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A model of anytime algorithm performance for bi-objective optimization

Author

Listed:
  • Alexandre D. Jesus

    (University of Coimbra, CISUC, DEI
    Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 - CRIStAL)

  • Luís Paquete

    (University of Coimbra, CISUC, DEI)

  • Arnaud Liefooghe

    (University of Tokyo)

Abstract

Anytime algorithms allow a practitioner to trade-off runtime for solution quality. This is of particular interest in multi-objective combinatorial optimization since it can be infeasible to identify all efficient solutions in a reasonable amount of time. We present a theoretical model that, under some mild assumptions, characterizes the “optimal” trade-off between runtime and solution quality, measured in terms of relative hypervolume, of anytime algorithms for bi-objective optimization. In particular, we assume that efficient solutions are collected sequentially such that the collected solution at each iteration maximizes the hypervolume indicator, and that the non-dominated set can be well approximated by a quadrant of a superellipse. We validate our model against an “optimal” model that has complete knowledge of the non-dominated set. The empirical results suggest that our theoretical model approximates the behavior of this optimal model quite well. We also analyze the anytime behavior of an $$\varepsilon $$ ε -constraint algorithm, and show that our model can be used to guide the algorithm and improve its anytime behavior.

Suggested Citation

  • Alexandre D. Jesus & Luís Paquete & Arnaud Liefooghe, 2021. "A model of anytime algorithm performance for bi-objective optimization," Journal of Global Optimization, Springer, vol. 79(2), pages 329-350, February.
    • Handle: RePEc:spr:jglopt:v:79:y:2021:i:2:d:10.1007_s10898-020-00909-9
    DOI: 10.1007/s10898-020-00909-9
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    References listed on IDEAS

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    1. Loudni, Samir & Boizumault, Patrice, 2008. "Combining VNS with constraint programming for solving anytime optimization problems," European Journal of Operational Research, Elsevier, vol. 191(3), pages 705-735, December.
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    3. Dubois-Lacoste, Jérémie & López-Ibáñez, Manuel & Stützle, Thomas, 2015. "Anytime Pareto local search," European Journal of Operational Research, Elsevier, vol. 243(2), pages 369-385.
    4. López-Ibáñez, Manuel & Dubois-Lacoste, Jérémie & Pérez Cáceres, Leslie & Birattari, Mauro & Stützle, Thomas, 2016. "The irace package: Iterated racing for automatic algorithm configuration," Operations Research Perspectives, Elsevier, vol. 3(C), pages 43-58.
    5. G. L. Nemhauser & Z. Ullmann, 1969. "Discrete Dynamic Programming and Capital Allocation," Management Science, INFORMS, vol. 15(9), pages 494-505, May.
    6. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions," LIDAM Reprints CORE 341, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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