Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T10:38:31.033Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Relaxation in Conic Formfor the Euclidean Steiner Problem in ℜ

Published online by Cambridge University Press:  15 August 2002

Marcia Fampa  and
Nelson Maculan
Marcia Fampa
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Departamento de Ciência da Computação, Caixa Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil; [email protected].
Nelson Maculan
Affiliation:
Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia de Sistemas e Computação, Caixa Postal 68511, Rio de Janeiro, RJ 21945-970, Brasil; [email protected].
Get access

Abstract

In this paper, we present anew mathematical programming formulation for the Euclidean SteinerTree Problem (ESTP) in ℜ. We relax the integralityconstrains on this formulation and transform the resultingrelaxation, which is convex, but not everywhere differentiable,into a standard convex programming problem in conic form. Weconsider then an efficient computation of an ϵ-optimalsolution for this latter problem using interior-point algorithm.

Keywords

Euclidean Steiner tree problemconicforminterior point algorithms.
Type
Research Article
Information
RAIRO - Operations Research , Volume 35 , Issue 4 , October 2001 , pp. 383 - 394
Copyright
© EDP Sciences, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alizadeh, F., Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5 (1995) 13-51. CrossRef
Gilbert, E.N. and Pollack, H.O., Steiner minimal trees. SIAM J. Appl. Math. 16 (1968) 323-345.
Goemans, M.X. and Williamson, D.P., Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 6 (1995) 1115-1145. CrossRef
Hwang, F.K., A linear time algorithm for full Steiner trees. Oper. Res. Lett. 4 (1986) 235-237. CrossRef
Hwang, F.K. and Weng, J.F., The shortest network under a given topology. J. Algorithms 13 (1992) 468-488. CrossRef
F.K. Hwang, D.S. Richards and P. Winter, The Steiner Tree Problem. Ann. Discrete Math. 53 (1992).
C. Lemaréchal and F. Oustry, Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization. Rapport de recherche, Institut National de Recherche en Informatique et en Automatique, INRIA (1999).
Maculan, N., Michelon, P. and Xavier, A.E., The Euclidean Steiner Problem in ℜ: A mathematical programming formulation. Ann. Oper. Res. 96 (2000) 209-220. CrossRef
Y.E. Nesterov and M.J. Todd, Self-Scaled Barriers and Interior-Point Methods for Convex Programming (manuscript).
Poljak, S., Rendl, F. and Wolkowicz, H., A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. Global Optim. 7 (1995) 51-73. CrossRef
Smith, W.D., How to find Steiner minimal trees in Euclidean d-space. Algorithmica 7 (1992) 137-177. CrossRef
Xue, G. and An Efficient Al, Y. Yegorithm for Minimizing a Sum of Euclidean Norms with Applications. SIAM J. Optim. 7 (1997) 1017-1036. CrossRef