Derivative-Free Optimization: Lifting Single-Objective to Multi-Objective Algorithm

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9075)

Abstract

Most of the derivative-free optimization (DFO) algorithms rely on a comparison function able to compare any pair of points with respect to a black-box objective function. Recently, new dedicated derivative-free optimization algorithms have emerged to tackle multi-objective optimization problems and provide a Pareto front approximation to the user. This work aims at reusing single objective DFO algorithms (such as Nelder-Mead) in the context of multi-objective optimization. Therefore we introduce a comparison function able to compare a pair of points in the context of a set of non-dominated points. We describe an algorithm, MOGEN, which initializes a Pareto front approximation composed of a population of instances of single-objective DFO algorithms. These algorithms use the same introduced comparison function relying on a shared Pareto front approximation. The different instances of single-objective DFO algorithms are collaborating and competing to improve the Pareto front approximation. Our experiments comparing MOGEN with the state-of the-art Direct Multi-Search algorithm on a large set of benchmarks shows the practicality of the approach, allowing to obtain high quality Pareto fronts using a reasonably small amount of function evaluations.

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References

  1. 1.
    Bandyopadhyay, S., Pal, S., Aruna, B.: Multiobjective gas, quantitative indices, and pattern classification. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 34(5), 2088–2099 (2004)CrossRefGoogle Scholar
  2. 2.
    Ben Abdelaziz, F., Lang, P., Nadeau, R.: Dominance and efficiency in multicriteria decision under uncertainty. Theory and Decision 47, 191–211 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Conn, A., Scheinberg, K., Vicente, L.: Introduction to Derivative-Free Optimization. Society for Industrial and Applied Mathematics, Mps-siam Series on Optimization (2009)Google Scholar
  4. 4.
    Custódio, A., Emmerich, M., Madeira, J.: Recent Developments in Derivative-Free Multiobjective Optimization (2012)Google Scholar
  5. 5.
    Custódio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM Journal on Optimization 21(3), 1109–1140 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)CrossRefGoogle Scholar
  7. 7.
    Dennis Jr, J.E., Torczon, V.: Direct search methods on parallel machines. SIAM Journal on Optimization 1, 448–474 (1991)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Mathematical programming 91(2), 201–213 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Echagüe, E., Delbos, F., Dumas, L.: A global derivative-free optimization method for expensive functions with bound constraints. In: Proceedings of Global Optimization Workshop, pp. 65–68 (2012)Google Scholar
  10. 10.
    Fourer, R., Gay, D.M., Kernighan, B.W.: A modeling language for mathematical programming. Management Sci. 36, 519–554 (1990)CrossRefMATHGoogle Scholar
  11. 11.
    Kollat, J., Reed, P., Kasprzyk, J.: A new epsilon-dominance hierarchical bayesian optimization algorithm for large multiobjective monitoring network design problems. Advances in Water Resources 31(5), 828–845 (2008)CrossRefGoogle Scholar
  12. 12.
    Mor, J., Wild, S.: Benchmarking derivative-free optimization algorithms. SIAM Journal on Optimization 20(1), 172–191 (2009)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. In Comput. J. 7 (1965)Google Scholar
  14. 14.
    Voorneveld, M.: Characterization of pareto dominance. Operations Research Letters 31(1), 7–11 (2003)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: On the design of pareto-compliant indicators via weighted integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  16. 16.
    Zitzler, E., Knowles, J., Thiele, L.: Quality assessment of pareto set approximations. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization. LNCS, vol. 5252, pp. 373–404. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  17. 17.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., da Fonseca, V.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Cyrille Dejemeppe
    • 1
  • Pierre Schaus
    • 1
  • Yves Deville
    • 1
  1. 1.ICTEAMUniversité Catholique de Louvain (UCLouvain)Louvain-la-NeuveBelgium

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