In applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, highly optimized tolerance (HOT) is a method of generating power law

Power law

A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

 behavior in systems by including a global optimization

Global optimization

Global optimization is a branch of applied mathematics and numerical analysis that deals with the optimization of a function or a set of functions to some criteria.- General :The most common form is the minimization of one real-valued function...

 principle. 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

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

, , 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

Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...

, 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.