Abstract
In continuous branch-and-bound algorithms, a very large number of boxes near global minima may be visited prior to termination. This so-called cluster problem (J Glob Optim 5(3):253–265, 1994) is revisited and a new analysis is presented. Previous results are confirmed, which state that at least second-order convergence of the relaxations is required to overcome the exponential dependence on the termination tolerance. Additionally, it is found that there exists a threshold on the convergence order pre-factor which can eliminate the cluster problem completely for second-order relaxations. This result indicates that, even among relaxations with second-order convergence, behavior in branch-and-bound algorithms may be fundamentally different depending on the pre-factor. A conservative estimate of the pre-factor is given for \(\alpha \)BB relaxations.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs-I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)
Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995)
Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Glob. Optim. 52(1), 1–28 (2012)
Du, K., Kearfott, R.B.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5(3), 253–265 (1994)
Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15, 550–569 (1969)
Hayes, B.: An adventure in the \(N\)th dimension. Am. Sci. 99(6), 442–446 (2011)
Heyl, P.R.: Properties of the locus \(r\)=constant in the space of \(n\) dimensions. In: Publications of the University of Pennsylvania, University of Pennsylvania, Philadelphia, PA, chap 2, pp 33–39, (1897) http://books.google.com/books?id=j5pQAAAAYAAJ
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)
Maranas, C.D., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4(2), 135–170 (1994)
Mayer, G.: Epsilon-inflation in verification algorithms. J. Comput. Appl. Math. 60(1–2), 147–169 (1995)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)
Neumaier, A.: Taylor forms–use and limits. Reliab. Comput. 9, 43–79 (2003)
Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. In: Iserles, A. (eds) Acta Numerica, vol. 13, pp. 271–370. Cambridge University Press, Cambridge (2004)
Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. Ellis Horwood, Chichester (1984)
Schöbel, A., Scholz, D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Glob. Optim. 48(3), 473–495 (2010)
Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51(4), 569–606 (2011)
Van Iwaarden, RJ: An improved unconstrained global optimization algorithm. Ph.D. thesis, University of Colorado at Denver, Denver, CO (1996)
Acknowledgments
The authors would like to acknowledge Kamil A. Khan for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wechsung, A., Schaber, S.D. & Barton, P.I. The cluster problem revisited. J Glob Optim 58, 429–438 (2014). https://doi.org/10.1007/s10898-013-0059-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0059-9