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Geometric Flows

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Convex relaxation and variational approximation of the Steiner problem: theory and numerics

M. Bonafini
Published Online: 2018-03-24 | DOI: https://doi.org/10.1515/geofl-2018-0003


We survey some recent results on convex relaxations and a variational approximation for the classical Euclidean Steiner tree problem and we see how these new perspectives can lead to effective numerical schemes for the identification of Steiner minimal trees.

Keywords: Calculus of Variations; Geometric Measure Theory; Gamma-convergence; Convex relaxation; Steiner problem

MSC 2010: 49J45; 49Q15; 49Q20; 49M20; 65K10


  • [1] MOSEK ApS. The MOSEK optimization software. Online at http://www.mosek.com.Google Scholar

  • [2] Mauro Bonafini, Giandomenico Orlandi, and Édouard Oudet. Variational approximation of functionals defined on 1- dimensional connected sets: the planar case. preprint arXiv:1610.03839v3, 2016.Google Scholar

  • [3] Matthieu Bonnivard, Antoine Lemenant, and Filippo Santambrogio. Approximation of length minimization problems among compact connected sets. SIAM J. Math. Anal., 47(2):1489-1529, 2015.CrossrefWeb of ScienceGoogle Scholar

  • [4] Matthieu Bonnivard, Vincent Millot, and Antoine Lemenant. On a phase field approximation of the planar steiner problem: existence, regularity, and asymptotic of minimizers. preprint, 2016.Google Scholar

  • [5] Richard H. Byrd, Jorge Nocedal, and Richard A. Waltz. KNITRO: An integrated package for nonlinear optimization. In Largescale nonlinear optimization, volume 83 of Nonconvex Optim. Appl., pages 35-59. Springer, New York, 2006.Google Scholar

  • [6] Antonin Chambolle, Daniel Cremers, and Thomas Pock. A convex approach to minimal partitions. SIAM J. Imaging Sci., 5(4):1113-1158, 2012.CrossrefGoogle Scholar

  • [7] Antonin Chambolle, Luca Ferrari, and Benoit Merlet. A phase-field approximation of the steiner problem in dimension two. Adv. Calc. Var., 2017.Google Scholar

  • [8] Iain Dunning, Joey Huchette, and Miles Lubin. JuMP: A Modeling Language for Mathematical Optimization. SIAM Review, 59(2):295-320, 2017.Google Scholar

  • [9] Herbert Federer. Geometric measure theory. Springer, 2014.Google Scholar

  • [10] Andrea Marchese and Annalisa Massaccesi. An optimal irrigation network with infinitely many branching points. ESAIM Control Optim. Calc. Var., 22(2):543-561, 2016.Google Scholar

  • [11] Andrea Marchese and Annalisa Massaccesi. The Steiner tree problem revisited through rectifiable G-currents. Adv. Calc. Var., 9(1):19-39, 2016.Google Scholar

  • [12] Edouard Oudet and Filippo Santambrogio. A Modica-Mortola approximation for branched transport and applications. Arch. Ration. Mech. Anal., 201(1):115-142, 2011.Web of ScienceGoogle Scholar

About the article

Received: 2017-09-15

Accepted: 2017-11-02

Published Online: 2018-03-24

Citation Information: Geometric Flows, Volume 3, Issue 1, Pages 19–27, ISSN (Online) 2353-3382, DOI: https://doi.org/10.1515/geofl-2018-0003.

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© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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