  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.
