Sparse basis pursuit for compliance minimization in the vanishing volume ratio limit

We consider a non-smooth convex variational problem appearing as a formal limit of compliance minimization in the vanishing volume ratio limit. The problem has a classical basis pursuit form, and several successful algorithms have been utilized to solve problems of this class in other application contexts. We discuss the well-posedness and regularity of solutions to these problems, possible solution algorithms, and their discretizations as relevant in this mechanical engineering context. We then test the algorithms on a few benchmark problems with available analytical solutions. We find that whereas many algorithms are successful in estimating the optimal objective value to the problem to a high accuracy, the same cannot be said about finding the optimal solutions themselves. In particular, in some examples the algorithms struggle to properly identify the areas where the solutions should vanish entirely. We also discuss an example where the found optimal solutions are not sparse even though sparse(r) solutions exist.