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On the Extension of the DIRECT Algorithm to Multiple Objectives

Author

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  • Alberto Lovison

    (Università di Padova)

  • Kaisa Miettinen

    (University of Jyvaskyla, Faculty of Information Technology)

Abstract

Deterministic global optimization algorithms like Piyavskii–Shubert, direct, ego and many more, have a recognized standing, for problems with many local optima. Although many single objective optimization algorithms have been extended to multiple objectives, completely deterministic algorithms for nonlinear problems with guarantees of convergence to global Pareto optimality are still missing. For instance, deterministic algorithms usually make use of some form of scalarization, which may lead to incomplete representations of the Pareto optimal set. Thus, all global Pareto optima may not be obtained, especially in nonconvex cases. On the other hand, algorithms attempting to produce representations of the globally Pareto optimal set are usually based on heuristics. We analyze the concept of global convergence for multiobjective optimization algorithms and propose a convergence criterion based on the Hausdorff distance in the decision space. Under this light, we consider the well-known global optimization algorithm direct, analyze the available algorithms in the literature that extend direct to multiple objectives and discuss possible alternatives. In particular, we propose a novel definition for the notion of potential Pareto optimality extending the notion of potential optimality defined in direct. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

Suggested Citation

  • Alberto Lovison & Kaisa Miettinen, 2021. "On the Extension of the DIRECT Algorithm to Multiple Objectives," Journal of Global Optimization, Springer, vol. 79(2), pages 387-412, February.
    • Handle: RePEc:spr:jglopt:v:79:y:2021:i:2:d:10.1007_s10898-020-00942-8
    DOI: 10.1007/s10898-020-00942-8
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    References listed on IDEAS

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    1. Alberto Lovison, 2013. "Global search perspectives for multiobjective optimization," Journal of Global Optimization, Springer, vol. 57(2), pages 385-398, October.
    2. A. Custódio & J. Madeira, 2015. "GLODS: Global and Local Optimization using Direct Search," Journal of Global Optimization, Springer, vol. 62(1), pages 1-28, May.
    3. Markus Hartikainen & Kaisa Miettinen & Margaret Wiecek, 2012. "PAINT: Pareto front interpolation for nonlinear multiobjective optimization," Computational Optimization and Applications, Springer, vol. 52(3), pages 845-867, July.
    4. Audet, Charles & Savard, Gilles & Zghal, Walid, 2010. "A mesh adaptive direct search algorithm for multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 204(3), pages 545-556, August.
    5. Daniela Lera & Yaroslav D. Sergeyev, 2018. "GOSH: derivative-free global optimization using multi-dimensional space-filling curves," Journal of Global Optimization, Springer, vol. 71(1), pages 193-211, May.
    6. E. F. Campana & M. Diez & G. Liuzzi & S. Lucidi & R. Pellegrini & V. Piccialli & F. Rinaldi & A. Serani, 2018. "A multi-objective DIRECT algorithm for ship hull optimization," Computational Optimization and Applications, Springer, vol. 71(1), pages 53-72, September.
    7. Panos M. Pardalos & Antanas Žilinskas & Julius Žilinskas, 2017. "Non-Convex Multi-Objective Optimization," Springer Optimization and Its Applications, Springer, number 978-3-319-61007-8, December.
    8. A. L. Custódio & J. F. A. Madeira, 2018. "MultiGLODS: global and local multiobjective optimization using direct search," Journal of Global Optimization, Springer, vol. 72(2), pages 323-345, October.
    9. M. Dellnitz & O. Schütze & T. Hestermeyer, 2005. "Covering Pareto Sets by Multilevel Subdivision Techniques," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 113-136, January.
    10. C. P. Stephens & W. Baritompa, 1998. "Global Optimization Requires Global Information," Journal of Optimization Theory and Applications, Springer, vol. 96(3), pages 575-588, March.
    11. Victor Gergel & Evgeny Kozinov, 2018. "Efficient multicriterial optimization based on intensive reuse of search information," Journal of Global Optimization, Springer, vol. 71(1), pages 73-90, May.
    12. Markus Hartikainen & Alberto Lovison, 2015. "PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization," Journal of Global Optimization, Springer, vol. 62(2), pages 243-261, June.
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