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

An overview of exact algorithms for the Euclidean Steiner tree problem in n‐space

“…In Maculan et al. (), the authors show that the extreme points of the set defined by constraints –, together with , for all , are integers (for a more combinatorial proof, see also D'Ambrosio et al., ). Therefore, if we fix the position of the Steiner points in MMX, we obtain the integer linear problem of determining the minimum spanning tree on the given nodes in V with an FST embedded in graph G , and this problem can be efficiently solved.…”

“…A few papers (Fampa and Maculan, ; Fampa et al., ; D'Ambrosio et al., ) considered the formulation MMX for the ESTP and proposed new formulations and specialized procedures based on it, with the goal of turning more practical the application of branch‐and‐bound algorithms to a mathematical programming formulation of the problem. The main drawbacks observed in implementing MMX in a spatial branch‐and‐bound algorithm are discussed in these papers and summarized as follows: (D4)The nonconvexity of the continuous relaxation of MMX introduces natural difficulties to the solution of the problem that requires global optimization methods. (D5)The objective function of MMX is nondifferentiable at points where the FSTs degenerate, hindering the application of algorithms to solve the nonlinear‐programming relaxations, that generally require the functions to be twice continuously differentiable. (D6)The lower bounds given by the relaxations at the nodes of the spatial branch‐and‐bound tree are extremely weak. (D7)A large number of isomorphic FSTs still belong to the feasible set of the model, leading to a very inefficient branch‐and‐bound algorithm that processes many isomorphic subproblems. …”

“…More recently, two other works (Fampa et al., ; D'Ambrosio et al., ) on the mathematical programming formulations for the ESTP were presented. In D'Ambrosio et al.…”

“…In D'Ambrosio et al. (), the authors propose enhancements to formulation MMX in order to mitigate the problems derived from drawbacks D5 and D6 and to turn more successful the solution of the problem by the global solver SCIP. The authors first propose and compare different smoothing techniques for the nondifferentiable Euclidean norm function, and implement in SCIP (via a new general feature) a smooth but piecewise‐defined concave function shown to have very good theoretical properties and numerical behavior.…”

“…() are the inclusion of linear valid inequalities in FM based on the geometric conditions on SMTs, that were also considered in Van Laarhoven and Anstreicher () and D'Ambrosio et al. (), and the use of a dynamic constraint set, where redundant nonlinear constraints are eliminated from the model. Finally, the authors propose a heuristic procedure to be executed during the branch and bound, to improve the upper bounds and lead to more fathoming.…”