A New Relaxation in Conic Form for the Euclidean Steiner Problem in ℜ

Marcia Fampa; Nelson Maculan

RAIRO - Operations Research (2010)

  • Volume: 35, Issue: 4, page 383-394
  • ISSN: 0399-0559

Abstract

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In this paper, we present a new mathematical programming formulation for the Euclidean Steiner Tree Problem (ESTP) in ℜ. We relax the integrality constrains on this formulation and transform the resulting relaxation, which is convex, but not everywhere differentiable, into a standard convex programming problem in conic form. We consider then an efficient computation of an ϵ-optimal solution for this latter problem using interior-point algorithm.

How to cite

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Fampa, Marcia, and Maculan, Nelson. "A New Relaxation in Conic Form for the Euclidean Steiner Problem in ℜ ." RAIRO - Operations Research 35.4 (2010): 383-394. <http://eudml.org/doc/197792>.

@article{Fampa2010,
abstract = { In this paper, we present a new mathematical programming formulation for the Euclidean Steiner Tree Problem (ESTP) in ℜ. We relax the integrality constrains on this formulation and transform the resulting relaxation, which is convex, but not everywhere differentiable, into a standard convex programming problem in conic form. We consider then an efficient computation of an ϵ-optimal solution for this latter problem using interior-point algorithm. },
author = {Fampa, Marcia, Maculan, Nelson},
journal = {RAIRO - Operations Research},
keywords = {Euclidean Steiner tree problem; conic form; interior point algorithms.; Euclidean Steiner type tree problem; conic form; interior point algorithms},
language = {eng},
month = {3},
number = {4},
pages = {383-394},
publisher = {EDP Sciences},
title = {A New Relaxation in Conic Form for the Euclidean Steiner Problem in ℜ },
url = {http://eudml.org/doc/197792},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Fampa, Marcia
AU - Maculan, Nelson
TI - A New Relaxation in Conic Form for the Euclidean Steiner Problem in ℜ
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 383
EP - 394
AB - In this paper, we present a new mathematical programming formulation for the Euclidean Steiner Tree Problem (ESTP) in ℜ. We relax the integrality constrains on this formulation and transform the resulting relaxation, which is convex, but not everywhere differentiable, into a standard convex programming problem in conic form. We consider then an efficient computation of an ϵ-optimal solution for this latter problem using interior-point algorithm.
LA - eng
KW - Euclidean Steiner tree problem; conic form; interior point algorithms.; Euclidean Steiner type tree problem; conic form; interior point algorithms
UR - http://eudml.org/doc/197792
ER -

References

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  8. N. Maculan, P. Michelon and A.E. Xavier, The Euclidean Steiner Problem in ℜ: A mathematical programming formulation. Ann. Oper. Res.96 (2000) 209-220.  Zbl0966.90064
  9. Y.E. Nesterov and M.J. Todd, Self-Scaled Barriers and Interior-Point Methods for Convex Programming (manuscript).  Zbl0871.90064
  10. S. Poljak, F. Rendl and H. Wolkowicz, A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. Global Optim.7 (1995) 51-73.  Zbl0843.90088
  11. W.D. Smith, How to find Steiner minimal trees in Euclidean d-space. Algorithmica7 (1992) 137-177.  Zbl0751.05028
  12. G. Xue and Y. Ye, An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications. SIAM J. Optim.7 (1997) 1017-1036.  Zbl0885.68074

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