NEO 2016 pp 159-182 | Cite as

Gradient-Based Multiobjective Optimization with Uncertainties

  • Sebastian PeitzEmail author
  • Michael Dellnitz
Part of the Studies in Computational Intelligence book series (SCI, volume 731)


In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated into a subdivision algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.



This work is supported by the Priority Programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems” of the German Research Foundation (DFG).


  1. 1.
    Banholzer, S., Beermann, D., Volkwein, S.: POD-based bicriterial optimal control by the reference point method. In: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations, pp. 210–215 (2016)Google Scholar
  2. 2.
    Basseur, M., Zitzler, E.: Handling uncertainty in indicator-based multiobjective optimization. Int. J. Comput. Intell. Res. 2(3), 255–272 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bosman, P.A.N.: On gradients and hybrid evolutionary algorithms. IEEE Trans. Evol. Comput. 16(1), 51–69 (2012)CrossRefGoogle Scholar
  4. 4.
    Coello Coello, C.A., Van Veldhuizen, D.A., Lamont, G.B.: Evolutionary Algorithms for Solving Multi-objective Problems, vol. 242. Kluwer Academic, New York (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Deb, K., Gupta, H.: Searching for robust Pareto-optimal solutions in multi-objective optimization. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) Evolutionary Multi-criterion Optimization, pp. 150–164. Springer, Berlin (2005)CrossRefGoogle Scholar
  6. 6.
    Deb, K., Mohan, M., Mishra, S.: Evaluating the \(\epsilon \)-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evol. Comput. 13(4), 501–525 (2005)Google Scholar
  7. 7.
    Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75(3), 293–317 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dellnitz, M., Schütze, O., Hestermeyer, T.: Covering Pareto sets by multilevel subdivision techniques. J. Optim. Theory Appl. 124(1), 113–136 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Desideri, J.A.: Mutiple-gradient descent algorithm for multiobjective optimization. In: Berhardsteiner, J. (ed.) European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS) (2012)Google Scholar
  10. 10.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar
  11. 11.
    Engau, A., Wiecek, M.M.: Generating \(\epsilon \)-efficient solutions in multiobjective programming. Eur. J. Oper. Res. 177(3), 1566–1579 (2007)Google Scholar
  12. 12.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fliege, J., Drummond, L.M.G., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hernández, C., Sun, J.Q., Schütze, O.: Computing the Set of Approximate Solutions of a Multi-objective Optimization Problem by Means of Cell Mapping Techniques, pp. 171–188. Springer International Publishing, Heidelberg (2013)Google Scholar
  15. 15.
    Hillermeier, C.: Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach. Birkhäuser, Basel (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hughes, E.J.: Evolutionary multi-objective ranking with uncertainty and noise. In: Zitzler, E., Thiele, L., Deb, K., Coello Coello, C.A., Corne, D. (eds.) Evolutionary Multi-criterion Optimization, pp. 329–343. Springer, Berlin (2001)Google Scholar
  17. 17.
    Jahn, J.: Multiobjective search algorithm with subdivision technique. Comput. Optim. Appl. 35(2), 161–175 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jin, Y.: Surrogate-assisted evolutionary computation: recent advances and future challenges. Swarm Evol. Comput. 1(2), 61–70 (2011)CrossRefGoogle Scholar
  19. 19.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the 2nd Berkeley Symposium on Mathematical and Statsitical Probability, pp. 481–492. University of California Press (1951)Google Scholar
  20. 20.
    Lara, A., Sanchez, G., Coello Coello, C.A., Schütze, O.: HCS: a new local search strategy for memetic multiobjective evolutionary algorithms. IEEE Trans. Evol. Comput. 14, 112–132. IEEE (2010)Google Scholar
  21. 21.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)zbMATHGoogle Scholar
  22. 22.
    Neri, F., Cotta, C., Moscato, P.: Handbook of Memetic Algorithms, vol. 379. Springer, Berlin (2012)Google Scholar
  23. 23.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Science & Business Media, New York (2006)zbMATHGoogle Scholar
  24. 24.
    Peitz, S., Ober-Blöbaum, S., Dellnitz, M.: Multiobjective optimal control methods for fluid flow using model order reduction (2015). arXiv:1510.05819
  25. 25.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer, Berlin (1998)zbMATHGoogle Scholar
  26. 26.
    Schäffler, S., Schultz, R., Weinzierl, K.: Stochastic method for the solution of unconstrained vector optimization problems. J. Optim. Theory Appl. 114(1), 209–222 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schütze, O., Coello Coello, C.A., Tantar, E., Talbi, E.G.: Computing the set of approximate solutions of an MOP with stochastic search algorithms. In: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, pp. 713–720. ACM (2008)Google Scholar
  28. 28.
    Schütze, O., Vasile, M., Coello Coello, C.A.: Computing the set of epsilon-efficient solutions in multi-objective space mission design. J. Aerosp. Comput. Inf. Commun. 8(3), 53–70 (2009)CrossRefGoogle Scholar
  29. 29.
    Schütze, O., Witting, K., Ober-Blöbaum, S., Dellnitz, M.: Set oriented methods for the numerical treatment of multiobjective optimization problems. In: Tantar, E., Tantar, A.A., Bouvry, P., Del Moral, P., Legrand, P., Coello Coello, C.A., Schütze, O. (eds.) EVOLVE- A Bridge Between Probability, Set Oriented Numerics and Evolutionary Computation. Studies in Computational Intelligence, vol. 447, pp. 187–219. Springer, Berlin (2013)Google Scholar
  30. 30.
    Schütze, O., Alvarado, S., Segura, C., Landa, R.: Gradient subspace approximation: a direct search method for memetic computing. Soft Comput. (2016)Google Scholar
  31. 31.
    Schütze, O., Martín, A., Lara, A., Alvarado, S., Salinas, E., Coello Coello, C.A.: The directed search method for multi-objective memetic algorithms. Comput. Optim. Appl. 63(2), 305–332 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Singh, A., Minsker, B.S.: Uncertainty-based multiobjective optimization of groundwater remediation design. Water Resour. Res. 44(2) (2008)Google Scholar
  33. 33.
    Teich, J.: Pareto-front exploration with uncertain objectives. In: Zitzler, E., Thiele, L., Deb, K., Coello Coello, C.A., Corne, D. (eds.) Evolutionary Multi-criterion Optimization, pp. 314–328. Springer, Berlin (2001)Google Scholar
  34. 34.
    Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49(2), 319–337 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsPaderborn UniversityPaderbornGermany

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