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Targeting solutions in Bayesian multi-objective optimization: sequential and batch versions

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Abstract

Multi-objective optimization aims at finding trade-off solutions to conflicting objectives. These constitute the Pareto optimal set. In the context of expensive-to-evaluate functions, it is impossible and often non-informative to look for the entire set. As an end-user would typically prefer a certain part of the objective space, we modify the Bayesian multi-objective optimization algorithm which uses Gaussian Processes and works by maximizing the Expected Hypervolume Improvement, to focus the search in the preferred region. The cumulated effects of the Gaussian Processes and the targeting strategy lead to a particularly efficient convergence to the desired part of the Pareto set. To take advantage of parallel computing, a multi-point extension of the targeting criterion is proposed and analyzed.

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References

  1. Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, pp 87–102. ACM (2009)

  2. Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Hypervolume-based multiobjective optimization: theoretical foundations and practical implications. Theor. Comput. Sci. 425, 75–103 (2012)

    Article  MathSciNet  Google Scholar 

  3. Auger, A., Hansen, N.: Performance evaluation of an advanced local search evolutionary algorithm. In: IEEE Congress on Evolutionary Computation, vol. 2, p 2005. IEEE (2005)

  4. Bechikh, S., Kessentini, M., Said, L.B., Ghédira, K.: Chap. 4: preference incorporation in evolutionary multiobjective optimization: a survey of the state-of-the-art. Adv. Comput. 98, 141–207 (2015)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Binois, M., Ginsbourger, D., Roustant, O.: Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations. Eur. J. Oper. Res. 243 (2), 386–394 (2015)

    Article  MathSciNet  Google Scholar 

  7. Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. In: International Conference on Parallel Problem Solving from Nature, pp 722–731. Springer (2004)

  8. Branke, J., Deb, K., Miettinen, K., Slowiński, R.: Multiobjective Optimization: Interactive and Evolutionary Approaches, vol. 5252. Springer, Berlin (2008)

    Book  Google Scholar 

  9. Buchanan, J., Gardiner, L.: A comparison of two reference point methods in multiple objective mathematical programming. Eur. J. Oper. Res. 149(1), 17–34 (2003)

    Article  MathSciNet  Google Scholar 

  10. Chevalier, C., Ginsbourger, D.: Fast computation of the multi-points expected improvement with applications in batch selection. In: International Conference on Learning and Intelligent Optimization, pp 59–69. Springer (2013)

  11. Couckuyt, I., Deschrijver, D., Dhaene, T.: Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization. J. Glob. Optim. 60(3), 575–594 (2014)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Deb, K., Sundar, J.: Reference point based multi-objective optimization using evolutionary algorithms. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, pp 635–642. ACM (2006)

  14. Emmerich, M., Deutz, A., Klinkenberg, J.W.: Hypervolume-based expected improvement: monotonicity properties and exact computation. In: IEEE Congress on Evolutionary Computation (CEC), 2011, pp 2147–2154. IEEE (2011)

  15. Emmerich, M., Yang, K., Deutz, A., Wang, H., Fonseca, C.M.: A multicriteria generalization of Bayesian global optimization. In: Advances in Stochastic and Deterministic Global Optimization, pp 229–242. Springer (2016)

  16. Feliot, P.: Une approche Bayesienne pour L’optimisation multi-objectif sous contraintes. PhD thesis, Universite Paris-Saclay (2017)

  17. Feliot, P., Bect, J., Vazquez, E.: User preferences in Bayesian multi-objective optimization: the expected weighted hypervolume improvement criterion. In: Giuseppe Nicosia, Panos Pardalos, Giovanni Giuffrida, Renato Umeton, and Vincenzo Sciacca, editors, Machine Learning, Optimization, and Data Science, pp 533–544. Springer, Cham (2019)

    Chapter  Google Scholar 

  18. Frazier, P.I., Clark, S.C.: Parallel global optimization using an improved multi-points expected improvement criterion. In: INFORMS Optimization Society Conference, Miami FL, vol. 26 (2012)

  19. Gal, T., Stewart, T., Hanne, T.: Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, vol. 21. Springer, Berlin (1999)

    Book  Google Scholar 

  20. Gaudrie, D., Le Riche, R., Enaux, B., Herbert, V.: Budgeted multi-objective optimization with a focus on the central part of the pareto front-extended version. arXiv:1809.10482 (2018)

  21. Ginsbourger, D., Janusevskis, J., Le Riche, R.: Dealing with asynchronicity in parallel gaussian process based global optimization. In: 4th International Conference of the ERCIM WG on computing and statistics (ERCIM’11) (2011)

  22. Ginsbourger, D., Riche, R.L.: Towards GP-based optimization with finite time horizon. Technical report, Centre d’Hydrogéologie et de Géothermie de Neuchâtel (2009)

  23. Ginsbourger, D., Le Riche, R., Carraro, L.: Kriging is well-suited to parallelize optimization. In: Computational Intelligence in Expensive Optimization Problems, pp 131–162. Springer (2010)

  24. Horn, D., Wagner, T., Biermann, D., Weihs, C., Bischl, B.: Model-based multi-objective optimization: taxonomy, multi-point proposal, toolbox and benchmark. In: International Conference on Evolutionary Multi-Criterion Optimization, pp 64–78. Springer (2015)

  25. Ishibuchi, H., Hitotsuyanagi, Y., Tsukamoto, N., Nojima, Y.: Many-objective test problems to visually examine the behavior of multiobjective evolution in a decision space. In: International Conference on Parallel Problem Solving from Nature, pp 91–100. Springer (2010)

