Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Geometric Flows

Ed. by Carfora, Mauro / Mantegazza, Carlo

Open Access
Online
ISSN
2353-3382
See all formats and pricing
More options …

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

Abstract

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

References

  • [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.

Export Citation

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

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Marcello Carioni and Alessandra Pluda
PAMM, 2018, Volume 18, Number 1, Page e201800085

Comments (0)

Please log in or register to comment.
Log in