Highly optimized tolerance


In applied mathematics, highly optimized tolerance is a method of generating power law behavior in systems by including a global optimization principle. It was developed by Jean M. Carlson in the early 2000s. For some systems that display a characteristic scale, a global optimization term could potentially be added that would then yield power law behavior. It has been used to generate and describe internet-like graphs, forest fire models and may also apply to biological systems.

Example

The following is taken from Sornette's book.
Consider a random variable,, that takes on values with probability. Furthmore, lets assume for another parameter
for some fixed. We then want to minimize
subject to the constraint
Using Lagrange multipliers, this gives
giving us a power law. The global optimization of minimizing the energy along with the power law dependence between and gives us a power law distribution in probability.