Mathlas repository
analytical package
The analytical package contains three analytical function definitions commonly used for optimization algorithm testing: the humps, Branin, and Rosembrock functions.
All these functions present some difficulties for numerical optimization routines like maxima at definition boundaries, so they make good candidates for testing the algorithms.
Humps function
The humps function is a one dimensional function with two maxima defined as: $$f(x) = \frac{1}{(x - 0.3)^2 + 0.01} + \frac{1}{(x - 0.9)^2 + 0.04} - 6$$
Branin function
The Branin function is a two dimensional function defined for \(x_1 ∈ [-5, 10]\), \(x_2 ∈ [0, 15]\) with three local minima (of 0.397887) defined as: $$f(x_1, x_2) = \left(x_2 - \frac{5.1 x_1^2}{4\pi^2} + \frac{5 x_1}{\pi} - 6 \right) ^ 2 + 10 \left(1 - \frac{1}{8 \pi}\right) \cos(x_1) + 10$$
Rosenbrock function
The Rosenbrock function is a two dimensional function with two parameters \(a\), \(b\) defined for \(x_1 ∈ [-5, 10]\), \(x_2 ∈ [-5, 10]\) with one minimum at \((x_1, x_2) = (a, a^2)\) defined as: $$f(x_1, x_2) = \left(a - x_1\right) ^ 2 + b \left(x_2 - x_1^2\right)^2$$
Our implementation defaults to \(a=1\), \(b=100\).