Journal of Global Optimization

, Volume 69, Issue 3, pp 629–676 | Cite as

The cluster problem in constrained global optimization



Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (Du and Kearfott in J Glob Optim 5(3):253–265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (Du and Kearfott 1994; Wechsung et al. in J Glob Optim 58(3):429–438, 2014) to the constrained setting based on a recently-developed notion of convergence order for convex relaxation-based lower bounding schemes. It is shown that clustering can occur both on nearly-optimal and nearly-feasible regions in the vicinity of a global minimizer. In contrast to the case of unconstrained optimization, where at least second-order convergent schemes of relaxations are required to mitigate the cluster problem when the minimizer sits at a point of differentiability of the objective function, it is shown that first-order convergent lower bounding schemes for constrained problems may mitigate the cluster problem under certain conditions. Additionally, conditions under which second-order convergent lower bounding schemes are sufficient to mitigate the cluster problem around a global minimizer are developed. Conditions on the convergence order prefactor that are sufficient to altogether eliminate the cluster problem are also provided. This analysis reduces to previous analyses of the cluster problem for unconstrained optimization under suitable assumptions.


Cluster problem Global optimization Constrained optimization Branch-and-bound Convergence order Convex relaxation Lower bounding scheme 

Mathematics Subject Classification

49M20 49M37 65K05 68Q25 90C26 90C46 



The authors would like to thank three anonymous reviewers and an associate editor for suggestions which helped improve the readability of this article, and Dr. Johannes Jäschke for helpful discussions (in particular, for bring references [5] and [12] to our attention). The first author would also like to thank Jaichander Swaminathan for helpful discussions regarding the proof of Theorem 2.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Process Systems Engineering Laboratory, Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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