Branin functio is a well-known test function for global optimization.

函数表达式：
$$\begin{array}{r}f(x)=a(x_2-b{x}_1^2+c{x}_1-r{)}^2+s(1-t)cos(x_1)+s\end{array}$$

函数描述：

1）Dimensions: 2
The Branin, or Branin-Hoo, function has three global minima. The recommended values of a, b, c, r, s and t are: a = 1, b = 5.1 ⁄ (4π2), c = 5 ⁄ π, r = 6, s = 10 and t = 1 ⁄ (8π).
2）Input Domain:
This function is usually evaluated on the square x1 ∈ [-5, 10], x2 ∈ [0, 15].
3）Global Minimum:
$$\begin{array}{r}f(x^{\ast })=0.397887,at,x^{\ast }=(-\pi ,12.275),(\pi ,2.275),and(9.42478,2.475)\end{array}$$

4）Modifications and Alternate Forms:
Picheny et al. (2012) use the following rescaled form of the Branin-Hoo function, on $[0,1{]}^2$:
$$\begin{array}{r}f(x)=\frac{1}{51.95}[(\overline{x_2}-\frac{5.1{\overline{x_1}}^2}{4{\pi }^2}+\frac{5\overline{x_1}}{\pi }-6{)}^2+(10-\frac{10}{8\pi })cos(\overline{x_1})-44.81]\end{array}$$
其中，$\overline{x_1}=15x_1-5,\overline{x_2}=15x_2$

This rescaled form of the function has a mean of zero and a variance of one. The authors also add a small Gaussian error term to the output.

For the purpose of Kriging prediction, Forrester et al. (2008) use a modified form of the Branin-Hoo function, in which they add a term 5x1 to the response. As a result, there are two local minima and only one global minimum, making it more representative of engineering functions.

函数三维图：

函数二维切面：（示意图）

References:
Dixon, L. C. W., & Szego, G. P. (1978). The global optimization problem: an introduction. Towards global optimization, 2, 1-15.

Forrester, A., Sobester, A., & Keane, A. (2008). Engineering design via surrogate modelling: a practical guide. Wiley.

Global Optimization Test Problems. Retrieved June 2013, from
http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO.htm.

Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf.

Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.