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A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

Author

Listed:
  • Lourdes Uribe

    (Instituto Politécnico Nacional, Mexico City 07738, Mexico)

  • 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)

  • Adriana Lara

    (Instituto Politécnico Nacional, Mexico City 07738, Mexico)

  • Günter Rudolph

    (Department of Computer Science, TU Dortmund University, 44227 Dortmund, Germany)

  • Oliver Schütze

    (Department of Computer Science, Cinvestav-IPN, Mexico City 07360, Mexico)

Abstract

Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance Δ p for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.

Suggested Citation

  • Lourdes Uribe & Johan M Bogoya & Andrés Vargas & Adriana Lara & Günter Rudolph & Oliver Schütze, 2020. "A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems," Mathematics, MDPI, vol. 8(10), pages 1-29, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1822-:d:430515
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    References listed on IDEAS

    as
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