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MultiGLODS: global and local multiobjective optimization using direct search

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Abstract

The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.

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Acknowledgements

We are thankful to the editor and to two anonymous reviewers whose suggestions helped us to improve the paper. Also, we would like to thank Professor Charles Audet and Professor Sébastien Le Digabel, from École Polytechnique de Montréal, for providing us the Matlab code of the styrene production problem.

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Authors and Affiliations

  1. Department of Mathematics, FCT-UNL-CMA, Campus de Caparica, 2829-516, Caparica, Portugal

    A. L. Custódio

  2. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1040-001, Lisbon, Portugal

    J. F. A. Madeira

  3. Department of Mathematics, ISEL, Rua Conselheiro Emídio Navarro, 1, 1959-007, Lisbon, Portugal

    J. F. A. Madeira

Authors
  1. A. L. Custódio
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  2. J. F. A. Madeira
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Correspondence to A. L. Custódio.

Additional information

Support for A.L. Custódio was provided by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) under the project UID/MAT/00297/2013 (CMA).

Support for J.F.A. Madeira was provided by ISEL, IPL, Lisboa, Portugal and by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through IDMEC, under LAETA, project UID/EMS/50022/2013.

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Custódio, A.L., Madeira, J.F.A. MultiGLODS: global and local multiobjective optimization using direct search. J Glob Optim 72, 323–345 (2018). https://doi.org/10.1007/s10898-018-0618-1

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  • DOI: https://doi.org/10.1007/s10898-018-0618-1

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