One tech to rule them all
Back in high school when you were battling polynomials with derivatives, you learnt one of your first all-purpose optimisation techniques. Given a function, take its derivative, set it to zero and solve for all optima. Voila! After you had coerced the equation to give up all its maxima secrets, you could then be safe in the knowledge that you had found the optimal values.
If you enjoyed computing the optimal speed to jump your calculus hurdles, then you probably went on to university to learn more about other mathematical tools. Once there you most likely met your second all-purpose optimisation technique, Linear Programming. Given a simple linear function with linear constraints, you learnt that you could use any of a number of tools such as the Simplex Method to easily and quickly solve such problem.
From there, if you had more of an applied orientation, you probably went on to learn about the exotic optimisation creatures. For example, Genetic Algorithms, Simulated Annealing as well as a host of other nature-inspired algorithms such as Ant Colony Optimisations, Swarm Optimisation, Artificial Bee Colony Optimisation techniques, etc. In the 1990s and 2000s, for almost any type of problem, it seemed that one could find a new nature-inspired optimisation technique to solve it better than previous more mainstream tools.
However, in spite of the swarm of new optimisation tools, most problems for which these new techniques were developed could be solved with existing tools. Although probably not as efficiently. Hence the question was which method was best? The answer to this question came in 1995 when two researchers, David Wolpert and William MacReady at the renowned Sante Fe Institute, proved a number of theorems collectively referred to as the “No free lunch” theorems. These results could be seen as implying that there is no one optimisation method that is best for all problems. In addition, when we average over all problems, we expect all methods to be equal.
This result has important and deep implications. It means that if you swear by a single generic optimisation method and try and use it to solve all your problems, then don’t expect it to perform better than your three year old son who guesses random solutions to all your different stochastic multi-objective constrained optimisation techniques.
Given this it would seem strange then that I am about to suggest the idea of solving numerous problems with a single optimisation technique. Besides the fact that a number of these problems don’t look like optimisation problems at all! The reason for doing this is to see the power of an interesting optimisation technique, as well as its flexibility and generality. In addition, one needs to bear in mind that the “No free lunch” theorem is a result about every imaginable optimisation problem, which is just a tad more than the few I will touch on here.
The class of optimisation technique that I want to discuss here is generally referred to as Genetic Algorithms. They have been successfully used on thousands of research and industrial problems and continue to amaze researchers with their potential to solve problems far beyond their original scope. For example, one of the most famous applications of Genetic Algorithms was by NASA in 2006 to develop a new antenna design for their ST5 antenna. It can be seen that the optimal design was anything but intuitive and most likely would not have been found by a “standard” optimisation technique based on initial human guesses.
So what type of problems would be considered inappropriate for Genetic Algorithms? First of all, you couldn’t do much worse than write some code to give to your son or daughter so that in their next algebra exam they can solve x + 5 = 6 with your tailored made Genetic Algorithm. They could just let your code chug away and sit there patiently while it aces the exam for them. Although not probably the most effective use of Genetic Algorithms, it is entirely possible.
So let’s take that thought one step further. What about solving the humble quadratic equation with a genetic algorithm? It has been done (and done again, and again). But the quadratic equation belongs to pure mathematics right? In addition, it’s an equation you can solve directly isn’t it? Yes and yes but interestingly enough Genetic Algorithms have started to make their way into some of the purest area of mathematics to help solve problems that are stumping pure mathematicians, This is truly one area where you would not expect the tools of applied mathematicians to come to the rescue of pure mathematicians.
We have only scratched the surface of unexpected applications of Genetic Algorithms. In fact, they have made an appearance in almost every endeavour of research from physics to economics, from architecture to chemistry and even all the way back to their nature-inspired beginnings with numerous applications in biology. So in spite of our knowledge that there is no one method to solve all problems, Genetic Algorithms present us with a versatile and powerful tool that seems to have a lot more in store for our future problem solvers.