Table 3 Benchmark test functions.

From: Multi-strategy hybrid adaptive dung beetle optimization for UAV photogrammetric 3D path planning under complex constraints

Test function

Dim

Domain

Fit

\({{\varvec{F}}}_{1}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{{\varvec{x}}}_{{\varvec{i}}}^{2}\)

30

[−100,100]

0

\({{\varvec{F}}}_{2}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}\left|{{\varvec{x}}}_{{\varvec{i}}}\right|+{\prod }_{{\varvec{i}}=1}^{{\varvec{D}}}\left|{{\varvec{x}}}_{{\varvec{i}}}\right|\)

30

[−10,10]

0

\({{\varvec{F}}}_{3}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}({{\sum }_{{\varvec{j}}=1}^{{\varvec{D}}}{{\varvec{x}}}_{{\varvec{j}}})}^{2}\)

30

[−100,100]

0

\({{\varvec{F}}}_{4}\left({\varvec{x}}\right)={{\varvec{max}}}_{{\varvec{i}}}\left\{\left|{{\varvec{x}}}_{{\varvec{i}}}\right|1\le {\varvec{i}}\le {\varvec{n}}\right\}\)

30

[−100,100]

0

\({{\varvec{F}}}_{5}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}-1}{{[100({{\varvec{x}}}_{{\varvec{i}}+1}-{\varvec{x}}}_{{\varvec{i}}}^{2})}^{2}+{\left({{\varvec{x}}}_{{\varvec{i}}}-1\right)}^{2}\)

30

[−30,30]

0

\({{\varvec{F}}}_{6}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{\left(\left|{{\varvec{x}}}_{{\varvec{i}}}+0.5\right|\right)}^{2}\)

30

[−100,100]

0

\({{\varvec{F}}}_{7}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{\varvec{i}}{{\varvec{x}}}_{{\varvec{i}}}^{4}+{\varvec{r}}{\varvec{a}}{\varvec{n}}{\varvec{d}}{\varvec{o}}{\varvec{m}}\left(\text{0,1}\right)\)

30

[−1.28,1.28]

0

\({{\varvec{F}}}_{8}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}-{{\varvec{x}}}_{{\varvec{i}}}{\varvec{s}}{\varvec{i}}{\varvec{n}}\left(\sqrt{\left|{{\varvec{x}}}_{{\varvec{i}}}\right|}\right)\)

30

[−500,500]

−12,569.5

\({{\varvec{F}}}_{9}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}\left[{{\varvec{x}}}_{{\varvec{i}}}^{2}-10{\varvec{c}}{\varvec{o}}{\varvec{s}}\left(2{\varvec{\pi}}{{\varvec{x}}}_{{\varvec{i}}}\right)+10\right]\)

30

[−5.12,5.12]

0

\({{\varvec{F}}}_{10}\left({\varvec{x}}\right)=-20{\varvec{exp}}\left(-0.2\sqrt{\frac{1}{{\varvec{D}}}{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{{\varvec{x}}}_{{\varvec{i}}}^{2}}\right)-{\varvec{e}}{\varvec{x}}{\varvec{p}}\left(\frac{1}{{\varvec{D}}}{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{\varvec{c}}{\varvec{o}}{\varvec{s}}\left(2{\varvec{\pi}}{{\varvec{x}}}_{{\varvec{i}}}\right)\right)+20+{\varvec{e}}\)

30

[−32,32]

0

\({{\varvec{F}}}_{11}\left({\varvec{x}}\right)=\frac{1}{4000}{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{{\varvec{x}}}_{{\varvec{i}}}^{2}-{\prod }_{{\varvec{i}}=1}^{{\varvec{D}}}{\varvec{c}}{\varvec{o}}{\varvec{s}}\left(\frac{{{\varvec{x}}}_{{\varvec{i}}}}{\sqrt{{\varvec{i}}}}\right)+1\)

30

[−600,600]

