Opt Art

Optimization is the branch of mathematics concerned with finding the best way to complete a task. Certainly, some tasks are easy, and we need not rely on optimization to tackle them. But many others are much more difficult, so much so that we may have very little hope of completing them satisfactorily—let alone optimally—without optimization. For example, suppose a friend of ours does volunteer work for a Meals-onWheels program. Once a week she bikes to the Meals-on-Wheels headquarters and picks up 20 meals and a list of 20 names and addresses. She then gets in the Meals-on-Wheels van, delivers the meals, and returns to headquarters. Her goal is to drop off the meals in an order that will minimize the number of miles she’ll travel, as this will minimize fuel consumption, pollutant emissions, and the amount of time she’ll spend on the job. Is her task—planning the route she’ll take—an easy one? It depends. If all of the addresses are on the same road, then it is extremely easy; the optimal route will be obvious to anyone who takes a look at a map. But if not, it can be extremely difficult, especially in the case in which the addresses appear to have been scattered about the city at random. (Why? Why not just list and evaluate every single route? The answer is that there are 20! ≈ 2.43 × 10 routes, one for every permutation of the 20 addresses. Even if our friend has a laptop that can evaluate one trillion (10) routes per second, she’ll have to run it for about 28 days if she wants to find the optimal route via complete enumeration!) Incidentally, this task is an instance of the Traveling Salesman Problem (TSP), one of the most difficult, important, and well-studied problems in the optimization field. Optimization has a seemingly unlimited number of applications. It has been put to good use in a great number of diverse disciplines: advertising, agriculture, biology, business, economics, engineering, manufacturing, medicine, telecommunications, and transportation (to name but a few). In this article, we showcase its amazing utility by describing some applications in the area of art, which at first glance would seem to have no use for it whatsoever!