Journal of Global Optimization

, Volume 36, Issue 3, pp 365–377 | Cite as

Bayesian Stopping Rules for Greedy Randomized Procedures

  • Carlotta Orsenigo
  • Carlo Vercellis
Original Article


A greedy randomized adaptive search procedure (GRASP) is proposed for the approximate solution of general mixed binary programming problems (MBP). Examples are provided of practical applications that can be formulated as MBP requiring the solution of a large number of problem instances. This justifies, from both a practical and a theoretical perspective, the development of stopping rules aimed at controlling the number of iterations in a GRASP. To this end, a bayesian framework is laid down, two different prior distributions are proposed and stopping conditions are explicitly derived in analytical form. Numerical evidence shows that the stopping rules lead to an optimal trade-off between accuracy and computational effort, saving from unneeded iterations and still achieving good approximations.


GRASP Bayesian stopping rules Heuristics Mixed binary programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Betrò B., Schoen F. (1987) Sequential stopping rules for the multistart algorithm in global optimisation. Math. Program. 38, 271–286CrossRefGoogle Scholar
  2. 2.
    Betrò B., Schoen F. (1992) Optimal and suboptimal stopping rules for the multistart algorithm in global optimisation. Math. Program. 57, 445–458CrossRefGoogle Scholar
  3. 3.
    Betrò B., Vercellis C. (1986) Bayesian nonparametric inference and Monte Carlo optimization. Optimization 17, 681–694CrossRefGoogle Scholar
  4. 4.
    Boender C., Rinnooy Kan A. (1987) Bayesian stopping rules for multistart global optimization methods. Math. Program. 37, 59–80CrossRefGoogle Scholar
  5. 5.
    Boender C., Rinnooy Kan A., Vercellis, C. Stochastic Optimization Methods. pp. 94–112. World Scientific (1987)Google Scholar
  6. 6.
    Canuto S., Resende M., Ribeiro C. (2001) Local search with perturbations for the prize-collecting steiner tree problem in graphs. Networks 38, 50–58CrossRefGoogle Scholar
  7. 7.
    Feo T., Resende M. (1995) Greedy randomized adaptative search procedures. J. Global Optimiz. 6, 109–133CrossRefGoogle Scholar
  8. 8.
    Festa, P., Resende, M. GRASP: An Annotated Bibliography. pp. 325–367. Kluwer Academic Publishers (2002)Google Scholar
  9. 9.
    Fumero F., Vercellis C. (1996) Capacity management through lagrangean relaxation: an application to tires production. Prod. Plan. Control 7, 604–614CrossRefGoogle Scholar
  10. 10.
    Fumero F., Vercellis C. (1997) Integrating distribution, lot-sizing and machine loading via lagrangean relaxation. Int. J. Prod. Econ. 49, 45–54CrossRefGoogle Scholar
  11. 11.
    Hart W. (1999) Sequential stopping rules for random optimization methods with applications to multistart local search. SIAM J. Optimiz. 9, 270–290CrossRefGoogle Scholar
  12. 12.
    Hettich, S., Blake C., Merz, C. UCI repository of machine learning databases. (1998). URL Scholar
  13. 13.
    Orsenigo C., Vercellis C. (2003) Multivariate classification trees based on minimum features discrete support vector machines. IMA J. Manage. Math. 14, 221–234CrossRefGoogle Scholar
  14. 14.
    Orsenigo C., Vercellis C. (2004a) Discrete support vector decision trees via tabu-search. J. Comput. Stat. Data Anal. 47, 311–322CrossRefGoogle Scholar
  15. 15.
    Orsenigo, C., Vercellis, C. One-against-all multicategory classification via discrete support vector machines. In: Ebecken N. et al. (eds.) Data Mining IV. pp. 255–264. WIT Press (2004b)Google Scholar
  16. 16.
    Prais M., Ribeiro C. (2000) Reactive grasp: An application to a matrix decomposition problem in TDMA traffic assignment. INFORMS J. Comput. 12, 164–176CrossRefGoogle Scholar
  17. 17.
    Resende, M., Ribeiro, C. Greedy Randomized Adaptive Search Procedures. pp. 219–249. Kluwer Academic Publishers (2003)Google Scholar
  18. 18.
    Wilks, S. Mathematical Statistics. Wiley (1962)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Dipartimento di Scienze Economiche, Aziendali e StatisticheUniversità di MilanoMilanoItaly
  2. 2.Dipartimento di Ingegneria GestionalePolitecnico di MilanoMilanoItaly

Personalised recommendations