The Traveling Salesman Problem with Neighborhoods: MINLP solution

The traveling salesman problem with neighborhoods extends the traveling salesman problem to the case where each vertex of the tour is allowed to move in a given region. This NP-hard optimization problem has recently received increasing attention in several technical fields such as robotics, unmanned aerial vehicles, or utility management. In this paper, the problem is formulated as a non-convex Mixed-Integer Nonlinear Program (MINLP) having the property that fixing all the integer variables to any integer values yields a convex nonlinear program. This property is used to modify the global MINLP optimizer Couenne, improving by orders of magnitude its performance and allowing the exact solution of instances large enough to be useful in applications. Computational results are presented where neighborhoods are either polyhedra or ellipsoids in R² or R³ and with the Euclidean norm as distance metric.