The cluster problem revisited

Achim Wechsung (), Spencer Schaber () and Paul Barton ()

Journal of Global Optimization, 2014, vol. 58, issue 3, 429-438

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. Copyright Springer Science+Business Media New York 2014

Keywords: Cluster problem; Global optimization; Convergence order; Convex relaxations; 49M20; 49M37; 65K05; 68Q25; 90C26 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (15)

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DOI: 10.1007/s10898-013-0059-9

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