A Shape-Newton approach to the problem of covering with identical balls

TL;DR

This paper introduces a shape-Newton method for optimizing the coverage of a region with identical balls, providing analytical derivatives and algorithms for convex polygons, supported by numerical experiments.

Contribution

It develops a shape-Newton approach with explicit bi-Lipschitz mappings and derivative computations for covering problems, including disjoint convex polygons.

Findings

Effective derivative computation for covering regions

Algorithms based on Voronoi diagrams without approximations

Numerical experiments demonstrating approach capabilities

Abstract

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to model small perturbations of the union of balls. This allows us to obtain analytical expressions for first- and second-order derivatives using nonsmooth shape optimization techniques under appropriate regularity assumptions. Singular cases are also studied using asymptotic analysis. For the case of regions given by the union of disjoint convex polygons, algorithms based on Voronoi diagrams that do not rely on approximations are given to compute the derivatives. Extensive numerical experiments illustrate the capabilities and limitations of the introduced approach.