A key issue in continuous global optimization is the so-called cluster problem where a largeÂ number of boxes may be visited in the vicinity of a global optimizer [1â??3]. While it is well-known that at least second-order Hausdorff convergence of the scheme of relaxations is usuallyÂ required to eliminate the cluster problem for unconstrained optimization, a similar result forÂ constrained optimization was unknown prior to this work. This work analyzes the cluster problemÂ for constrained optimization based on a recently proposed definition of convergence order ofÂ bounding schemes for constrained problems. Conditions under which first-order and second-order convergent bounding schemes are sufficient to mitigate the cluster problem are provided,Â based on suitable assumptions.
 K. Du and R. B. Kearfott, â??The cluster problem in multivariate global optimization,â? JournalÂ of Global Optimization, vol. 5, no. 3, pp. 253â??265, 1994.
 A. Neumaier, â??Complete search in continuous global optimization and constraint satisfaction,â? Acta Numerica, vol. 13, pp. 271â??369, 2004.
 A. Wechsung, S. D. Schaber, and P. I. Barton, â??The cluster problem revisited,â? Journal ofÂ Global Optimization, vol. 58, no. 3, pp. 429â??438, 2014.