Journal of Optimization Theory and Applications

, Volume 153, Issue 3, pp 633-649

First online:

Hull-Volume with Applications to Convergence Analysis

  • Adam B. LevyAffiliated withDepartment of Mathematics, Bowdoin College Email author 

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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).


Polytope Convex hull Simplex Volume Numerical optimization Convergence analysis