  26. Janusevskis, J., Riche, R.L., Ginsbourger, D.: Parallel expected improvements for global optimization: summary, bounds and speed-up. Technical report, Institut Fayol, École des Mines de Saint-Étienne (2011)

  27. Janusevskis, J., Le Riche, R., Ginsbourger, D., Girdziusas, R.: Expected improvements for the asynchronous parallel global optimization of expensive functions: potentials and challenges. In: Learning and Intelligent Optimization, pp 413–418. Springer (2012)

  28. Jones, D.R, Schonlau, M., Welch, W.: Efficient Global Optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)

    Article  MathSciNet  Google Scholar 

  29. Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica: J. Econometric Society, 513–518 (1975)

    Article  MathSciNet  Google Scholar 

  30. Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44(4), 879–891 (2006)

    Article  Google Scholar 

  31. Knowles, J.: ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans. Evol. Comput. 10(1), 50–66 (2006)

    Article  Google Scholar 

  32. Marmin, S., Chevalier, C., Ginsbourger, D.: Differentiating the multipoint expected improvement for optimal batch design. In: International Workshop on Machine Learning, Optimization and Big Data, pp 37–48. Springer (2015)

  33. Marmin, S., Chevalier, C., Ginsbourger, D.: Efficient batch-sequential bayesian optimization with moments of truncated gaussian vectors. arXiv:1609.02700 (2016)

  34. Miettinen, K.: Nonlinear Multiobjective Optimization, vol. 12. Springer, Berlin (1998)

    Book  Google Scholar 

  35. Molchanov, I.: Theory of Random Sets, vol. 19. Springer, Berlin (2005)

    Google Scholar 

  36. Pardalos, P.M, žilinskas, A., žilinskas, J.: Non-convex Multi-Objective Optimization. Springer, Berlin (2017)

    Book  Google Scholar 

  37. Parr, J.: Improvement criteria for constraint handling and multiobjective optimization. PhD thesis, University of Southampton (2013)

  38. Picheny, V.: Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction. Stat. Comput. 25(6), 1265–1280 (2015)

    Article  MathSciNet  Google Scholar 

  39. Ponweiser, W., Wagner, T., Biermann, D., Vincze, M.: Multiobjective optimization on a limited budget of evaluations using model-assisted S-metric selection. In: International Conf. on Parallel Problem Solving from Nature, pp 784–794. Springer (2008)

  40. Ribaud, M.: Krigeage pour la conception de turbomachines: grande dimension et optimisation robuste. PhD thesis Université de Lyon (2018)

  41. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization, vol. 176. Elsevier, Amsterdam (1985)

    MATH  Google Scholar 

  42. Schonlau, M.: Computer experiments and global optimization. PhD thesis, University of Waterloo (1997)

  43. Svenson, J.: Computer experiments: multiobjective optimization and sensitivity analysis. PhD thesis, The Ohio State University (2011)

  44. Svenson, J., Santner, T.J.: Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement, p 32. The Ohio State University Columbus, Ohio (2010)

    Google Scholar 

  45. Triantaphyllou, E.: Multi-criteria decision making methods. In: Multi-Criteria Decision Making Methods: a Comparative Study, pp 5–21. Springer (2000)

  46. While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE Trans. Evol. Comput. 16(1), 86–95 (2012)

    Article  Google Scholar 

  47. Wierzbicki, A.: The use of reference objectives in multiobjective optimization. In: Multiple Criteria Decision Making Theory and Application, pp 468–486. Springer (1980)

  48. Wierzbicki, A.: Reference point approaches. In: Gal, T., Stewart, T., Hanne, T. (eds.) Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, pp 237–275. Springer (1999)

  49. Yang, K., Emmerich, M., Deutz, A., Bäck, T.: Multi-objective Bayesian global optimization using expected hypervolume improvement gradient. Swarm Evol. Comput. 44, 945–956 (2019)

    Article  Google Scholar 

  50. Yang, K., Emmerich, M., Deutz, A., Fonseca, C.M.: Computing 3-D expected hypervolume improvement and related integrals in asymptotically optimal time. In: International Conference on Evolutionary Multi-Criterion Optimization, pp 685–700. Springer (2017)

  51. Yang, K., Li, L., Deutz, A., Bäck, T., Emmerich, M.: Preference-based multiobjective optimization using truncated expected hypervolume improvement. In: 2016 12th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD), pp 276–281. IEEE (2016)

  52. Zeleny, M.: The theory of the displaced ideal. In: Multiple Criteria Decision Making Kyoto 1975, pp 153–206. Springer (1976)

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

    Article  Google Scholar 

  54. Zitzler, E.: Evolutionary algorithms for multiobjective optimization: methods and applications. Citeseer (1999)

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Acknowledgments

This research was performed within the framework of a CIFRE grant (convention #2016/0690) established between the ANRT and the Groupe PSA for the doctoral work of David Gaudrie.

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Authors and Affiliations

  1. Groupe PSA, Vélizy-Villacoublay, France

    David Gaudrie, Benoît Enaux & Vincent Herbert

  2. CNRS LIMOS, École Nationale Supérieure des Mines de Saint-Étienne, Saint-Étienne, France

    David Gaudrie & Rodolphe Le Riche

  3. Prowler.io, Cambridge, UK

    Victor Picheny

Authors
  1. David Gaudrie
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  2. Rodolphe Le Riche
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  3. Victor Picheny
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  4. Benoît Enaux
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  5. Vincent Herbert
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Correspondence to David Gaudrie.

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Gaudrie, D., Le Riche, R., Picheny, V. et al. Targeting solutions in Bayesian multi-objective optimization: sequential and batch versions. Ann Math Artif Intell 88, 187–212 (2020). https://doi.org/10.1007/s10472-019-09644-8

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