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Multimodality in Multi-objective Optimization – More Boon than Bane?

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Evolutionary Multi-Criterion Optimization (EMO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11411))

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Abstract

This paper addresses multimodality of multi-objective (MO) optimization landscapes. Contrary to common perception of local optima, according to which they are hindering the progress of optimization algorithms, it will be shown that local efficient sets in a multi-objective setting can assist optimizers in finding global efficient sets. We use sophisticated visualization techniques, which rely on gradient field heatmaps, to highlight those insights into landscape characteristics. Finally, the MO local optimizer MOGSA is introduced, which exploits those observations by sliding down the multi-objective gradient hill and moving along the local efficient sets.

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Notes

  1. 1.

    If no initial point is given, it will be sampled randomly within the search space.

  2. 2.

    Note that the current implementation of MOGSA only enables the optimization of bi-objective problems.

References

  1. Beume, N., Naujoks, B., Emmerich, M.T.M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. EJOR 181(3), 1653–1669 (2007)

    Article  Google Scholar 

  2. Bossek, J.: ecr 2.0: a modular framework for evolutionary computation in R. In: Proceedings of GECCO Companion, pp. 1187–1193. ACM (2017)

    Google Scholar 

  3. Bossek, J.: smoof: single- and multi-objective optimization test functions. R J. (2017). https://journal.r-project.org/archive/2017/RJ-2017-004/

  4. Brockhoff, D., Tran, T.D., Hansen, N.: Benchmarking numerical multiobjective optimizers revisited. In: Proceedings of GECCO, pp. 639–646. ACM (2015)

    Google Scholar 

  5. Burden, R.L., Faires, D.J.: Numeric Analysis, 3rd edn. Prindle, Weber & Schmidt Publishing Company, Boston (1985)

    Google Scholar 

  6. Daolio, F., Liefooghe, A., Verel, S., Aguirre, H.E., Tanaka, K.: Global vs local search on multi-objective NK-landscapes: contrasting the impact of problem features. In: Proceedings of GECCO, pp. 369–376. ACM (2015)

    Google Scholar 

  7. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE TEVC 6(2), 182–197 (2002)

    Google Scholar 

  8. Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable test problems for evolutionary multiobjective optimization. In: Abraham, A., Jain, L., Goldberg, R. (eds.) Evolutionary Multiobjective Optimization, pp. 105–145. Springer, London (2005)

    Chapter  Google Scholar 

  9. Ehrgott, M., Klamroth, K.: Connectedness of efficient solutions in multiple criteria combinatorial optimization. EJOR 97(1), 159–166 (1997)

    Article  Google Scholar 

  10. Emmerich, M.T.M., Deutz, A.H.: Test problems based on Lamé superspheres. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 922–936. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70928-2_68

    Chapter  Google Scholar 

  11. da Fonseca, C.M.M.: Multiobjective genetic algorithms with application to control engineering problems. Ph.D. thesis, University of Sheffield (1995)

    Google Scholar 

  12. Gerstl, K., Rudolph, G., Schtze, O., Trautmann, H.: Finding evenly spaced fronts for multiobjective control via averaging Hausdorff-measure. In: 2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control, pp. 1–6 (2011). https://doi.org/10.1109/ICEEE.2011.6106656

  13. Grimme, C., Kerschke, P., Emmerich, M.T.M., Preuss, M., Deutz, A.H., Trautmann, H.: Sliding to the global optimum: how to benefit from non-global optima in multimodal multi-objective optimization. In: Proceedings of LeGO (2018, accepted)

    Google Scholar 

  14. Grimme, C., Lepping, J., Papaspyrou, A.: Adapting to the habitat: on the integration of local search into the predator-prey model. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 510–524. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01020-0_40

    Chapter  Google Scholar 

  15. Hansen, N., Finck, S., Ros, R., Auger, A.: Real-parameter black-box optimization benchmarking 2009: noiseless functions definitions. Technical report, INRIA (2009)

    Google Scholar 

  16. Jin, Y., Sendhoff, B.: Connectedness, regularity and the success of local search in evolutionary multi-objective optimization. In: Proceedings of the IEEE CEC, vol. 3, pp. 1910–1917. IEEE (2003)

    Google Scholar 

  17. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays, Courant Anniversary Volume, pp. 187–204. Interscience (1948)

    Google Scholar 

  18. Kerschke, P., Grimme, C.: An expedition to multimodal multi-objective optimization landscapes. In: Trautmann, H., et al. (eds.) EMO 2017. LNCS, vol. 10173, pp. 329–343. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-54157-0_23

