A successive relaxation algorithm to solve a MILP involving piecewise linear functions with application to road design

This paper presents a new algorithm to build feasible solutions to a MILP formulation of the vertical alignment problem in road design. This MILP involves a large number of special ordered set of type 2 variables used to describe piecewise linear functions. The principle of the algorithm is to successively solve LPs adapted from the MILP by replacing the special ordered set of type 2 constraints by linear constraints. Proof that the solutions to the successive linear relaxations of the MILP converge to a feasible solution to the MILP is provided. Numerical results emphasize that the algorithm performs better than CPLEX for large scale vertical alignment problems.