The cluster problem in constrained global optimization
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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.
KeywordsCluster problem Global optimization Constrained optimization Branch-and-bound Convergence order Convex relaxation Lower bounding scheme
Mathematics Subject Classification49M20 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  and  to our attention). The first author would also like to thank Jaichander Swaminathan for helpful discussions regarding the proof of Theorem 2.
- 8.Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of test problems in local and global optimization. In: Nonconvex Optimization and Its Applications, 1st edn, vol. 33. Springer, Berlin (1999)Google Scholar
- 13.Kearfott, R.B., Du, K.: The cluster problem in global optimization: the univariate case. In: Validation Numerics, Computing Supplementum, vol. 9, pp. 117–127. Springer, Berlin (1993)Google Scholar
- 14.Khan, K.A., Watson, H.A.J., Barton, P.I.: Differentiable McCormick relaxations. J. Glob. Optim. 67(4), 687–729 (2017)Google Scholar
- 27.Van Iwaarden, R.J.: An improved unconstrained global optimization algorithm. Ph.D. thesis, University of Colorado at Denver (1996)Google Scholar
- 28.Wechsung, A.: Global optimization in reduced space. Ph.D. thesis, Massachusetts Institute of Technology (2014)Google Scholar