Abstract
We present a new mathematical programming formulation for the Steiner minimal tree problem. We relax the integrality constraints on this formulation and transform the resulting problem (which is convex, but not everywhere differentiable) into a standard convex programming problem in conic form. We consider an efficient computation of an ε-optimal solution for this latter problem using an interior-point algorithm.
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References
F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5 (1995) 13–51.
K.D. Andersen, E. Christiansen, A.R. Conn and M.L. Overton, An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms, SIAM J. Sci. Comput. 22(1) (2000) 243–262.
K.M. Anstreicher, Eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem, SIAM J. Optim. 11 (2000) 254–265.
E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 323–345.
M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 6 (1995) 1115–1145.
F.K. Hwang, A linear time algorithm for full Steiner trees, Oper. Res. Lett. 4 (1986) 235–237.
F.K. Hwang, D.S. Richards and P. Winter, The Steiner tree problem, in: Annals of Discrete Mathematics, Vol. 53 (North-Holland, Amsterdam, 1992).
F.K. Hwang and J.F. Weng, The shortest network under a given topology, J. Algorithms 13 (1992) 468–488.
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).
N. Maculan, P. Michelon and A.E. Xavier, The Euclidean Steiner problem in ℝn: A mathematical programming formulation, Ann. Oper. Res. 96 (2000) 209–220.
M. Minoux, Programmation Mathématique, Vol. 2 (Dunod, Paris, 1983) pp. 80–89.
F. Montenegro, N. Maculan, G. Plateau and P. Boucher, New heuristics for the Euclidean Steiner problem in ℝn, in: Essays and Surveys in Metaheuristics, eds. C.C. Ribeiro and P. Hansen (Kluwer Academic, Dordrecht, 2002) pp. 509–524.
Y.E. Nesterov and A. Nemirovskii, Interior Polynomial Algorithm in Convex Programming (SIAM, Philadelphia, PA, 1994).
Y.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res. 22 (1997) 1–42.
S. Poljak, F. Rendl and H. Wolkowicz, A recipe for semidefinite relaxation for (0, 1)-quadratic programming, J. Global Optim. 7 (1995) 51–73.
W.D. Smith, How to find Steiner minimal trees in Euclidean d-space, Algorithmica 7 (1992) 137–177.
G. Xue, J. Rosen and P. Pardalos, A polynomial time dual algorithm for the Euclidean multifacility location problem, in: Proc. of the 2nd Conf. on Integer Programming and Combinatorial Optimization (Pittsburg, PA, 1992) pp. 227–236.
G. Xue and Y. Ye, An Efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM J. Optim. 7 (1997) 1017–1036.
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Fampa, M., Maculan, N. Using a Conic Formulation for Finding Steiner Minimal Trees. Numerical Algorithms 35, 315–330 (2004). https://doi.org/10.1023/B:NUMA.0000021765.17831.bc
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DOI: https://doi.org/10.1023/B:NUMA.0000021765.17831.bc