Skip to main content
SpringerLink
Log in

On local optima in multiobjective combinatorial optimization problems

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their acceptance criterion is based on outperformance relations. Proofs of the soundness and completeness of these algorithms are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aarts, E. H. L., & Lenstra, J. K. (Eds.). (1997). Local search in combinatorial optimization. Chichester: Wiley.

    Google Scholar 

  • Angel, E., Bampis, E., & Gourvés, L. (2004a). Approximating the Pareto curve with local search for the bicriteria TSP(1,2) problem. Theoretical Computer Science, 310, 135–146.

    Article  Google Scholar 

  • Angel, E., Bampis, E., & Gourvés, L. (2004b). A dynasearch neighborhood for the bicriteria traveling salesman problem. In X. Gandibleux, M. Sevaux, K. Sörensen & V. T’kindt (Eds.), Lecture notes in economics and mathematical systems : Vol. 535. Metaheuristics for multiobjective optimisation (pp. 153–176). Berlin: Springer.

    Google Scholar 

  • Deb, K. (1999). Multi-objective genetic algorithms: problem difficulties and construction of test problems. Evolutionary Computation, 7(3), 205–230.

    Google Scholar 

  • Ehrgott, M. (2000). Lecture notes in economics and mathematical systems: Vol. 491. Multicriteria optimization. Heidelberg: Springer.

    Google Scholar 

  • Ehrgott, M., & Gandibleux, X. (2004). Approximative solution methods for combinatorial multicriteria optimization. TOP, 12(1), 1–90.

    Article  Google Scholar 

  • Emelichev, V., & Perepelitsa, V. (1991). Complexity of vector optimization problems on graphs. Optimization, 6, 903–918.

    Google Scholar 

  • Emelichev, V., & Perepelitsa, V. (1992). On the cardinality of the set of alternatives in discrete many-criterion problems. Discrete Mathematics and Applications, 2(5), 461–471.

    Article  Google Scholar 

  • Erlebach, T., Kellerer, H., & Pferschy, U. (2002). Approximating multiobjective knapsack problems. Management Science, 48(12), 1603–1612.

    Article  Google Scholar 

  • Fonseca, C., Fleming, P., Zitzler, E., Deb, K., & Thiele, L. (Eds.). (2003). Lecture notes in computer science: Vol. 2632. Evolutionary multi-criterion optimization (EMO 2003). Berlin: Springer.

    Google Scholar 

  • Hamacher, H., & Ruhe, G. (1994). On spanning tree problems with multiple objectives. Annals of Operations Research, 5, 209–230.

    Article  Google Scholar 

  • Hansen, M., & Jaszkiewicz, A. (1998). Evaluating the quality of approximations to the non-dominated set (Technical report IMM-REP-1998-7). Institute of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark.

  • Hansen, P. (1979). Bicriterion path problems. In Lecture notes in economics and mathematical systems : Vol. 177. Multiple criteria decision making theory and application (pp. 109–127). Berlin: Springer.

    Google Scholar 

  • Hansen, P., & Mladenović, N. (2002). Variable neighborhood search. In F. Glover & G. Kochenberger (Eds.), Handbook of metaheuristics (pp. 145–184). Norwell: Kluwer Academic.

    Google Scholar 

  • Hoos, H. H., & Stützle, T. (2004). Stochastic local search—foundations and applications. San Francisco: Morgan Kaufmann.

    Google Scholar 

  • Jörnsten, K., Andersen, K., & Lind, M. (1996). On bicriterion minimal spanning trees: an approximation. Computers & Operations Research, 23(12), 1171–1182.

    Article  Google Scholar 

  • Knowles, J., & Corne, D. (1999). The Pareto archived evolution strategy: a new base line algorithm for multiobjective optimisation. In Proceedings of the 1999 congress on evolutionary computation (CEC’99) (pp. 98–105). Washington.

  • Knowles, J., & Corne, D. (2000). Approximating the nondominated front using the Pareto archived evolution strategy. Evolutionary Computation, 8(2), 149–172.

    Article  Google Scholar 

  • Knowles, J., & Corne, D. (2004). Bounded Pareto archiving: theory and practice. In X. Gandibleux, M. Sevaux, K. Sörensen & V. T’kindt (Eds.), Lecture notes in economics and mathematical systems : Vol. 535. Metaheuristics for multiobjective optimisation (pp. 39–64). Berlin: Springer.

    Google Scholar 

  • Kung, H., Luccio, F., & Preparata, F. (1975). On finding the maxima of a set of vectors. Journal of the Association for Computing Machinery, 22(4), 469–476.

