Table 3 Test functions for multimodal, unimodal, and multimodal with fixed dimensions.

From: Improved optimization based on parrot’s chaotic optimizer for solving complex problems in engineering and medical image segmentation

 

Functions

Descriptions

Dimensions

Range

Unimodal functions

F1

\(f(x)=\sum\nolimits_{{i=1}}^{n} {x_{i}^{2}}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 100,100]\)

F2

\(f(x)=\sum\nolimits_{{i=1}}^{n} {\left| {{x_i}} \right|} +\prod\nolimits_{{i=0}}^{n} {\left| {{x_i}} \right|}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 10,10]\)

F3

\(f(x)={\sum\nolimits_{{i=1}}^{d} {\left( {\sum\nolimits_{{j=1}}^{i} {{x_j}} } \right)} ^2}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 100,100]\)

F4

\(f(x)={\hbox{max} _i}\left\{ {\left| {{x_i}} \right|,\,\,1 \leqslant i \leqslant n} \right\}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 100,100]\)

F5

\(f(x)=\sum\nolimits_{{i=1}}^{{n - 1}} {\left[ {100{{\left( {x_{i}^{2} - {x_{i+1}}} \right)}^2}+{{\left( {1 - {x_i}} \right)}^2}} \right]}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 30,30]\)

F6

\(f(x)={\sum\nolimits_{{i=1}}^{n} {\left( {{x_i}+0.5} \right)} ^2}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 100,100]\)

F7

\(f(x)=\sum\nolimits_{{i=0}}^{n} {ix_{i}^{4}+random[0,1)}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 128,128]\)

Multimodal functions

F8

\(f(x)=\sum\nolimits_{{i=1}}^{n} {({x_i}} \sin (\sqrt {\left| {{x_i}} \right|} ))\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 500,500]\)

F9

\(f(x)=\sum\nolimits_{{i=1}}^{n} [ x_{i}^{2} - 10\cos (2\pi {x_i})+10]\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 5.12,\,\,5.12]\)

F10

\(\begin{gathered} f(x)=20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{{i=1}}^{n} {x_{i}^{2}} } } \right) - \hfill \\ \exp \left( {\frac{1}{n}\sum\nolimits_{{i=1}}^{n} {\cos \left( {2\pi {x_i}} \right)} } \right)+20+e \hfill \\ \end{gathered}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 32,32]\)

F11

\(f(x)=1+\frac{1}{{4000}}\sum\nolimits_{{i=1}}^{n} {x_{i}^{n} - \prod\nolimits_{{i=1}}^{n} {\cos \left( {\frac{{{x_i}}}{{\sqrt i }}} \right)} }\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 600,600]\)

F12

\(\begin{gathered} f(x) = \frac{\pi }{n}\left[ {10\sin (\pi y_{1} )} \right] \hfill \\ + \sum\nolimits_{{i = 1}}^{n} {(y_{i} - 1)^{2} \left[ {1 + 10\sin ^{2} \left( {\pi y_{{i + 1}} } \right)} \right.} \hfill \\ \left. { + \sum\nolimits_{{i = 1}}^{n} {u(x_{i} ,10,100,4)} } \right] \hfill \\ \end{gathered}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 50,50]\)

F13

\(\begin{gathered} f(x) = 0.1\left( {\sin ^{2} \left( {3\pi x_{1} } \right)} \right. \hfill \\ + \sum\nolimits_{{i = 1}}^{n} {(x_{i} - 1)^{2} \left[ {1 + \sin ^{2} \left( {3\pi x_{i} + 1} \right)} \right]} \hfill \\ + \left( {x_{n} - 1} \right)^{2} \left( {1 + \sin ^{2} \left( {2\pi x_{n} } \right)} \right) \hfill \\ \left. { + \sum\nolimits_{{i = 1}}^{n} {u\left( {x_{i} ,5,100,4} \right)} } \right) \hfill \\ \end{gathered}\)

\(30,\,\,100,\,\,500,\,\,1000\)

\([ - 50,50]\)

Multimodal functions with a fixed dimension

F14

\(f(x)={\left( {\frac{1}{{500}}+\sum\nolimits_{{j=1}}^{{25}} {\frac{1}{{j+\sum\nolimits_{{i=1}}^{2} {({x_i} - {a_{i,j}})} }}} } \right)^{ - 1}}\)

2

\([ - 65,65]\)

F15

\(f(x)={\sum\nolimits_{{i=1}}^{{11}} {\left[ {{a_i} - \frac{{{x_1}\left( {b_{i}^{2}+{b_i}{x_2}} \right)}}{{b_{i}^{2}+{b_i}{x_3}+{x_4}}}} \right]} ^2}\)

4

\([ - 5,5]\)

F16

\(f(x)=4x_{1}^{2} - 2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+{x_1}{x_2} - 4x_{2}^{2}+4x_{2}^{4}\)

2

\([ - 5,5]\)

F17

\(\begin{gathered} f(x) = \left( {x_{2} - \frac{{5.1}}{{4\pi ^{2} }}x_{1}^{2} + \frac{5}{4}x_{1} - 6} \right)^{2} \hfill \\ + 10\left( {1 - \frac{1}{{8\pi }}} \right)\cos x_{1} + 10 \hfill \\ \end{gathered}\)

2

\([ - 5,5]\)

F18

\(\begin{gathered} f(x) = \left[ {1 + \left( {x_{1} + x_{2} + 1} \right)^{2} } \right. \hfill \\ \left. {\left( {19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} } \right)} \right] \hfill \\ \times \left[ {30 + \left( {2x_{1} - 3x_{2} } \right)^{2} } \right. \hfill \\ \left. { \times \left( {18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} } \right)} \right] \hfill \\ \end{gathered}\)

2

\([ - 2,2]\)

F19

\(f(x)= - \sum\nolimits_{{i=1}}^{4} {{c_i}\exp \left( { - \sum\nolimits_{{i=1}}^{3} {{a_{ij}}{{\left( {{x_{ij}} - {p_{ij}}} \right)}^2}} } \right)}\)

3

\([ - 1,2]\)

F20

\(f(x)= - \sum\nolimits_{{i=1}}^{4} {{c_i}\exp \left( { - \sum\nolimits_{{i=1}}^{6} {{a_{ij}}{{\left( {{x_{ij}} - {p_{ij}}} \right)}^2}} } \right)}\)

6

\([0,1]\)

F21

\(f(x)= - {\sum\nolimits_{{i=1}}^{5} {\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^T}+{c_i}} \right]} ^{ - 1}}\)

4

\([0,1]\)

F22

\(f(x)= - {\sum\nolimits_{{i=1}}^{7} {\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^T}+{c_i}} \right]} ^{ - 1}}\)

4

\([0,1]\)

F23

\(f(x)= - {\sum\nolimits_{{i=1}}^{{10}} {\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^T}+{c_i}} \right]} ^{ - 1}}\)

4

\([0,1]\)