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Open AccessReview

The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

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Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, Colombia
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Computer Science Department, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, Mexico City 07360, Mexico
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Dr. Rodolfo Quintero Ramirez Chair, UAM Cuajimalpa, Mexico City 05348, Mexico
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 894; https://doi.org/10.3390/math7100894
Received: 27 August 2019 / Revised: 10 September 2019 / Accepted: 17 September 2019 / Published: 24 September 2019
(This article belongs to the Special Issue Recent Trends in Multiobjective Optimization and Optimal Control)
A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide a stat-of-the-art overview of their behavior from a theoretical point of view. The presentation treats separately the definitions of the ( p , q ) -distances GD p , q , IGD p , q , and Δ p , q for finite sets and their generalization for arbitrary measurable sets that covers as an important example the case of continuous sets. Among the presented results, we highlight the rigorous consideration of metric properties of these definitions, including a proof of the triangle inequality for distances between disjoint subsets when p , q 1 , and the study of the behavior of associated indicators with respect to the notion of compliance to Pareto optimality. Illustration of these results in particular situations are also provided. Finally, we discuss a collection of examples and numerical results obtained for the discrete and continuous incarnations of these distances that allow for an evaluation of their usefulness in concrete situations and for some interesting conclusions at the end, justifying their use and further study. View Full-Text
Keywords: Averaged Hausdorff distance; evolutionary multi-objective optimization; Pareto compliance; performance indicator; power means Averaged Hausdorff distance; evolutionary multi-objective optimization; Pareto compliance; performance indicator; power means
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Bogoya, J.M.; Vargas, A.; Schütze, O. The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review. Mathematics 2019, 7, 894.

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