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Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization

Author

Listed:
  • Ana Luísa Custódio

    (FCT-UNL-CMA, Campus de Caparica)

  • Youssef Diouane

    (Université de Toulouse)

  • Rohollah Garmanjani

    (FCT-UNL-CMA, Campus de Caparica)

  • Elisa Riccietti

    (Université de Toulouse)

Abstract

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.

Suggested Citation

  • Ana Luísa Custódio & Youssef Diouane & Rohollah Garmanjani & Elisa Riccietti, 2021. "Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 73-93, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01781-z
    DOI: 10.1007/s10957-020-01781-z
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    References listed on IDEAS

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    1. Kely D. V. Villacorta & Paulo R. Oliveira & Antoine Soubeyran, 2014. "A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 865-889, March.
    2. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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