Bellow You'll find the definitions of all the test functions implemented in this package.
Function name: ackley
$$f(x) = -20 e^{-0.2 \sqrt{D^{-1} \sum\nolimits_{i=1}^D x_i^2}} - e^{D^{-1} \sum\nolimits_{i=1}^D \cos(2 \pi x_i)} + 20 + e$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: alpine1
$$f(x) = \sum_{i=1}^{D} \lvert {x_i \sin \left( x_i \right) + 0.1 x_i} \rvert$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: alpine2
$$f(x) = \prod_{i=1}^{D} \sqrt{x_i} \sin(x_i)$$
Dimensions: $D$
Global optimum: $f(x^*) = 2.808^D$ for $x_i^* = 7.917$
Function name: cigar
$$f(x) = x_1^2 + 10^6\sum_{i=2}^{D} x_i^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: cosine_mixture
$$f(x) = -0.1 \sum_{i=1}^D \cos (5 \pi x_i) - \sum_{i=1}^D x_i^2$$
Dimensions: $D$
Global optimum: $f(x^*) = -0.1 D$ for $x_i^* = 0$
Function name: csendes
$$f(x) = \sum_{i=1}^D x_i^6 \left( 2 + \sin \frac{1}{x_i}\right)$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: dixon_price
$$f(x) = (x_1 - 1)^2 + \sum_{i = 2}^D i (2x_i^2 - x_{i - 1})^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 2^{- \frac{(2^i - 2)}{2^i}}$
Function name: griewank
$$f(x) = \sum_{i=1}^D \frac{x_i^2}{4000} - \prod_{i=1}^D \cos(\frac{x_i}{\sqrt{i}}) + 1$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: katsuura
$$\prod_{i=1}^D \left(1 + i \sum_{j=1}^{32} \frac{\lvert 2^j x_i - round\left(2^j x_i \right) \rvert}{2^j} \right)$$
Dimensions: $D$
Global optimum: $f(x^*) = 1$ for $x_i^* = 0$
Function name: levy
$$\begin{gather}
\sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)),\,\text{where}\\\
w_i = 1 + \frac{x_i - 1}{4},\, \text{for all } i = 1, \ldots, D
\end{gather}$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 1$
Function name: michalewicz
$$f(x) = - \sum_{i = 1}^{D} \sin(x_i) \sin^{2m}\left( \frac{ix_i^2}{\pi} \right)$$
Dimensions: $D$
Global optimum: $\text{at } D=2,\,f(x^*) = -1.8013$ for $x^* = (2.20, 1.57)$
Function name: perm1
$$f(x) = \sum_{i = 1}^D \left( \sum_{j = 1}^D (j^i + \beta) \left( \left(\frac{x_j}{j}\right)^i - 1 \right) \right)^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = i$
Function name: perm2
$$f(x) = \sum_{i = 1}^D \left( \sum_{j = 1}^D (j - \beta) \left( x_j^i - \frac{1}{j^i} \right) \right)^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = \frac{1}{i}$
Function name: pinter
$$f(x) = \sum_{i=1}^D ix_i^2 + \sum_{i=1}^D 20i \sin^2 A + \sum_{i=1}^D i \log_{10} (1 + iB^2),\, \text{where}$$
$$\begin{align}
A &= (x_{i-1}\sin(x_i)+\sin(x_{i+1})) \\\
B &= (x_{i-1}^2 - 2x_i + 3x_{i+1} - \cos(x_i) + 1)
\end{align}$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: powell
$$f(x) = \sum_{i = 1}^{D/4} \left[ (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right]$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: qing
$$f(x) = \sum_{i=1}^D \left(x_i^2 - i\right)^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = \pm \sqrt{i}$
Function name: quintic
$$f(x) = \sum_{i=1}^D \left| x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4\right|$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = -1\quad \text{or} \quad x_i^* = 2$
Function name: rastrigin
$$f(x) = 10D + \sum_{i=1}^D \left[x_i^2 -10\cos(2\pi x_i)\right]$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: rosenbrock
