# The cluster problem revisited

## Author

Listed:
• Achim Wechsung

()

• Spencer Schaber

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• Paul Barton

()

## 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

## Suggested Citation

• Achim Wechsung & Spencer Schaber & Paul Barton, 2014. "The cluster problem revisited," Journal of Global Optimization, Springer, vol. 58(3), pages 429-438, March.
• Handle: RePEc:spr:jglopt:v:58:y:2014:i:3:p:429-438
DOI: 10.1007/s10898-013-0059-9
as

File URL: http://hdl.handle.net/10.1007/s10898-013-0059-9

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## References listed on IDEAS

as
1. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
2. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
3. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. repec:spr:mathme:v:86:y:2017:i:1:d:10.1007_s00186-017-0591-3 is not listed on IDEAS
2. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
3. repec:spr:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0474-9 is not listed on IDEAS
4. repec:spr:jglopt:v:69:y:2017:i:3:d:10.1007_s10898-017-0531-z is not listed on IDEAS
5. Jaromił Najman & Alexander Mitsos, 2016. "Convergence analysis of multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 66(4), pages 597-628, December.
6. repec:spr:jglopt:v:67:y:2017:i:4:d:10.1007_s10898-016-0440-6 is not listed on IDEAS

### Keywords

Cluster problem; Global optimization; Convergence order; Convex relaxations; 49M20; 49M37; 65K05; 68Q25; 90C26;

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