On a linear programming approach to the discrete Willmore boundary value problem and generalizations

TL;DR

This paper formulates the discrete Willmore boundary value problem as an integer linear program, analyzes its relaxation, and provides insights into the challenges of guaranteeing integral solutions for minimal energy surfaces.

Contribution

It introduces a novel integer linear programming approach for approximating solutions to the discrete Willmore boundary value problem and investigates the relaxation's properties.

Findings

Relaxation may produce fractional solutions in some cases.

Total unimodularity of the constraint matrix cannot be guaranteed.

Future work could develop specialized solvers for better approximations.

Abstract

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature Our purpose is to address the delicate task of approximating global minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that: - One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well; - Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution. Due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem.