Sensitivity analysis and tailored design of minimization diagrams

TL;DR

This paper introduces a general framework for sensitivity analysis and tailored design of minimization diagrams, including Voronoi diagrams, enabling customized diagram construction through optimization techniques.

Contribution

It develops an abstract perturbation theory and precise conditions to avoid singularities, facilitating optimization-based customization of minimization diagrams.

Findings

Framework successfully constructs customized Voronoi diagrams

Method handles various constraints on cell properties

Numerical experiments validate the approach's effectiveness

Abstract

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.