Braninfunction简介weixin43509834的博客-

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

函数表达式：

$begin{aligned} f(x)=a(x_2-bx_1^2+cx_1-r)^2+s(1-t)cos(x_1)+s end{aligned}$

f(x)=a(x2​−bx12​+cx1​−r)2+s(1−t)cos(x1​)+s​

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:

$begin{aligned} f(x^*)=0.397887, at,x^*=(-pi ,12.275), (pi ,2.275), and (9.42478,2.475) end{aligned}$

f(x∗)=0.397887,at,x∗=(−π,12.275),(π,2.275),and(9.42478,2.475)​

4）Modifications and Alternate Forms:
Picheny et al. (2012) use the following rescaled form of the Branin-Hoo function, on
$[0, 1]^2$

[0,1]2:

$begin{aligned} f(x)=frac{1}{51.95}[(bar {x_2}-frac{5.1bar {x_1}^2}{4pi^2}+frac{5bar {x_1}}{pi}-6)^2+(10-frac{10}{8pi})cos(bar {x_1})-44.81] end{aligned}$

f(x)=51.951​[(x2​ˉ​−4π25.1x1​ˉ​2​+π5x1​ˉ​​−6)2+(10−8π10​)cos(x1​ˉ​)−44.81]​
其中，
$bar {x_1}=15x_1-5,bar {x_2}=15x_2$

x1​ˉ​=15x1​−5,x2​ˉ​=15x2​

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 5×1 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
https://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 https://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.