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Computing Optimal Distances to Pareto Sets of Multi-Objective Optimization Problems in Asymmetric Normed Lattices

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Abstract

Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric “distance” among a subset and a point. Our analysis has its roots and finds its applications in the current development of effective algorithms for multi-objective optimization programs. We are interested in providing the fundamental theoretical results for the associated convex analysis, fixing in this way the framework for this new optimization tool. The fact that the associated topology is not Hausdorff forces us to define a new setting and to use a new point of view for this analysis. Existence and uniqueness theorems for this optimization are shown. Our main result is the translation of the original abstract optimal distance problem to a clear optimization scheme. Actually, this justifies the algorithms and shows new aspects of the numerical and computational methods that have been already used in visualization of multi-objective optimization problems.

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Authors and Affiliations

  1. Instituto Universitario de Automática e Informática Industrial, Universitat Politècnica de València, Valencia, 46022, Spain

    X. Blasco

  2. Industrial and Systems Engineering Graduate Program (PPGEPS), Polytechnic School, Pontifical Catholic University of Paraná (PUCPR), Curitiba, 80215-901, Brazil

    G. Reynoso-Meza

  3. Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain

    E. A. Sánchez-Pérez

  4. Centro de Tecnologías Físicas: Acústica, Materiales y Astrofísica, Universitat Politècnica de València, Valencia, 46022, Spain

    J. V. Sánchez-Pérez

Authors
  1. X. Blasco
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  2. G. Reynoso-Meza
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  3. E. A. Sánchez-Pérez
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  4. J. V. Sánchez-Pérez
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Corresponding author

Correspondence to E. A. Sánchez-Pérez.

Additional information

This work was supported by the Ministerio de Economía y Competitividad (Spain) under grants DPI2015-71443-R and MTM2016-77054-C2-1-P.

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Blasco, X., Reynoso-Meza, G., Sánchez-Pérez, E.A. et al. Computing Optimal Distances to Pareto Sets of Multi-Objective Optimization Problems in Asymmetric Normed Lattices. Acta Appl Math 159, 75–93 (2019). https://doi.org/10.1007/s10440-018-0184-z

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  • DOI: https://doi.org/10.1007/s10440-018-0184-z

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