A global constrained optimization method for designing road networks with small diameters

The road network design problem is to optimize the road network by selecting paths to improve or adding paths in the existing road network, under certain constraints, e.g., the weighted sum of modifying costs. Since its multi-objective nature, the road network design problem is often challenging for designers. Empirically, the smaller diameter a road network has, the more connected and efficient the road network is. Based on this observation, we propose a set of constrained convex models for designing road networks with small diameters. To be specific, we theoretically prove that the diameter of the road network, which is evaluated w.r.t the travel times in the network, can be bounded by the algebraic connectivity in spectral graph theory since that the upper and lower bounds of diameter are inversely proportional to algebraic connectivity. Then we can focus on increasing the algebraic connectivity instead of reducing the network diameter, under the budget constraints. The above formulation leads to a semi-definite program, in which we can get its global solution easily. Then, we present some simulation experiments to show the correctness of our method. At last, we compare our method with an existing method based on the genetic algorithm.