Efficient Method for Estimating Globally Optimal Simple Vertical Curves

Different methods for estimating simple vertical curves that optimally fit observed profile data have been developed. In 1999, the author developed a linear programming (LP) method for estimating simple vertical curves using LINGO optimization software. To obtain the global optimal solution, the LP formulation was manually solved for different combinations of the two unknown nonlinear variables (using $5\mathrm{m}$ increments). In 2004, an improved method that automates the iterations using Visual-Basic in Excel Solver was published. The global optimal solution required $10\mathrm{h}$ for an increment of $0.1\mathrm{m}$ . This technical note presents an extension of the previously developed LP formulation that converges to the global optimal solution in a minute. The formulation involves no iterations of the nonlinear variables. Instead, the start and end points of the parabolic curve were modeled using three binary variables, and the resulting mixed-integer nonlinear model was solved using LINGO global option that has been recently developed. The proposed method, which is applicable to both crest and sag vertical curves, should be of interest to surveying professionals.