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Discrete representation of the non-dominated set for multi-objective optimization problems using kernels

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  • Bazgan, Cristina
  • Jamain, Florian
  • Vanderpooten, Daniel

Abstract

In this paper, we are interested in producing discrete and tractable representations of the set of non-dominated points for multi-objective optimization problems, both in the continuous and discrete cases. These representations must satisfy some conditions of coverage, i.e. providing a good approximation of the non-dominated set, spacing, i.e. without redundancies, and cardinality, i.e. with the smallest possible number of points. This leads us to introduce the new concept of (ε, ε′)-kernels, or ε-kernels when ɛ′=ɛ is possible, which correspond to ε-Pareto sets satisfying an additional condition of ε′-stability. Among these, the kernels of small, or possibly optimal, cardinality are claimed to be good representations of the non-dominated set.

Suggested Citation

  • Bazgan, Cristina & Jamain, Florian & Vanderpooten, Daniel, 2017. "Discrete representation of the non-dominated set for multi-objective optimization problems using kernels," European Journal of Operational Research, Elsevier, vol. 260(3), pages 814-827.
    • Handle: RePEc:eee:ejores:v:260:y:2017:i:3:p:814-827
    DOI: 10.1016/j.ejor.2016.11.020
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    References listed on IDEAS

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    1. Shao, Lizhen & Ehrgott, Matthias, 2016. "Discrete representation of non-dominated sets in multi-objective linear programming," European Journal of Operational Research, Elsevier, vol. 255(3), pages 687-698.
    2. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
    3. Bazgan, Cristina & Hugot, Hadrien & Vanderpooten, Daniel, 2009. "Implementing an efficient fptas for the 0-1 multi-objective knapsack problem," European Journal of Operational Research, Elsevier, vol. 198(1), pages 47-56, October.
    4. Przybylski, Anthony & Gandibleux, Xavier & Ehrgott, Matthias, 2008. "Two phase algorithms for the bi-objective assignment problem," European Journal of Operational Research, Elsevier, vol. 185(2), pages 509-533, March.
    5. Matthias Ehrgott & Lizhen Shao & Anita Schöbel, 2011. "An approximation algorithm for convex multi-objective programming problems," Journal of Global Optimization, Springer, vol. 50(3), pages 397-416, July.
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    Cited by:

    1. Arne Herzel & Stefan Ruzika & Clemens Thielen, 2021. "Approximation Methods for Multiobjective Optimization Problems: A Survey," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1284-1299, October.
    2. Nathan Adelgren & Akshay Gupte, 2022. "Branch-and-Bound for Biobjective Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 909-933, March.
    3. Lakmali Weerasena, 2022. "Advancing local search approximations for multiobjective combinatorial optimization problems," Journal of Combinatorial Optimization, Springer, vol. 43(3), pages 589-612, April.
    4. Cristina Bazgan & Arne Herzel & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2024. "Approximating multiobjective optimization problems: How exact can you be?," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 5-25, August.

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