# Rosenbrock函數

Rosenbrock函數的定義如下：

${\displaystyle f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}.\quad }$

Rosenbrock函數的每个等高线大致呈抛物线形，其全域最小值也位在抛物线形的山谷中（香蕉型山谷）。很容易找到這個山谷，但由於山谷內的值變化不大，要找到全域的最小值相當困難。

## 多變數下的擴展

${\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\dots ,x_{N})=\sum _{i=1}^{N/2}\left[100(x_{2i-1}^{2}-x_{2i})^{2}+(x_{2i-1}-1)^{2}\right].}$[2]

${\displaystyle f(\mathbf {x} )=\sum _{i=1}^{N-1}\left[(1-x_{i})^{2}+100(x_{i+1}-x_{i}^{2})^{2}\right]\quad \forall \mathbf {x} \in \mathbb {R} ^{N}.}$[3]

## 随机函数

${\displaystyle f(\mathbf {x} )=\sum _{i=1}^{n-1}{\Big [}(1-x_{i})^{2}+100\epsilon _{i}(x_{i+1}-x_{i}^{2})^{2}{\Big ]},}$

## 參考資料

1. ^ Rosenbrock, H.H. An automatic method for finding the greatest or least value of a function. The Computer Journal. 1960, 3: 175–184. ISSN 0010-4620. doi:10.1093/comjnl/3.3.175.
2. ^ L C W Dixon, D J Mills. Effect of Rounding errors on the Variable Metric Method. Journal of Optimization Theory and Applications 80, 1994. [1]页面存档备份，存于互联网档案馆
3. ^ Generalized Rosenbrock's function. [2008-09-16]. （原始内容存档于2008-09-26）.
4. ^ Schalk Kok, Carl Sandrock. Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function. Evolutionary Computation 17, 2009. [2]页面存档备份，存于互联网档案馆
5. ^ Yang X.-S. and Deb S., Engineering optimization by cuckoo searc１h, Int. J. Math. Modelling Num. Optimisation, Vol. 1, No. 4, 330-343 (2010)