On Optimization

As I sat for an hour on an unmoving train about 15 minutes from my destinaton while the conductors waited for freight traffic, I couldn't help but think that the train schedules could be optimized to prevent these sorts of delays. We know how fast trains can go, how many there are, and the set routes for every train on the tracks at any given time. Since trains infrequently leave enter stretches of track, we can model the trains with the classic transport equation, and predict when and where shocks will occur with relative ease. With this information and desired departure/arrival times, we can plug the data into standard optimization routines to minimize the delay. This would probably even speed up the major freight railways that pay money for track priority.

Rants about train delays aside, I've been thinking recently about how optimization appears almost everywhere in science and engineering. Optimal controllers help us drive our airplanes and fly our cars. Our numerical methods used for physics simulations are optimized for accuracy and in many cases speed of computation. Most technology is optimized to meet size/cost/aesthetic constraints. Even outside of "traditional" hard science, optimization can be found behind the scenes, and for the rest of this blog post I'll talk about some interesting and personally relevant optimization applications.

One of the most important optimization routines to a twenty-something graduate student is how Netflix decides which movies that we like. Essentially, they create a sparse matrix (which is also low rank because it's pretty much physically impossible to watch everything on Netflix) out of all the movies you've watched. Then they compare the average ratings of everybody else who has watched those movies, and movies in the same genre (organized as rows or columns in the "movie space" matrix). Based off of this they use optimization routines to fill in the empty rows of your movie matrix while keeping it as similar to what you've said about your rating preferences as possible. In 2006 Netflix initiated the Netflix Prize to find the best algorithm, with a grand prize of $1M. The moral of the story? Rate everything you watch on netlfix to get better recommendations.

Another personally important relevant topic is building climate control. Most basic thermostats work in an on/off fashion. When it's too hot/cold, the system goes on. When it's within an acceptable range, the system turns off. But what is the best range for personal comfort? Or for energy cost? Now it becomes a problem of decided the right thresholds to maximize personal comfort while minimizing the client's energy bill. Going up a step in the process, buildings have to be designed in the best way to heat/cool relevant areas. This requires knowledge of air and heat dynamics, and there are several interesting optimization papers trying to improve the efficiency of climate control in buildings.

As a student of applied mathematics, optimization is everywhere. Every problem we want to solve by computer turns into optimizing the accuracy subject to the constraints of your algorithm and processor. I've been learning a lot of theory this semester in graduate school, but at the end of the semester almost every class has turned into "What's the best we can do when we can't write the solution down anymore?" As I progress in research, I will probably find a problem, proceed as far as I can analytically, and then utilize or develop the optimization routines necessary to solve the problem numerically. This proces hopefully results in something new.