Hull‐Volume with Applications to Convergence Analysis

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).