Non-linear Continuous Multi-Extremal Optimization

$\displaystyle S(\mathbf{x}) = \sum_{i=1}^{n-1} 100 (x_{i+1} - x_i^2)^2 + (x_i-1)^2$

(1)

$\displaystyle S(\mathbf{x}) = 1 + \sum_{i=1}^n 8\sin^2(\eta(x_i-x_i^*)^2) + 6\sin^2(2\eta(x_i-x_i^*)^2)+\mu(x_i-x_i^*)^2 \;.$

(2)

Paviani's function

$\displaystyle \sum_{i=1}^n \bigg(\ln ^2(x_i-2) + \ln ^2(10-x_i)\bigg) - \left( \prod_{i=1}^n x_i\right)^{0.2}.$

(3)

Sphere model, first De Jong's function

Third De Jong's function

$\displaystyle \sum_{i=1}^{n}\vert x_i\vert , \quad -2.048\le x_i\le 2.048$

(5)

Fourth De Jong's function, Quartic function with noise, Quartic Gaussian function

$\displaystyle \sum_{i=1}^{n}ix_{i}^4 ,\quad -1.28\leqslant x_i \leqslant 1.28$

(6)

Rastrigin's function

$\displaystyle n*10 + \sum_{i=1}^{n}(x_i^2-10\cos(2\pi x_i)) , \quad -5.12\leqslant x_i \leqslant 5.12$

(7)

Schwefel's function

$\displaystyle \sum_{i=1}^{n}-x_i\sin(\sqrt{\vert x_i\vert}) , \quad -512\leqslant x_i \leqslant 512$

(8)

Griewangk's function

$\displaystyle -\prod_{i=1}^{n}\cos\left(\frac{x_i}{\sqrt{i}}\right)+\sum_{i=1}^{n}\frac{x_i^2}{4000}+1 , \quad -600\leqslant x_i \leqslant 600$

(9)

Sine envelope sine wave function

$\displaystyle -\sum_{i=1}^{n-1}\left(\frac{\sin^2(\sqrt{x_{i+1}^2+x_{i}^2}-0.5)... ...01(x_{i+1}^2+x_{i}^2)+1)^2}+0.5\right) ,\quad -100\leqslant x_i \leqslant 100$

(10)

Stretch V sine wave function (Ackley)

$\displaystyle \sum_{i=1}^{n-1}(x_{i+1}^2+x_{i}^2)^{0.25}(\sin^2(50(x_{i+1}^2+x_{i}^2)^{0.1})+1) , \quad -10\leqslant x_i \leqslant 10$

(11)

Test function (Ackley)

$\displaystyle \sum_{i=1}^{n-1}\left(3(\cos(2x_{i})+\sin(2x_{i+1}))+\frac{\sqrt{x_{i+1}^2+x_{i}^2}}{e^{0.2}}\right) ,\quad -30\leqslant x_i \leqslant 30$

(12)

Ackley's function

$\displaystyle \sum_{i=1}^{n-1}\left(20+e^{-20}e^{-0.2\sqrt{0.5(x_{i+1}^2+x_{i}^... ...cos(2\pi x_{i+1})+\cos(2\pi x_i))}\right) ,\quad -30\leqslant x_i \leqslant 30$

(13)

Egg Holder

$\begin{displaymath}\begin{split}& \sum_{i=1}^{n-1}\left(-(x_{i+1}+47)\sin\left(\... ...-x_{i})\right) \ & -512\leqslant x_i \leqslant 512 \end{split}\end{displaymath}$

(14)

Rana's function

$\begin{displaymath}\begin{split}&\sum_{i=1}^{n-1}\Big((x_{i+1}+1)\cos\left(\sqrt... ...rt}\right)\Big) , -500\leqslant x_i \leqslant 500 \end{split}\end{displaymath}$

(15)

Pathological test function

$\displaystyle \sum_{i=1}^{n-1}\left(\frac{\sin^2\left(\sqrt{x_{i+1}^2+100x_{i}^... ...{i+1}x_{i}+x_{i}^2)^2+1.0}+0.5\right) , \quad -100\leqslant x_i \leqslant 100$

(16)

Michalewicz's function

$\displaystyle \sum_{i=1}^{n-1}\left(\sin(x_{i+1})\sin^{20}\left(\frac{2x_{i+1}^... ...left(\frac{x_{i+1}^2}{\pi}\right)\right) , \quad 0\leqslant x_i \leqslant \pi$

(17)

Master's cosine wave function

$\displaystyle -\sum_{i=1}^{n-1}e^{-\frac{1}{8}\left(x_{i+1}^2+0.5x_{i}x_{i+1}+x... ...x_{i+1}^2+0.5x_{i}x_{i+1}+x_{i}^2}\right) , \quad -5\leqslant x_i \leqslant 5$

(18)

Keane's function

$\displaystyle \left\vert\frac{ \left(\sum_{i=1}^{n}\cos^4\left(x_{i}\right)-2\p... ...{n}\cos^2\left(x_{i}\right)\right)}{ \sqrt{\sum_{i=1}^{n}ix_{i}^2}} \right\vert$

(19)

Constraints: $\prod_{i=1}^{n}x_{i}\ge0.75, \sum_{i=1}^{n}x_{i}\leqslant 7.5 n, 0\le x_{i} \le 10$ .

Hougen function (non-linear regression)

rate $\displaystyle = \frac{\beta_1 x_2 - x_3/\beta_5}{ 1 + \beta_2 x_1 + \beta_3 x_2 + \beta_4 x_3},$

Table 1: Input data for the Hougen function

x(1)	x(2)	x(3)	rate
470	300	10	8.55
285	80	10	3.79
470	300	120	4.82
470	80	120	0.02
470	80	10	2.75
100	190	10	14.39
100	80	65	2.54
470	190	65	4.35
100	300	54	13.00
100	300	120	8.50
100	80	120	0.05
285	300	10	11.32
285	190	120	3.13