Archiving”)

  • Knowles, J.D., Corne, D.: Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Trans. Evolut. Comput. 7(2), 100–116 (2003)

    Article  Google Scholar 

  • Knowles, J., Corne, D.: On metrics for comparing nondominated sets. In: Proceedings of the 2002 Congress on Evolutionary Computation (CEC’02), vol. 1, pp. 711–716. IEEE (2002)

  • Lara, A., Coello Coello, C. A., Schütze, O.: A painless gradient-assisted multi-objective memetic mechanism for solving continuous bi-objective optimization problems. In: 2010 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE Press, IEEE (2010)

  • Lara, A., Sanchez, G., Coello Coello, C.A., Schütze, O.: HCS: a new local search strategy for memetic multiobjective evolutionary algorithms. IEEE Trans. Evolut. Comput. 14(1), 112–132 (2010)

    Article  Google Scholar 

  • Moscato, P.: On evolution, search, optimization, genetic algorithms and martial arts: toward memetic algorithms. Technical Report C3P Report 826, Caltech Concurrent Computation Program (1989)

  • Rozenberg, G., Bäck, T., Kok, J.N.: Handbook of Natural Computing. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  • Schäffler, S., Schultz, R., Weinzierl, K.: A stochastic method for the solution of unconstrained vector optimization problems. J. Optim. Theory Appl. 114(1), 209–222 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Schütze, O., Martín, A., Lara, A., Alvarado, S., Salinas, E., Coello, C.A.: The directed search method for multiobjective memetic algorithms. J. Comput. Optim. Appl. 63, 305–332 (2016)

    Article  MATH  Google Scholar 

  • Shukla, P.: On gradient based local search methods in unconstrained evolutionary multi-objective optimization. In: Obayashi, S. et al. (ed), EMO 2007, pp. 96–110 (2007)

  • Shukla, P.K.: Gradient based stochastic mutation operators in evolutionary multi-objective optimization. Adaptive and Natural Computing Algorithms, pp. 58–66. Springer, Berlin (2007)

    Chapter  Google Scholar 

  • Vasile, M., Zuiani, F.: Multi-agent collaborative search: an agent-based memetic multi-objective optimization algorithm applied to space trajectory design. Proc. Inst. Mech. Eng. G 225(11), 1211–1227 (2011)

    Article  Google Scholar 

  • Wagner, T., Beume, N., Naujoks, B.: Pareto-, aggregation-, and indicator-based methods in many-objective optimization. In: Proceedings of the 4th International Conference on Evolutionary Multi-criterion Optimization, EMO’07, pp. 742–756 (2007). Springer, Berlin

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

    Article  Google Scholar 

  • Zitzler, E.: Evolutionary algorithms for multiobjective optimization: methods and applications. PhD thesis, ETH Zurich (1999)

  • Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X. et al., (eds) Proc. Int’l Conf. on Parallel problem Solving from Nature (PPSN VIII), pp. 832–842. Springer, Berlin (2004)

  • Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evolut. Comput. 7(2), 117–132 (2003)

    Article  Google Scholar 

Download references

Acknowledgments

Victor Adrian Sosa Hernández acknowledges support by the Consejo Nacional de Ciencia y Tecnología (CONACYT). Heike Trautmann acknowledges support by the European Center of Information Systems (ERCIS). All authors acknowledge support from CONACYT Project no. 174443 and DFG Project no. TR 891/5-1

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Oliver Schütze.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schütze, O., Hernández, V.A.S., Trautmann, H. et al. The hypervolume based directed search method for multi-objective optimization problems. J Heuristics 22, 273–300 (2016). https://doi.org/10.1007/s10732-016-9310-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-016-9310-0

Keywords

Navigation