On a Nonconvex MINLP Formulation of the Euclidean Steiner Tree Problem in n-Space

The Euclidean Steiner Tree Problem in dimension greater than 2 is notoriously difficult. Successful methods for exact solution are not based on mathematical-optimization -- rather, they involve very sophisticated enumeration. There are two types of mathematical-optimization formulations in the literature, and it is an understatement to say that neither scales well enough to be useful. We focus on a known nonconvex MINLP formulation. Our goal is to make some first steps in improving the formulation so that large instances may eventually be amenable to solution by a spatial branch-and-bound algorithm. Along the way, we developed a new feature which we incorporated into the global-optimization solver SCIP and made accessible via the modeling language AMPL, for handling piecewise-smooth univariate functions that are globally concave.

[1]  Nelson Maculan,et al.  Using a Conic Formulation for Finding Steiner Minimal Trees , 2004, Numerical Algorithms.

[2]  K. Chao,et al.  Steiner Minimal Trees , 2005 .

[3]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[4]  Andreas Wächter,et al.  A Global-Optimization Algorithm for Mixed-Integer Nonlinear Programs Having Separable Non-convexity , 2009, ESA.

[5]  David M. Warme,et al.  Exact Algorithms for Plane Steiner Tree Problems: A Computational Study , 2000 .

[6]  Ding-Zhu Du,et al.  Steiner tree problems in computer communication networks , 2008 .

[7]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[8]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[9]  Kenji Shimada,et al.  The travelling salesman problem with neighbourhoods: MINLP solution , 2013, Optim. Methods Softw..

[10]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[11]  L. Cavalli-Sforza,et al.  PHYLOGENETIC ANALYSIS: MODELS AND ESTIMATION PROCEDURES , 1967, Evolution; international journal of organic evolution.

[12]  A. Cenedese,et al.  Optimal design of water distribution networks , 1978 .

[13]  Kurt M. Anstreicher,et al.  Geometric conditions for Euclidean Steiner trees in ℜd , 2013, Comput. Geom..

[14]  Warren D. Smith.  How to find Steiner minimal trees in euclideand-space , 1992, Algorithmica.

[15]  Nelson Maculan,et al.  The Euclidean Steiner tree problem in Rn: A mathematical programming formulation , 2000, Ann. Oper. Res..

[16]  im Fach Mathematik , .

[17]  Kurt M. Anstreicher,et al.  An improved algorithm for computing Steiner minimal trees in Euclidean d-space , 2008, Discret. Optim..

[18]  A. Lodi,et al.  On the optimal design of water distribution networks: a practical MINLP approach , 2012 .

[19]  F. Hwang,et al.  The Steiner Tree Problem , 2012 .