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

Author

Listed:
  • Johan M. Bogoya

    (Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, Colombia)

  • Andrés Vargas

    (Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, Colombia)

  • Oliver Schütze

    (Computer Science Department, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, Mexico City 07360, Mexico
    Dr. Rodolfo Quintero Ramirez Chair, UAM Cuajimalpa, Mexico City 05348, Mexico)

Abstract

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.

Suggested Citation

  • Johan M. Bogoya & Andrés Vargas & Oliver Schütze, 2019. "The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review," Mathematics, MDPI, vol. 7(10), pages 1-35, September.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:894-:d:270299
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    References listed on IDEAS

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