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

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. Most works in the literature have been devoted to the approximation of a surface evolving by the Willmore flow and, in particular, to the approximation of the so-called Willmore surfaces, i.e., the critical points of the Willmore energy. 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. The reason for such strategy is the well-known existence of algorithms that can compute global minimizers of a large class of linear optimization problems, however at a significant computational and memory cost. The case of integer linear programming is particularly delicate and usual strategies consist in relaxing the integral constraint x ∈ {0,1} into x ∈ [0,1] which is easier to handle. 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. These facts indicate that the problem cannot be tackled with classical linear programming solvers, but only with pure integer linear solvers. Nevertheless, 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.