Journal of Optimization Theory and Applications

, Volume 153, Issue 3, pp 633–649

Hull-Volume with Applications to Convergence Analysis


DOI: 10.1007/s10957-011-9959-3

Cite this article as:
Levy, A.B. J Optim Theory Appl (2012) 153: 633. doi:10.1007/s10957-011-9959-3


We introduce and study decompositions of finite sets as well as coverings of their convex hulls, and use these objects to develop various estimates of and formulas for the “hull-volume” of the sets (i.e., the volume of their convex hull). We apply our results to the convergence analysis of the “iterate-sets” associated with each iteration of a reduce-or-retreat optimization method (including pattern-search methods like Nelder–Mead as well as model-based methods).


PolytopeConvex hullSimplexVolumeNumerical optimizationConvergence analysis

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsBowdoin CollegeBrunswickUSA