The cluster problem revisited
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Author(s)
Wechsung, Achim
Schaber, Spencer D.
Barton, Paul I.
Date Issued
March 2013
Journal
Journal of Global Optimization
Publisher
Springer US
Citation
Wechsung, Achim, Spencer D. Schaber, and Paul I. Barton. “The Cluster Problem Revisited.” J Glob Optim 58, no. 3 (March 19, 2013): 429–438.
Version
Author's final manuscript
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 α BB relaxations.
MIT Department
Massachusetts Institute of Technology. Department of Chemical Engineering
Massachusetts Institute of Technology. Process Systems Engineering Laboratory
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s10898-013-0059-9
