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An algorithmic approach to multiobjective optimization with decision uncertainty

Journal of Global Optimization volume 77, pages 3–25 (2020)Cite this article

Abstract

In real life applications, optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one decision variable is a whole set, which includes all possible outcomes of this decision variable. We choose a robust approach and thus these sets have to be compared using the so-called upper-type less order relation. We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branch-and-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization, as convex underestimators, as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.

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References

  1. Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs: I-theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  2. Barrico, C., Antunes, C.: Robustness analysis in multi-objective optimization using a degree of robustness concept. In: IEEE Congress on Evolutionary Computation. CEC 2006, pp. 1887–1892. IEEE Computer Society (2006)

  3. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  Google Scholar 

  4. Ben-Tal, A., Nemirovski, A.: Robust optimization—methodology and applications. Math. Program. 92(3), 453–480 (2002)

    Article  MathSciNet  Google Scholar 

  5. Benson, H.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13(1), 1–24 (1998)

    Article  MathSciNet  Google Scholar 

  6. Binh, T., Korn, U.: MOBES: A multiobjective evolution strategy for constrained optimization problems. In: Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 176–182 (1999)

  7. Deb, K., Gupta, H.: Introducing robustness in multi-objective optimization. Evol. Comput. 14(4), 463–494 (2006)

    Article  Google Scholar 

  8. Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Glob. Optim. 50(3), 397–416 (2011)

    Article  MathSciNet  Google Scholar 

  9. Eichfelder, G., Gerlach, T.: On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances. In: Khan, A., Köbis, E., Tammer, C. (eds.) Variational Analysis and Set Optimization, Chap. 10, pp. 265–290. CRC Press, Boca Raton (2019)

    MATH  Google Scholar 

  10. Eichfelder, G., Krüger, C., Schöbel, A.: Decision uncertainty in multiobjective optimization. J. Glob. Optim. 69(2), 485–510 (2017)

    Article  MathSciNet  Google Scholar 

  11. Fliege, J., Werner, R.: Robust multiobjective optimization & applications in portfolio optimization. Eur. J. Oper. Res. 234(2), 422–433 (2014). https://doi.org/10.1016/j.ejor.2013.10.028

    Article  MathSciNet  MATH  Google Scholar 

  12. Goberna, M.A., Jeyakumar, V., Li, G., Vicente-Pérez, J.: Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty. SIAM J. Optim. 24(3), 1402–1419 (2014)

    Article  MathSciNet  Google Scholar 

  13. Ide, J., Köbis, E., Kuroiwa, D., Schöbel, A., Tammer, C.: The relationship between multi-objective robustness concepts and set-valued optimization. Fixed Point Theory Appl. 2014(1), 83 (2014)

    Article  MathSciNet  Google Scholar 

  14. Jahn, J.: Vectorization in set optimization. J. Optim. Theory Appl. 167(3), 783–795 (2013)

    Article  MathSciNet  Google Scholar 

  15. Jahn, J.: A derivative-free descent method in set optimization. Comput. Optim. Appl. 60(2), 393–411 (2015)

    Article  MathSciNet  Google Scholar 

  16. Jahn, J.: A derivative-free rooted tree method in nonconvex set optimization. Pure Appl. Funct. Anal. 3, 603–623 (2018)

    MathSciNet  Google Scholar 

  17. Jahn, J., Ha, T.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    Article  MathSciNet  Google Scholar 

  18. Khan, A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)

    Book  Google Scholar 

  19. Köbis, E., Köbis, M.: Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization. Optimization 65(10), 1805–1827 (2016)

    Article  MathSciNet  Google Scholar 

  20. Krüger, C.: On minmax robustness for multiobjective optimization with decision or parameter uncertainty. Ph.D. thesis, Georg-August Universität Göttingen (2018)

  21. Krüger, C.: Peat and pots: analysis of robust solutions for a biobjective problem in agriculture. Preprint-Reihe, Institut für Numerische und Angewandte Mathematik, Georg-August Universität Göttingen (2018)

  22. Krüger, C., Castellani, F., Geldermann, J., Schöbel, A.: Peat and pots: an application of robust multiobjective optimization to a mixing problem in agriculture. Comput. Electron. Agric. 154, 265–275 (2018)

    Article  Google Scholar 

  23. Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1031, 85–90 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Kuroiwa, D., Lee, G.M.: On robust multiobjective optimization. Vietnam J. Math. 40(2&3), 305–317 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Lewis, A., Pang, C.: Lipschitz behavior of the robust regularization. SIAM J. Control Optim. 48(5), 3080–3105 (2009)

    Article  MathSciNet  Google Scholar 

  26. Löhne, A., Rudloff, B., Ulus, F.: Primal and dual approximation algorithms for convex vector optimization problems. J. Glob. Optim. 60(4), 713–736 (2014)

    Article  MathSciNet  Google Scholar 

  27. Löhne, A., Schrage, C.: An algorithm to solve polyhedral convex set optimization problems. Optimization 62(1), 131–141 (2013)

    Article  MathSciNet  Google Scholar 

  28. Maranas, C.D., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4(2), 135–170 (1994)

    Article  MathSciNet  Google Scholar 

  29. Niebling, J., Eichfelder, G.: A branch-and-bound-based algorithm for nonconvex multiobjective optimization. SIAM J. Optim. 29(1), 794–821 (2019)

    Article  MathSciNet  Google Scholar 

  30. Rodríguez-Marín, L., Sama, M.: \((\Lambda, C)\)-contingent derivatives of set-valued maps. J. Math. Anal. Appl. 335(2), 974–989 (2007)

    Article  MathSciNet  Google Scholar 

  31. Rump, S.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  32. Schulze Darup, M., Mönnigmann, M.: Improved automatic computation of Hessian matrix spectral bounds. SIAM J. Sci. Comput. 38(4), A2068–A2090 (2016)

    Article  MathSciNet  Google Scholar 

  33. Zhou-Kangas, Y., Miettinen, K., Sindhya, K.: Solving multiobjective optimization problems with decision uncertainty: an interactive approach. J. Bus. Econ. 89(1), 25–51 (2019). https://doi.org/10.1007/s11573-018-0900-1

    Article  Google Scholar 

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Acknowledgements

The second author thanks the Carl-Zeiss-Stiftung and the DFG-founded Research Training Group 1567 “Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing” for financial support. The work of the third author is funded by the Deutsche Forschungsgemeinschaft under Grant No. EI 821/4. Additionally, we thank the two anonymous referees for their helpful remarks and comments, which helped us to improve this article.

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Authors and Affiliations

  1. Technische Universität Ilmenau, Ilmenau, Germany

    Gabriele Eichfelder, Julia Niebling & Stefan Rocktäschel

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  1. Gabriele Eichfelder
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  2. Julia Niebling
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  3. Stefan Rocktäschel
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Correspondence to Julia Niebling.

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Eichfelder, G., Niebling, J. & Rocktäschel, S. An algorithmic approach to multiobjective optimization with decision uncertainty. J Glob Optim 77, 3–25 (2020). https://doi.org/10.1007/s10898-019-00815-9

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  • DOI: https://doi.org/10.1007/s10898-019-00815-9

Keywords

  • Multiobjective optimization
  • Decision uncertainty
  • Branch-and-bound algorithm