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# 常見23種群智能演算法測試函數整理
## 目錄
[TOC]
## 前言
最近在改良演算法，但是套用在工程模型上測試一次就要4個小時才知道結果，沒效果又要再改，然後+4hr，所以就在網路上找一些Benchmark來測試，但是網路上的資源比較零散，所以就寫篇文整理一下。
## Function
## unimodal
### Sphere
$$
f(x) = \sum\limits_{i = 1}^{30} {\mathop x\nolimits_i^2 } \quad\quad - 100 \le {x_i} \le 100 \\
\ \quad \quad \quad min (f) = f(0, \ldots ,0) = 0.
$$
### Schwefel’s 2.22
$$
f(x) = \sum\limits_{i = 1}^{30} {\left| {\mathop x\nolimits_i^2 } \right|} + \prod\limits_{i = 1}^{30} {\left| {\mathop x\nolimits_i } \right|} \quad \quad - 10 \le {x_i} \le 10\\
\quad \quad \quad {\rm{m}}in(f) = f(0, \ldots ,0) = 0.
$$
### Schwefel’s 1.20
$$
f(x) = {\sum\limits_{i = 1}^{30} {\left( {\sum\limits_{j = 1}^i {\mathop x\nolimits_j } } \right)} ^2} \quad \quad - 100 \le x_i^{} \le 100\\
\quad \quad \quad\min (f) = f(0, \ldots ,0) = 0.
$$
### Schwefel’s 2.21
$$
f(x) = \max \{ \left| {\mathop x\nolimits_i } \right|,1 \le i \le 30\} \quad \quad - 100 \le \mathop x\nolimits_i \le 100\\
\quad \quad \quad\min (f) = f(0, \ldots ,0) = 0.
$$
### Generalized Rosenbrock

$$
f(x) = \sum\limits_{i = 1}^{29} {\left[ {100{{\left( {\mathop x\nolimits_{i + 1} - \mathop x\nolimits_i^2 } \right)}^2}} \right]}\quad \quad - 30 \le \mathop x\nolimits_i \le 30\\
\min (f) = f(1, \ldots ,1) = 0.
$$
### Step

$$
f(x) = \sum\limits_{i = 1}^{30} {{{\left( {\left\lfloor {\mathop x\nolimits_i + 0.5} \right\rfloor } \right)}^2}}\quad \quad - 100 \le \mathop x\nolimits_i \le 100\\
\ \quad \quad \quad min (f) = f(0, \ldots ,0) = 0.
$$
### Quartic Noise

$$
f(x) = \sum\limits_{i = 1}^{30} {i\mathop x\nolimits_i^4 } + random\left[ {0,1} \right) \quad \quad - 128 \le \mathop x\nolimits_i \le 128\\
\ \quad \quad \quad min (f) = f(0, \ldots ,0) = 0.
$$
## multimodal
### Generalized Schwefel's Problem 2.26

$$
f(x) = - \sum\limits_{i = 1}^{30} {\left( {\mathop x\nolimits_i \sin \left( {\sqrt {\left| {\mathop x\nolimits_i } \right|} } \right)} \right)} \quad \quad - 500 \le \mathop x\nolimits_i \le 500 \\ \quad \\
\ \quad \quad \quad min (f) = f(420.9687, \ldots ,420.8687) = -12569.5.
$$
### Generalized Rastrigin

$$
f(x) = \sum\limits_{i = 1}^{30} {\left[ {\mathop x\nolimits_i^2 - 10\cos \left( {2\pi \mathop x\nolimits_i } \right) + 10} \right]} \quad \quad - 5.12 \le \mathop x\nolimits_i \le 5.12\\ \quad
\\ \quad \quad \quad min (f) = f(0, \ldots ,0) = 0.
$$
### Ackley

$$
f(x) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{{30}}\sum\limits_{i = 1}^{30} {\mathop x\nolimits_i^2 } } } \right) \\- \exp \left( {\frac{1}{{30}}\sum\limits_{i = 1}^{30} {\cos 2\pi \mathop x\nolimits_i } } \right) + 20 + e \\ \quad \\ - 32 \le \mathop x\nolimits_i \le 32 \quad
\min (f) = f(0, \ldots ,0) = 0.
$$
### Generalized Griewank

$$
f(x) = \frac{1}{{4000}}\sum\limits_{i = 1}^{30} {\mathop x\nolimits_i^2 } - \prod\limits_{i = 1}^{30} {\cos \left( {\frac{{\mathop x\nolimits_i }}{{\sqrt i }}} \right) + 1}
\\ \quad \\ - 600 \le \mathop x\nolimits_i \le 600 \quad
\min (f) = f(0, \ldots ,0) = 0.
$$
### Generalized Penalized 1

$$
f(x) = \frac{\pi }{{30}}\left\{ {10{{\sin }^2}\left( {\pi \mathop y\nolimits_1 } \right) + {{\sum\limits_{i = 1}^{29} {{{\left( {\mathop y\nolimits_i - 1} \right)}^2}\cdot
\left[ {1 + 10{{\sin }^2}\left( {\pi \mathop y\nolimits_1 + 1} \right)} \right] + \left( {\mathop y\nolimits_n - 1} \right)} }^2}} \right\} \\ + \sum\limits_{i = 1}^{30} {u\left( {\mathop x\nolimits_i ,10,100,4} \right)}
\\
\quad \\ - 50 \le \mathop x\nolimits_i \le 50 \quad
\min (f) = f(1, \ldots ,1) = 0.
$$
### Generalized Penalized 2

