A convex approach to the Gilbert-Steiner problem

Abstract

We describe a convex relaxation for the Gilbert-Steiner problem both in R d and on manifolds, extending the framework proposed in [9], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.

The oriented mailing problem and its convex relaxation

“…Such approach was initiated in [16] and [15] in the framework of the Steiner tree problem and the Gilbert-Steiner problem, respectively, in order to tackle the difficult task of proving the optimality of a candidate minimizer. Similar approaches were recently presented in [2][3][4]21]. More precisely, our strategy allows to prove the equivalence between the original problem and the minimization of a convex functional, which is defined on a non-convex set, though.…”

“…More precisely, our strategy allows to prove the equivalence between the original problem and the minimization of a convex functional, which is defined on a non-convex set, though. Yet an interesting by-product is the possibility to define a notion of calibration, which proved to be an efficient tool to validate the minimality both from the theoretical [7,8,15,16,18] and from the numerical point of view [2][3][4]19].…”

The oriented mailing problem and its convex relaxation

“…Such approach was initiated in [9] and [10] in the framework of the Steiner tree problem and the Gilbert-Steiner problem, respectively, in order to tackle the difficult task of proving the optimality of a concrete configuration. Similar approaches were recently presented in [11,12,13,14]. This strategy allowed to define a notion of calibration, which proved to be an ⋆ The second and third author received partial support by the 2018 INdAM-GNAMPA project "Geometric Measure Theoretical approaches to Optimal Networks".…”