Local Optimal Sets and Bounded Archiving on Multi-objective NK-Landscapes with Correlated Objectives

  • Manuel López-Ibáñez
  • Arnaud Liefooghe
  • Sébastien Verel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8672)


The properties of local optimal solutions in multi-objective combinatorial optimization problems are crucial for the effectiveness of local search algorithms, particularly when these algorithms are based on Pareto dominance. Such local search algorithms typically return a set of mutually nondominated Pareto local optimal (PLO) solutions, that is, a PLO-set. This paper investigates two aspects of PLO-sets by means of experiments with Pareto local search (PLS). First, we examine the impact of several problem characteristics on the properties of PLO-sets for multi-objective NK-landscapes with correlated objectives. In particular, we report that either increasing the number of objectives or decreasing the correlation between objectives leads to an exponential increment on the size of PLO-sets, whereas the variable correlation has only a minor effect. Second, we study the running time and the quality reached when using bounding archiving methods to limit the size of the archive handled by PLS, and thus, the maximum size of the PLO-set found. We argue that there is a clear relationship between the running time of PLS and the difficulty of a problem instance.


Local Search Problem Instance Objective Space Local Search Algorithm Local Optimal Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aguirre, H.E., Tanaka, K.: Working principles, behavior, and performance of MOEAs on MNK-landscapes. Eur. J. Oper. Res. 181(3), 1670–1690 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bringmann, K., Friedrich, T.: Convergence of hypervolume-based archiving algorithms I: Effectiveness. In: Krasnogor, N., et al. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2011, pp. 745–752. ACM Press, New York (2011)Google Scholar
  3. 3.
    Drugan, M.M., Thierens, D.: Stochastic Pareto local search: Pareto neighbourhood exploration and perturbation strategies. J. Heuristics 18(5), 727–766 (2012)CrossRefGoogle Scholar
  4. 4.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems. Comput. Oper. Res. 38(8), 1219–1236 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kauffman, S.A.: The Origins of Order. Oxford University Press (1993)Google Scholar
  6. 6.
    Knowles, J.D.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. Ph.D. thesis, University of Reading, UK (2002)Google Scholar
  7. 7.
    Knowles, J., Corne, D.: Bounded Pareto archiving: Theory and practice. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation. LNEMS, vol. 535, pp. 39–64. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Laumanns, M., Zenklusen, R.: Stochastic convergence of random search methods to fixed size Pareto front approximations. Eur. J. Oper. Res. 213(2), 414–421 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    López-Ibáñez, M., Knowles, J., Laumanns, M.: On sequential online archiving of objective vectors. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 46–60. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Lust, T., Teghem, J.: Two-phase Pareto local search for the biobjective traveling salesman problem. J. Heuristics 16(3), 475–510 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Paquete, L., Chiarandini, M., Stützle, T.: Pareto local optimum sets in the biobjective traveling salesman problem: An experimental study. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation. LNEMS, vol. 535, pp. 177–200. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Paquete, L., Schiavinotto, T., Stützle, T.: On local optima in multiobjective combinatorial optimization problems. Annals of Operations Research 156, 83–97 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Paquete, L., Stützle, T.: A study of stochastic local search algorithms for the biobjective QAP with correlated flow matrices. Eur. J. Oper. Res. 169(3), 943–959 (2006)CrossRefzbMATHGoogle Scholar
  14. 14.
    Verel, S., Liefooghe, A., Jourdan, L., Dhaenens, C.: On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives. Eur. J. Oper. Res. 227(2), 331–342 (2013)CrossRefGoogle Scholar
  15. 15.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Manuel López-Ibáñez
    • 1
  • Arnaud Liefooghe
    • 2
  • Sébastien Verel
    • 3
  1. 1.IRIDIAUniversité Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.LIFL, UMR CNRS 8022, Inria Lille-Nord EuropeUniversité Lille 1France
  3. 3.LISICUniversité du Littoral Côte d’OpaleFrance

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