System, methods, and other embodiments described herein relate to generating a path for a vehicle to travel. In one embodiment, a method includes receiving left and right boundary points of a corridor in which the vehicle is to travel and creating point pairs from the left and right boundary points. Respective ones of the point pairs include one of the left boundary points and one of the right boundary points that is a closest corresponding point of the right boundary points. The method further includes generating an output path according to an optimization problem that determines path points, with a respective one of the path points for each of the point pairs. The point pairs function as hard constraints for the corresponding respective ones of the path point.