Locating network trees by a bilevel scheme

In this paper we investigate how to choose an optimal position of a specific facility that is constrained to a network tree connecting some given demand points in a given area. A bilevel formulation is provided and existence results are given together with some properties when a density describes the construction cost of the networks in the area. This includes the presence of an obstacle or of free regions. To prove existence of a solution of the bilevel problem, that is framed in Euclidean spaces, a lower semicontinuity property is required. This is obtained proving an extension of Golab's theorem in the general setting of metric spaces, which allows for considering a density function.