  Minimization diagrams encompass a large class of diagrams of interest in the
literature, such as generalized Voronoi diagrams. We develop an abstract
perturbation theory and perform a sensitivity analysis for functions depending
on sets defined through intersections of smooth sets, and formulate precise
conditions to avoid singular situations. This allows us to define a general
framework for solving optimization problems depending on minimization diagrams.
The particular case of Voronoi diagrams is discussed to illustrate the general
theory. A variety of numerical experiments is presented. The experiments
include constructing Voronoi diagrams with cells of equal size, cells
satisfying conditions on the relative size of their edges or their internal
angles, cells with the midpoints of pairs of Voronoi and Delaunay edges as
close as possible, or cells of varying sizes governed by a given function.
Overall, the experiments show that the proposed methodology allows the
construction of customized Voronoi diagrams using off-the-shelf
well-established optimization algorithms.
