Abstract
AbstractIn 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 Goła̧b’s theorem in the general setting of metric spaces, which allows for considering a density function.
Funder
Università degli Studi di Napoli Federico II
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Adams, R. (1975). Sobolev spaces. Academic Press.
2. Ambrosio, L., & Tilli, P. (2004). Topics on analysis in metric spaces., Oxford Lecture Ser. Math. Appl. 25, Oxford University Press, Oxford.
3. Aubin, J. P., & Frankowska, H. (1990). Set-valued analysis. Birkhauser.
4. Başar, T., & Olsder, G- J. (1999). Dynamic noncooperative game theory, Reprint of the second (1995) edition. Classics in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
5. Carlier, G., & Mallozzi, L. (2022). Softening bi-level problems via two-scale Gibbs measures. Set-Valued and Variational Analysis, 30, 573–595.