Optimisation test functions

The OneMax function returns the sum of the individual. For an individual of length $n$, maximum is achieved with $n$ ones.

\[\text{OneMax}(\mathbf{x}) = \sum_{i=1}^n x_i\]

The LeadingOnes function returns the number of uninterrupted ones from the start of the chromosome. The maximum is achieved with $n$ ones, but the landscape is a bit more difficult to traverse.

\[\text{LO}(\mathbf{x}) = \sum_{i=1}^n \prod_j^i x_j\]

jumpk(x; k=6)

The JumpK function is a modification of the onemax function, with a valley of size $k$ right before the maximum. The only way for a hill climber to reach the maximum is with a perfect flip of $k$ bits, which is considered extremely difficult.

\[\text{JUMP}_k(x) = \begin{cases} \lVert{x}\rVert_1 & \quad \text{if } \lVert x \rVert_1 \in [0..n-k] \cup \{n\},\\ -\lVert x \rVert_1 & \quad \text{otherwise} \end{cases}\]

where $\lVert x \rVert_1 = \sum_{i=1}^n x_i$ is the number of 1-bits in $x \in \{0, 1\}^n$.

ackley(x; a=20, b=0.2, c=2π)

Ackley's benchmark function. Global minimum is at the origin, with value of 0. Parameters are typically $a=20$, $b=0.2$, and $c=2\pi$.

\[f(x) = -a \exp\left(-b\sqrt{\frac{1}{d} \sum_{i=1}^d x_i^2}\right) - \exp\left(\frac{1}{d} \sum_{i=1}{d} \cos (cx_i) \right) + a + \exp(1)\]

The Booth function is a 2-dimensional quadratic function with global minimum $x^* = [1, 3]$ and optimal value $f(x^*) = 0$.

\[f(x) = (x_1 + 2x_2 - 7)^2 + (2 x_1 + x_2 - 5)^2\]

branin(x; a=1, b=5.1/(4π^2), c=5/π, r=6, s=10, t=1/(8π))

The Branin (a.k.a. Branin-Hoo) function has six optional parameters, and features multiple global minima. Some of them are at $x^* = [-\pi, 12.275]$, $x^* = [\pi, 2.275]$, $x^* = [3\pi, 2.475]$ and $x^* = [5\pi, 12.875]$, with $f(x^*) \approx 0.397887$.

\[f(x) = a(x_2 - bx_1^2 + cx_1 - r)^2 + s(1 - t)\cos(x_1) + s\]

circle(x)

The circle function is a multiobjective test function, given by

\[f(x) = \begin{bmatrix} 1 - r\cos(\theta) \\ 1 - r\sin(\theta) \end{bmatrix}\]

where $\theta=x_1$ and $r$ is obtained by

\[r = \frac{1}{2} + \frac{1}{2} \left(\frac{2x_2}{1+x_2^2}\right)\]

flower(x; a=1, b=1, c=4)

The flower function is a two-dimensional test function with flower-like contour lines coming out from the origin. Typically, optional parameters are set at $a=1$, $b=1$ and $c=4$. The function is minimised near the origin, although it does not have a global minimum due to atan bein undefined at $[0, 0]$.

\[f(x) = a\lVert\mathbb{x}\rVert + b \sin(c\arctan(x_2, x_1))\]

michalewicz(x; m=10)

The Michalewicz function is a $d$-dimensional function with several steep valleys, where m controls the steepness. m is usually set at 10. For 2 dimensions, $x^* = [2.20, 1.57]$, with $f(x^*) = -1.8011$.

\[f(x) = -\sum_{i=1}^{d}\sin(x_i) \sin^{2m}\left(\frac{ix_i^2}{\pi}\right)\]

rosenbrock(x; b=100)

The $d$-dimensional Rosenbrock banana benchmark function. With $b=100$, minimum is at $f([1, \dots, 1]) = 0$

\[f(x) = \sum_{i=1}^{d-1} \left[b(x_{i+1} - x_i^2)^2 + (x_i - 1)^2 \right]\]

wheeler(x, a=1.5)

The Wheeler's ridge is a 2-d function with a single global minimum in a curved peak. With $a$ (by default at 1.5) $x^* = [1, 1.5]$, with $f(x^*) = -1$.

\[f(x) = - \exp(- (x_1 x_2 - a)^2 - (x_2 - a)^2 )\]