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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access March 24, 2018

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

  • M. Bonafini EMAIL logo
From the journal Geometric Flows

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.

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

References

[1] MOSEK ApS. The MOSEK optimization software. Online at http://www.mosek.com.Search in 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.Search in 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, 2015.10.1137/14096061XSearch in Google 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.Search in 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, 2006.10.1007/0-387-30065-1_4Search in Google Scholar

[6] Antonin Chambolle, Daniel Cremers, and Thomas Pock. A convex approach to minimal partitions. SIAM J. Imaging Sci., 5(4):1113-1158, 2012.10.1137/110856733Search in Google Scholar

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

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

[9] Herbert Federer. Geometric measure theory. Springer, 2014.Search in 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, 2016.10.1051/cocv/2015028Search in 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.Search in 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, 2011.10.1007/s00205-011-0402-6Search in Google Scholar

Received: 2017-9-15
Accepted: 2017-11-2
Published Online: 2018-3-24

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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