    Google Scholar 

  • Laumanns, M., Thiele, L., Zitzler, E., Welzl, E., & Deb, K. (2002). Running time analysis of multi-objective evolutionary algorithms on a simple discrete optimization problem. In J. M. Guervos et al. (Eds.), Lecture notes in computer science : Vol. 2439. Proceedings of PPSN-VII (pp. 44–53). Berlin: Springer.

    Google Scholar 

  • Moscato, P. (1999). Memetic algorithms: a short introduction. In D. Corne & M. Dorigo (Eds.), New ideas in optimization (pp. 219–234). London: McGraw-Hill.

    Google Scholar 

  • Papadimitriou, C. H., & Yannakakis, M. (2000). On the approximability of trade-offs and optimal access of web sources. In Proceedings of the 41st annual symposium on foundations of computer science (FOCS 2000) (pp. 86–92).

  • Paquete, L., Chiarandini, M., & Stützle, T. (2002). A study of local optima in the multiobjective traveling salesman problem (Technical report AIDA-02-07). Fachgebiet Intellektik, Fachbereich Informatik, Technische Universität Darmstadt. Presented at the multiobjective metaheuristics workshop (MOMH 2002), Paris, 4–5 November 2002.

  • Paquete, L., Chiarandini, M., & Stützle, T. (2004). Pareto local optimum sets in the biobjective traveling salesman problem: an experimental study. In X. Gandibleux, M. Sevaux K. Sörensen & V. T’kindt (Eds.), Lecture notes in economics and mathematical systems : Vol. 535. Metaheuristics for multiobjective optimisation (pp. 177–200). Berlin: Springer.

    Google Scholar 

  • Paquete, L., & Stützle, T. (2003). A two-phase local search for the biobjective traveling salesman problem. In C. Fonseca, P. Fleming, E. Zitzler, K. Deb & L. Thiele (Eds.), Lecture notes in computer science : Vol. 2632. Evolutionary multi-criterion optimization (EMO 2003) (pp. 479–493). New York: Springer.

    Google Scholar 

  • Paquete, L., & Stützle, T. (2006). A study of stochastic local search algorithms for the biobjective QAP with correlated flow matrices. European Journal of Operational Research, 169(3), 943–959.

    Article  Google Scholar 

  • Ruhe, G., & Fruhwirth, B. (1990). ε-optimality for bicriteria programs and its application to minimum cost flows. Computing, 44(1), 21–34.

    Article  Google Scholar 

  • Schott, J. R. (1995). Fault tolerant design using single and multicriteria genetic algorithm optimization. Master’s thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA.

  • Serafini, P. (1986). Some considerations about computational complexity for multiobjective combinatorial problems. In Lecture notes in economics and mathematical systems : Vol. 294. Recent advances and historical development of vector optimization (pp. 222–231). Berlin: Springer.

    Google Scholar 

  • Talbi, E. G. (2003). A hybrid evolutionary approach for multicriteria optimization problems: application to the flow shop. In C. Fonseca, P. Fleming, E. Zitzler K. Deb & L. Thiele (Eds.), Lecture notes in computer science : Vol. 2632. Evolutionary multi-criterion optimization (EMO 2003) (pp. 416–428). Berlin: Springer.

    Google Scholar 

  • Warburton, A. (1987). Approximation of Pareto optima in multi-objective shortest-path problems. Operations Research, 35(1), 70–79.

    Article  Google Scholar 

  • Zitzler, E., Deb, K., Thiele, L., Coello, C. A., & Corne, D. (Eds.). (2001). In Lecture notes in computer science: Vol. 1993. Evolutionary multi-criterion optimization (EMO 2001). Berlin: Springer.

  • Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C. M., & Fonseca, V. G. (2003). Performance assessment of multiobjective optimizers: an analysis and review. IEEE Transactions on Evolutionary Computation, 7(2), 117–132.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculdade de Economia, Centro de Sistemas Inteligentes, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    Luis Paquete

  2. Signum, Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126, Pisa, Italy

    Tommaso Schiavinotto

  3. CoDE, IRIDIA, Université Libre de Bruxelles, CP 194/6, Av. F. Roosevelt 50, 1050, Brussels, Belgium

    Thomas Stützle

Authors
  1. Luis Paquete
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tommaso Schiavinotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Stützle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Paquete.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paquete, L., Schiavinotto, T. & Stützle, T. On local optima in multiobjective combinatorial optimization problems. Ann Oper Res 156, 83–97 (2007). https://doi.org/10.1007/s10479-007-0230-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-007-0230-0

Keywords

Search

Navigation