$$f(x) = \sum_{i=1}^{D-1} \left[100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2 \right]$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 1$
Function name: salomon
$$f(x) = 1 - \cos\left(2\pi\sqrt{\sum\nolimits_{i=1}^D x_i^2} \right)+ 0.1 \sqrt{\sum\nolimits_{i=1}^D x_i^2}$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: schaffer2
$$f(x) = 0.5 + \frac{ \sin^2 \left( x_1^2 - x_2^2 \right) - 0.5 }{ \left[ 1 + 0.001 \left( x_1^2 + x_2^2 \right) \right]^2 }$$
Dimensions: 2
Global optimum: $f(x^*) = 0$ for $x^* = (0, 0)$
Function name: schaffer4
$$f(x) = 0.5 + \frac{ \cos^2 \left( \sin \left( \vert x_1^2 - x_2^2\vert \right) \right)- 0.5 }{ \left[ 1 + 0.001 \left( x_1^2 + x_2^2 \right) \right]^2 }$$
Dimensions: 2
Global optimum: $f(x^*) = 0.292579$ for $x^* = (0, \pm 1.25313) \text{or} (\pm 1.25313, 0)$
Function name: schwefel
$$f(x) = 418.9829D - \sum_{i=1}^{D} x_i \sin(\sqrt{\lvert x_i \rvert})$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 420.9687$
Function name: schwefel221
$$f(x) = \max_{1 \leq i \leq D} \vert x_i\vert$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: schwefel222
$$f(x) = \sum_{i=1}^{D} \lvert x_i \rvert +\prod_{i=1}^{D} \lvert x_i \rvert$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: sphere
$$f(x) = \sum_{i=1}^D x_i^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: step
$$f(x) = \sum_{i=1}^D \left( \lfloor \lvert x_i \rvert \rfloor \right)$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: step2
$$f(x) = \sum_{i=1}^D \left( \lfloor x_i + 0.5 \rfloor \right)^2$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = -0.5$
Function name: styblinski_tang
$$$$
Dimensions: $D$
Global optimum: $f(x^*) = -39.16599 D$ for $x_i^* = -2.903534$
Function name: trid
$$f(x) = \sum_{i = 1}^D \left( x_i - 1 \right)^2 - \sum_{i = 2}^D x_i x_{i - 1}$$
Dimensions: $D$
Global optimum: $f(x^*) = \frac{-D (D + 4) (D - 1)}{6}$ for $x_i^* = i (d + 1 - i)$
Function name: weierstrass
$$f(x) = \sum_{i=1}^D \left[ \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right] - D \sum_{k=0}^{k_{max}} a^k \cos \left(\pi b^k \right)$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
Function name: whitley
$$f(x) = \sum_{i=1}^D \sum_{j=1}^D \left[\frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - \cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1\right]$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 1$
Function name: zakharov
$$f(x) = \sum_{i = 1}^D x_i^2 + \left( \sum_{i = 1}^D 0.5 i x_i \right)^2 + \left( \sum_{i = 1}^D 0.5 i x_i \right)^4$$
Dimensions: $D$
Global optimum: $f(x^*) = 0$ for $x_i^* = 0$
[1] P. Ernesto and U. Diliman, “MVF–Multivariate Test Functions Library in C for Unconstrained Global Optimization,” University of the Philippines Diliman, Quezon City, 2005.
[2] M. Jamil and X.-S. Yang, “A literature survey of benchmark functions for global optimisation problems,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, p. 150, Jan. 2013, doi: 10.1504/ijmmno.2013.055204.
[3] J. J. Liang, B. Y. Qu, and P. N. Suganthan, “Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization,” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, vol. 635, no. 2, 2013.
[4] S. Surjanovic and D. Bingham, Virtual Library of Simulation Experiments: Test Functions and Datasets. Retrieved November 7, 2023, from https://www.sfu.ca/~ssurjano/.