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Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 75–93 | Cite as

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

  • X. Blasco
  • G. Reynoso-Meza
  • E. A. Sánchez-PérezEmail author
  • J. V. Sánchez-Pérez
Article

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.

Keywords

Multi-objective Optimization Asymmetric norm Nearest point 

Mathematics Subject Classification (2010)

47N10 46B85 46L85 

References

  1. 1.
    Alegre, C., Ferrer, J., Gregori, V.: On the Hahn-Banach theorem in certain linear quasi-uniform structures. Acta Math. Hung. 82, 315–320 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alegre, C., Ferrando, I., García-Raffi, L.M., Sánchez-Pérez, E.A.: Compactness in asymmetric normed spaces. Topol. Appl. 155, 527–539 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics. American Mathematical Soc., Providence (2003) CrossRefzbMATHGoogle Scholar
  4. 4.
    Blasco, X., Reynoso-Meza, G., Sánchez-Pérez, E.A., Sánchez-Pérez, J.V.: Asymmetric distances to improve n-dimensional Pareto fronts graphical analysis. Inf. Sci. 340, 228–249 (2016) CrossRefGoogle Scholar
  5. 5.
    Cobzaş, S.: Separation of convex sets and best approximation in spaces with asymmetric norm. Quaest. Math. 27, 275–296 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cobzaş, S.: Geometric properties of Banach spaces and the existence of nearest and farthest points. Abstr. Appl. Anal. 2005(3), 259–285 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cobzaş, S.: Functional Analysis in Asymmetric Normed Spaces. Birkhäuser, Basel (2013) CrossRefzbMATHGoogle Scholar
  8. 8.
    Cobzaş, S., Mustăţa, C.: Extension of bounded linear functionals and best approximation in spaces with asymmetric norm. Rev. Anal. Numér. Théor. Approx. 33, 39–50 (2004) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Conradie, J.J.: Asymmetric norms, cones and partial orders. Topol. Appl. 193, 100–115 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conradie, J.J., Mabula, M.D.: Completeness, precompactness and compactness in finite-dimensional asymmetrically normed lattices. Topol. Appl. 160, 2012–2024 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, Hoboken (2001) zbMATHGoogle Scholar
  12. 12.
    Ferrer, J., Gregori, V., Alegre, A.: Quasi-uniform structures in linear lattices. Rocky Mt. J. Math. 23, 877–884 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    García Raffi, L.M., Romaguera, S., Sánchez Pérez, E.A.: On Hausdorff asymmetric normed linear spaces. Houst. J. Math. 29, 717–728 (2003) MathSciNetzbMATHGoogle Scholar
  14. 14.
    García Raffi, L.M., Romaguera, S., Sánchez-Pérez, E.A.: The dual space of an asymmetric normed linear space. Quaest. Math. 26, 83–96 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    García Raffi, L.M., Romaguera, S., Sánchez Pérez, E.A.: Weak topologies on asymmetric normed linear spaces and non-asymptotic criteria in the theory of complexity analysis of algorithms. J. Anal. Appl. 2, 125–138 (2004) MathSciNetzbMATHGoogle Scholar
  16. 16.
    García-Raffi, L.M.: Compactness and finite dimension in asymmetric normed linear spaces. Topol. Appl. 153, 844–853 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    García-Raffi, L.M., Sánchez-Pérez, E.A.: Asymmetric norms and optimal distance points in linear spaces. Topol. Appl. 155, 1410–1419 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jonard-Pérez, N., Sánchez-Pérez, E.A.: Extreme points and geometric aspects of convex compact sets in asymmetric normed spaces. Topol. Appl. 203, 12–21 (2016) CrossRefzbMATHGoogle Scholar
  19. 19.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1996) zbMATHGoogle Scholar
  20. 20.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces. North Holland, Amsterdam (1971) zbMATHGoogle Scholar
  21. 21.
    Martin, J., Mayor, G., Valero, O.: On aggregation of normed structures. Math. Comput. Model. 54, 815–827 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Massanet, S., Valero, O.: On aggregation of metric structures: the extended quasi-metric case. Int. J. Comput. Intell. Syst. 6, 115–126 (2013) CrossRefGoogle Scholar
  23. 23.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Springer, Berlin (2012) zbMATHGoogle Scholar
  24. 24.
    Reynoso-Meza, G., Blasco, X., Sanchis, J., Herrero, J.M.: Comparison of design concepts in multi-criteria decision-making using level diagrams. Inf. Sci. 221, 124–141 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yu, P.-L.: Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions. Springer, Berlin (2013) Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Instituto Universitario de Automática e Informática IndustrialUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Industrial and Systems Engineering Graduate Program (PPGEPS), Polytechnic SchoolPontifical Catholic University of Paraná (PUCPR)CuritibaBrazil
  3. 3.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  4. 4.Centro de Tecnologías Físicas: Acústica, Materiales y AstrofísicaUniversitat Politècnica de ValènciaValenciaSpain

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