A1 Journal article (refereed)
On the Extension of the DIRECT Algorithm to Multiple Objectives (2020)

Lovison, Alberto; Miettinen, Kaisa (2020). On the Extension of the DIRECT Algorithm to Multiple Objectives. Journal of Global Optimization, Early online. DOI: 10.1007/s10898-020-00942-8

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Publication details

All authors or editors: Lovison, Alberto; Miettinen, Kaisa

Journal or series: Journal of Global Optimization

ISSN: 0925-5001

eISSN: 1573-2916

Publication year: 2020

Volume: Early online

Publisher: Springer Science and Business Media LLC

Publication country: Netherlands

Publication language: English

DOI: https://doi.org/10.1007/s10898-020-00942-8

Open Access: Open access publication published in a hybrid channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/71664

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.

Keywords: multi-objective optimisation; optimisation; algorithms; determinants; multivariable methods

Free keywords: global convergence; multiobjective optimization; multiple criteria optimization; DIRECT algorithm; deterministic optimization algorithms

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