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. 2021 Dec 20;23(12):1700. doi: 10.3390/e23121700

Table 3.

Fixed-dimension multimodal benchmark functions.

Function	Dim	Range	f_min
$F_{14} (x) = {(\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} {(x_{i} - a_{i j})}^{6}})}^{- 1}$	2	[−65,65]	0.998
$F_{15} (x) = \sum_{i = 1}^{11} {[a_{i} - \frac{x_{1} (b_{i}^{2} + b_{i} x_{2})}{b_{i}^{2} + b_{i} x_{3} + x_{4}}]}^{2}$	4	[−5,5]	0.00030
$F_{16} (x) = 4 x_{1}^{2} - 2.1 x_{1}^{4} + \frac{1}{3} x_{1}^{6} + x_{1} x_{2} - 4 x_{2}^{2} + x_{2}^{4}$	2	[−5,5]	−1.0316
$F_{17} (x) = {(x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)}^{2} + 10 (1 - \frac{1}{8 π}) \cos x_{1} + 10$	2	[−5,5]	0.398
$\begin{array}{l} F_{18} (x) = [1 + {(x_{1} + x_{2} + 1)}^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})] \\ \times [30 + {(2 x_{1} - 3 x_{2})}^{2} \times (18 - 32 x_{2} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})] \end{array}$	2	[−2,2]	3
$F_{19} (x) = - \sum_{i = 1}^{4} c_{i} \exp (- \sum_{j = 1}^{3} a_{i j} {(x_{j} - p_{i j})}^{2})$	3	[−1,2]	−3.86
$F_{20} (x) = - \sum_{i = 1}^{4} c_{i} \exp (- \sum_{j = 1}^{6} a_{i j} {(x_{j} - p_{i j})}^{2})$	6	[0,1]	−3.32
$F_{21} (x) = - {\sum_{i = 1}^{5} [(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	4	[0,10]	−10.1532
$F_{22} (x) = - {\sum_{i = 1}^{7} [(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	4	[0,10]	−10.4028
$F_{23} (x) = - {\sum_{i = 1}^{10} [(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	4	[0,10]	−10.5363