Numerical Algorithms 35: 315–330, 2004.  2004 Kluwer Academic Publishers. Printed in the Netherlands. Using a conic formulation for finding Steiner minimal trees ∗ Marcia Fampa a and Nelson Maculan b a Universidade Federal do Rio de Janeiro, Instituto de Matemática, Departamento de Ciência da Computação, Caixa Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil E-mail: fampa@cos.ufrj.br b Universidade Federal do Rio de Janeiro, COPPE,Programa de Engenharia de Sistemas e Computação (PESC), Caixa Postal 68511, Rio de Janeiro, RJ 21945-970, Brazil E-mail: maculan@cos.ufrj.br Received 16 December 2001; accepted 27 April 2002 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 program- ming problem in conic form. We consider an efficient computation of an ε-optimal solution for this latter problem using an interior-point algorithm. Keywords: Steiner minimal trees, conic formulation, interior-point algorithms AMS subject classification: 90C27, 90C11, 90C51 1. Introduction The Steiner minimal tree problem (SMTP) can be defined as follows: Given p points in R n , find a minimum tree that spans these points using or not extra points, which are called Steiner points. The distances considered between points are Euclidean. This is a very well known problem in combinatorial optimization [7], which goes back to an ancient problem studied by Fermat in the 17th century. Fermat considered the following challenge: “Given three points in the plane, find a fourth point such that the sum of its distance to the three given points is at minimum.” Torricelli, in 1647, proved that the circle circumscribing the equilateral triangles constructed on the sides of and outside the given triangle intersect at the desired point. Heinen, in 1837, seems to be the first to prove that, for a triangle in which one angle is greater than or equal to 120 ◦ , the vertex associated with this angle is the minimizing point. Some examples ∗ This work was partially supported by CNPq, FUJB, FAPERJ and CTPETRO.