Case studies in trajectory optimization: trains, planes, and other pastimes

This is first in a series of papers presenting case studies in modern large-scale constrained optimization, the purpose of which is to illustrate how recent advances in algorithms and modeling languages have made it easy to solve difficult optimization problems using off-the-shelf software. In this first paper, we consider four trajectory optimization problems: (a) how to operate a train efficiently, (b) how to putt a golf ball on an uneven green so that it arrives at the cup with minimal speed, (c) how to fly a hang glider so as to maximize or minimize the range of the glide, and (d) how to design a slide to make a toboggan go from beginning to end as quickly as possible. In addition to the tutorial aspects of this paper, we also present evidence suggesting that the widely used trapezoidal discretization method is inferior in several ways to a certain simpler midpoint discretization method.