Trust-Region Based Multi-objective Optimization for Low Budget Scenarios

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11411)


In many practical multi-objective optimization problems, evaluation of objectives and constraints are computationally time-consuming, because they require expensive simulation of complicated models. Researchers often use a comparatively less time-consuming surrogate or metamodel (model of models) to drive the optimization task. Effectiveness of the metamodeling method relies not only on how it manages the search process (to find infill sampling) but also how it deals with associated error uncertainty between metamodels and the true models during an optimization run. In this paper, we propose a metamodel-based multi-objective evolutionary algorithm that adaptively maintains regions of trust in variable space to make a balance between error uncertainty and progress. In contrast to other trust-region methods for single-objective optimization, our method aims to solve multi-objective expensive problems where we incorporate multiple trust regions, corresponding to multiple non-dominated solutions. These regions can grow or shrink in size according to the deviation between metamodel prediction and high-fidelity computed values. We introduce two performance indicators based on hypervolume and achievement scalarization function (ASF) to control the size of the trust regions. The results suggest that our proposed trust-region based methods can effectively solve test and real-world problems using a limited budget of solution evaluations with increased accuracy.


Surrogate modeling Metamodel Trust-region method Multi-objective optimization 


  1. 1.
    Alexandrov, N.M., Dennis, J.E., Lewis, R.M., Torczon, V.: A trust-region framework for managing the use of approximation models in optimization. Struct. Optim. 15(1), 16–23 (1998)CrossRefGoogle Scholar
  2. 2.
    Bhattacharjee, K.S., Singh, H.K., Ray, T.: Multi-objective optimization with multiple spatially distributed surrogates. J. Mech. Des. 138(9), 091401-091401-10 (2016)CrossRefGoogle Scholar
  3. 3.
    Bhattacharjee, K.S., Singh, H.K., Ray, T., Branke, J.: Multiple surrogate assisted multiobjective optimization using improved pre-selection. In: IEEE CEC (2016)Google Scholar
  4. 4.
    Chugh, T., Jin, Y., Miettinen, K., Hakanen, J., Sindhya, K.: A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization. IEEE Trans. Evol. Comput. 22(1), 129–142 (2018)CrossRefGoogle Scholar
  5. 5.
    Deb, K., Hussein, R., Roy, P.C., Toscano, G.: A taxonomy for metamodeling frameworks for evolutionary multi-objective optimization. IEEE Trans. Evol. Comput. (in Press)Google 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.
    Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186, 311–338 (2000)CrossRefGoogle Scholar
  8. 8.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)zbMATHGoogle Scholar
  9. 9.
    Deb, K., Hussein, R., Roy, P., Toscano, G.: Classifying metamodeling methods for evolutionary multi-objective optimization: first results. In: Trautmann, H., et al. (eds.) EMO 2017. LNCS, vol. 10173, pp. 160–175. Springer, Cham (2017). Scholar
  10. 10.
    Emmerich, M.T.M., Deutz, A.H., Klinkenberg, J.W.: Hypervolume-based expected improvement: monotonicity properties and exact computation. In: 2011 IEEE Congress of Evolutionary Computation, CEC, pp. 2147–2154 (2011)Google Scholar
  11. 11.
    Emmerich, M.T.M., Giannakoglou, K.C., Naujoks, B.: Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans. Evol. Comput. 10(4), 421–439 (2006)CrossRefGoogle Scholar
  12. 12.
    Fliege, J., Vaz, A.I.F.: A method for constrained multiobjective optimization based on SQP techniques. SIAM J. Optim. 26(4), 2091–2119 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hussein, R., Deb, K.: A generative kriging surrogate model for constrained and unconstrained multi-objective optimization. In: GECCO 2016. ACM Press (2016)Google Scholar
  14. 14.
    Jin, Y.: Surrogate-assisted evolutionary computation: recent advances and future challenges. Swarm Evol. Comput. 1(2), 61–70 (2011)CrossRefGoogle Scholar
  15. 15.
    Knowles, J.: ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans. Evol. Comput. 10, 50–66 (2006)CrossRefGoogle Scholar
  16. 16.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  17. 17.
    Pedrielli, G., Ng, S.: G-STAR: a new kriging-based trust region method for global optimization. IEEE Press, United States, January 2017Google Scholar
  18. 18.
    Ponweiser, W., Wagner, T., Biermann, D., Vincze, M.: Multiobjective optimization on a limited budget of evaluations using model-assisted \(\cal{S}\)-metric selection. In: Rudolph, G., Jansen, T., Beume, N., Lucas, S., Poloni, C. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 784–794. Springer, Heidelberg (2008). Scholar
  19. 19.
    Roy, P., Deb, K.: High dimensional model representation for solving expensive multi-objective optimization problems. In: IEEE CEC, pp. 2490–2497 (2016)Google Scholar
  20. 20.
    Roy, P.C., Deb, K., Islam, M.M.: An efficient nondominated sorting algorithm for large number of fronts. IEEE Trans. Cyber. 1–11 (2018)Google Scholar
  21. 21.
    Roy, P., Hussein, R., Deb, K.: Metamodeling for multimodal selection functions in evolutionary multi-objective optimization. In: GECCO 2017. ACM Press (2017)Google Scholar
  22. 22.
    Roy, P.C., Blank, J., Hussein, R., Deb, K.: Trust-region based algorithms with low-budget for multi-objective optimization. In: GECCO, pp. 195–196. ACM (2018)Google Scholar
  23. 23.
    Roy, P.C., Islam, M.M., Deb, K.: Best order sort: a new algorithm to non-dominated sorting for evolutionary multi-objective optimization. In: GECCO (2016)Google Scholar
  24. 24.
    Ryu, J.H., Kim, S.: A derivative-free trust-region method for biobjective optimization. SIAM J. Optim. 24(1), 334–362 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Viana, F.A.C., Haftka, R.T., Watson, L.T.: Efficient global optimization algorithm assisted by multiple surrogate techniques. J. Global Optim. 56, 669–689 (2013)CrossRefGoogle Scholar
  26. 26.
    Wierzbicki, A.P.: The use of reference objectives in multiobjective optimization. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making Theory and Application. LNE, vol. 177, pp. 468–486. Springer, Heidelberg (1980). Scholar
  27. 27.
    Zhang, Q., Liu, W., Tsang, E., Virginas, B.: Expensive multiobjective optimization by MOEA/D With Gaussian process model. IEEE Trans. Evol. Comput. 14(3), 456–474 (2010)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

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