    Chapter  Google Scholar 

  19. Kerschke, P., et al.: Cell mapping techniques for exploratory landscape analysis. In: Tantar, A.-A., et al. (eds.) EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation V. AISC, vol. 288, pp. 115–131. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07494-8_9

    Chapter  Google Scholar 

  20. Kerschke, P., et al.: Towards analyzing multimodality of continuous multiobjective landscapes. In: Handl, J., Hart, E., Lewis, P.R., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds.) PPSN 2016. LNCS, vol. 9921, pp. 962–972. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45823-6_90

    Chapter  Google Scholar 

  21. Kerschke, P., et al.: Search dynamics on multimodal multi-objective problems. Evol. Comput. 1–33 (2018). https://doi.org/10.1162/evco_a_00234

  22. Preuss, M.: Multimodal Optimization by Means of Evolutionary Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-07407-8. https://www.springer.com/de/book/9783319074061

    Book  MATH  Google Scholar 

  23. Rosenthal, S., Borschbach, M.: A concept for real-valued multi-objective landscape analysis characterizing two biochemical optimization problems. In: Mora, A.M., Squillero, G. (eds.) EvoApplications 2015. LNCS, vol. 9028, pp. 897–909. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16549-3_72

    Chapter  Google Scholar 

  24. Schütze, O., Hernández, V.A., Trautmann, H., Rudolph, G.: The hypervolume based directed search method for multi-objective optimization problems. J. Heuristics 22(3), 273–300 (2016)

    Article  Google Scholar 

  25. Schütze, O., Martín, A., Lara, A., Alvarado, S., Salinas, E., Coello, C.A.: The directed search method for multi-objective memetic algorithms. Comput. Optim. Appl. 63(2), 305–332 (2016)

    Article  MathSciNet  Google Scholar 

  26. Schütze, O., Sanchez, G., Coello Coello, C.A.: A new memetic strategy for the numerical treatment of multi-objective optimization problems. In: Proceedings of GECCO, pp. 705–712. ACM (2008)

    Google Scholar 

  27. Spall, J.C.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 37(3), 332–341 (1992)

    Article  MathSciNet  Google Scholar 

  28. Stein, M.: Large sample properties of simulations using latin hypercube sampling. Technometrics 29, 143–151 (1987)

    Article  MathSciNet  Google Scholar 

  29. Tušar, T., Filipič, B.: Visualization of Pareto front approximations in evolutionary multiobjective optimization: a critical review and the prosection method. IEEE TEVC 19(2), 225–245 (2015)

    Google Scholar 

  30. Tušar, T., Brockhoff, D., Hansen, N., Auger, A.: COCO: the bi-objective black box optimization benchmarking (bbob-biobj) test suite. arXiv preprint (2016)

    Google Scholar 

  31. Ulrich, T., Bader, J., Thiele, L.: Defining and optimizing indicator-based diversity measures in multiobjective search. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN 2010. LNCS, vol. 6238, pp. 707–717. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15844-5_71

    Chapter  Google Scholar 

  32. Verel, S., Liefooghe, A., Jourdan, L., Dhaenens, C.: On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives. Eur. J. Oper. Res. 227(2), 331–342 (2013)

    Article  MathSciNet  Google Scholar 

  33. Wessing, S.: Two-stage methods for multimodal optimization. Ph.D. thesis, Technische Universität Dortmund (2015). http://hdl.handle.net/2003/34148

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Authors and Affiliations

  1. Information Systems and Statistics, ERCIS, University of Münster, Leonardo-Campus 3, 48149, Münster, Germany

    Christian Grimme, Pascal Kerschke & Heike Trautmann

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  1. Christian Grimme
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  2. Pascal Kerschke
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  3. Heike Trautmann
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Correspondence to Christian Grimme .

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Editors and Affiliations

  1. Michigan State University, East Lansing, MI, USA

    Kalyanmoy Deb

  2. Michigan State University, East Lansing, MI, USA

    Erik Goodman

  3. CINVESTAV-IPN, Mexico, Mexico

    Carlos A. Coello Coello

  4. University of Wuppertal, Wuppertal, Germany

    Kathrin Klamroth

  5. University of Jyvaskyla, Jyväskylä, Finland

    Kaisa Miettinen

  6. Otto von Guericke University Magdeburg, Magdeburg, Germany

    Sanaz Mostaghim

  7. Cornell University, Ithaca, NY, USA

    Patrick Reed

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Grimme, C., Kerschke, P., Trautmann, H. (2019). Multimodality in Multi-objective Optimization – More Boon than Bane?. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_11

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  • DOI: https://doi.org/10.1007/978-3-030-12598-1_11

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