$$
f(x) = 0.1\left\{ {{{\sin }^2}\left( {\pi 3\mathop x\nolimits_1 } \right) + \sum\limits_{i = 1}^{29} {{{\left( {\mathop x\nolimits_i - 1} \right)}^2}\left[ {1 + {{\sin }^2}\left( {3\pi \mathop x\nolimits_{i + 1} } \right)} \right]} } \right\} \\+ {\left( {\mathop x\nolimits_n - 1} \right)^2}\left[ {1 + {{\sin }^2}\left( {2\pi \mathop x\nolimits_{30} } \right)} \right]+\sum\limits_{i = 1}^{30} {u\left( {\mathop x\nolimits_i ,5,100,4} \right)}\\
\quad \\ - 50 \le \mathop x\nolimits_i \le 50 \quad
\min (f) = f(1, \ldots ,1) = 0.
$$

where

$$
u\left( {\mathop x\nolimits_i ,a,k,m} \right) = \left\{ \begin{array}{l}
k{\left( {{x_i} - a} \right)^m},\quad \quad{\rm{ }}\mathop x\nolimits_i > a\\
0, \quad \quad \quad \quad \quad {\rm{ }} - a \le \mathop x\nolimits_i \le a\\
k{\left( { - {x_i} - a} \right)^m}\quad \quad {\rm{ }}\mathop x\nolimits_i {\rm{ < }} - a{\rm{ }}
\end{array} \right.
\\
\quad \\
{\mathop y\nolimits_i = 1 + \frac{1}{4}}\left( {\mathop x\nolimits_i + 1} \right)
$$
## fixed-dimenstion multimodal
### Shekel’s Foxholes

$$
f(x) = {\left[ {\frac{1}{{500}} + \sum\limits_{j = 1}^{25} {\frac{1}{{j + \sum\nolimits_{i = 1}^2 {{{\left( {\mathop x\nolimits_i - \mathop a\nolimits_{ij} } \right)}^6}} }}} } \right]^{ - 1}}\\ \quad\\- 65.536 \le \mathop x\nolimits_i \le 65.536 \quad \min \left( f \right) = \left( { - 32,32} \right) \approx 1
$$
where
$$
\left( {\mathop a\nolimits_{ij} } \right)=
\left( \begin{array}{l}-32 \quad-16\quad\quad0\quad\quad\quad16\quad\quad32\quad-32\quad\cdots\quad0\quad16\quad32 \\-32 \quad-32\quad-32\quad-32\quad-32\quad-16\quad\cdots\quad32\quad32\quad32
\end{array} \right)
$$

### Kowalik

$$
f(x) = \sum\limits_{i = 1}^{11} {{{\left[ {\mathop a\nolimits_i - \frac{{\mathop x\nolimits_{1\left( {\mathop b\nolimits_i^2 + \mathop b\nolimits_i \mathop x\nolimits_2 } \right)} }}{{\mathop b\nolimits_i^2 + \mathop b\nolimits_i \mathop x\nolimits_3 + \mathop x\nolimits_4 }}} \right]}^2}} \quad - 5 \le \mathop x\nolimits_i \le 5 \\ \quad \\
\quad \min (f) \approx f\left( {0.1928,0.1908,0.1231,0.1358} \right) \approx 0.0003075
$$
### Six-Hump Camel-Back

$$
f = 4\mathop x\nolimits_1^2 - 2.1\mathop x\nolimits_1^4 + \frac{1}{3}\mathop x\nolimits_1^6 + \mathop x\nolimits_1 \mathop x\nolimits_2 - 4\mathop x\nolimits_2^2 + 4\mathop x\nolimits_2^4 \\- 5 \le \mathop x\nolimits_i \le 5\\\mathop x\nolimits_{\min } = \left( {0.08983, - 0.7126} \right),\left( { - 0.08983,0.7126} \right)\\\min (f) = - 1.0316285.
$$
### Branin

$$
f(x) = {\left( {\mathop x\nolimits_2 - \frac{{5.1}}{{4{\pi ^2}}}\mathop x\nolimits_1^2 + \frac{5}{\pi }\mathop x\nolimits_1 - 6} \right)^2} + 10\left( {1 - \frac{1}{{8\pi }}} \right)\cos \mathop x\nolimits_1 + 10\\
\quad\\ \quad - 5 \le \mathop x\nolimits_1 \le 10 \quad \quad 0 \le \mathop x\nolimits_1 \le 15\\
{x_{\min }} = \left( { - 3.142,12.275} \right),(3.142,2.275),(9.425,2.425)\\
\min (f) = 0.398.
$$
### Goldstein-Price

$$
f(x) = \left[ {1{\rm{ + }}{{\left( {\mathop x\nolimits_1 + \mathop x\nolimits_2 + 1} \right)}^2}\left( {19 - 14\mathop x\nolimits_1 + 3\mathop x\nolimits_1^2 - 14\mathop x\nolimits_2 + 6\mathop x\nolimits_1 \mathop x\nolimits_2 + 3\mathop x\nolimits_2^2 } \right)} \right]\\
\times \left[ {30 + {{\left( {2\mathop x\nolimits_1 - 3\mathop x\nolimits_2 } \right)}^2} \times \left( {18 - 32\mathop x\nolimits_1 + 12\mathop x\nolimits_1^2 + 48\mathop x\nolimits_2 - 36\mathop x\nolimits_1 \mathop x\nolimits_2 + 27\mathop x\nolimits_2^2 } \right)} \right]\\ - 2 \le \mathop x\nolimits_i \le 2,\quad \min (f) = f(0, - 1) = 3.
$$
### Hartman Family
### Shekel Family