# Rosenbrock函數

Rosenbrock函數的定義如下：

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

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

$f(\mathbf{x}) = f(x_1, x_2, \dots, x_N) = \sum_{i=1}^{N/2} \left[100(x_{2i-1}^2 - x_{2i})^2 + (x_{2i-1} - 1)^2 \right].$[2]

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

## 随机函数

$f(\mathbf{x}) =\sum_{i=1}^{n-1} \Big[(1-x_i)^2+100 \epsilon_i (x_{i+1}-x_i^2)^2 \Big],$

## 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. doi:10.1093/comjnl/3.3.175. ISSN 0010-4620.
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]
3. ^ Generalized Rosenbrock's function. [2008-09-16].
4. ^ Schalk Kok, Carl Sandrock. Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function. Evolutionary Computation 17, 2009. [2]
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)