0

\({{\varvec{F}}}_{12}\left({\varvec{x}}\right)=\frac{{\varvec{\pi}}}{{\varvec{D}}}\left\{10{{\varvec{s}}{\varvec{i}}{\varvec{n}}}^{2}\left({\varvec{\pi}}{{\varvec{y}}}_{{\varvec{i}}}\right)+\sum_{{\varvec{i}}=1}^{{\varvec{D}}-1}{\left({{\varvec{y}}}_{{\varvec{i}}}-1\right)}^{2}\left[1+10{{\varvec{s}}{\varvec{i}}{\varvec{n}}}^{2}\left({\varvec{\pi}}{{\varvec{y}}}_{{\varvec{i}}+1}\right)\right]\right.\left.+{\left({{\varvec{y}}}_{{\varvec{D}}}-1\right)}^{2}\right\}+{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}}{\varvec{u}}\left({{\varvec{x}}}_{{\varvec{i}}},\text{10,100,4}\right)\)

\({{\varvec{y}}}_{{\varvec{i}}}=1+\frac{1}{4}\left({{\varvec{x}}}_{{\varvec{i}}}+1\right)\)

\({\varvec{u}}\left({{\varvec{x}}}_{{\varvec{i}}},{\varvec{a}},{\varvec{k}},{\varvec{m}}\right)=\left\{\begin{array}{ll}{\varvec{k}}{\left({{\varvec{x}}}_{{\varvec{i}}}-{\varvec{a}}\right)}^{{\varvec{m}}},& {{\varvec{x}}}_{{\varvec{i}}}>{\varvec{a}},\\ 0,& -{\varvec{a}}\le {{\varvec{x}}}_{{\varvec{i}}}\le {\varvec{a}},\\ {\varvec{k}}{\left(-{{\varvec{x}}}_{{\varvec{i}}}-{\varvec{a}}\right)}^{{\varvec{m}}},& {{\varvec{x}}}_{{\varvec{i}}}<-{\varvec{a}}\end{array}\right.\)

30

[−50,50]

0

\({{\varvec{F}}}_{13}\left({\varvec{x}}\right)=0.1\left\{{{\varvec{s}}{\varvec{i}}{\varvec{n}}}^{2}\left(3{\varvec{\pi}}{{\varvec{x}}}_{1}\right)+{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}-1}{({\varvec{x}}}_{{\varvec{i}}}-1{)}^{2}\left[1+{\varvec{s}}{\varvec{i}}{{\varvec{n}}}^{2}\left(3{\varvec{\pi}}{{\varvec{x}}}_{{\varvec{i}}+1}\right)\right]\left.+{\left({{\varvec{x}}}_{{\varvec{D}}}-1\right)}^{2}\left[1+{{\varvec{s}}{\varvec{i}}{\varvec{n}}}^{2}\left(2{\varvec{\pi}}{{\varvec{x}}}_{{\varvec{D}}}\right)\right]\right\}+{\sum }_{{\varvec{i}}=1}^{{\varvec{D}}-1}{\varvec{u}}\left({{\varvec{x}}}_{{\varvec{i}}},\text{5,100,4}\right)\right.\)

30

[−500,500]

0

\({{\varvec{F}}}_{14}\left({\varvec{x}}\right)={\left[\begin{array}{c}\frac{1}{500}+\sum_{{\varvec{j}}=1}^{25}\frac{1}{{\varvec{j}}+\sum_{{\varvec{i}}=1}^{2}({{\varvec{x}}}_{{\varvec{i}}}-{{\varvec{a}}}_{{\varvec{i}}{\varvec{j}}}{)}^{6}}\end{array}\right]}^{-1}\)

2

[−65.536, 65.536]

1

\(\begin{array}{l}{{\varvec{F}}}_{15}\left({\varvec{x}}\right)={\sum }_{{\varvec{i}}=1}^{11}{\left[{{\varvec{a}}}_{{\varvec{i}}}-\frac{{{\varvec{x}}}_{1}\left({{\varvec{b}}}_{{\varvec{i}}}^{2}+{{\varvec{b}}}_{{\varvec{i}}}{{\varvec{x}}}_{2}\right)}{{{\varvec{b}}}_{{\varvec{i}}}^{2}+{{\varvec{b}}}_{{\varvec{i}}}{{\varvec{x}}}_{3}+{{\varvec{x}}}_{4}}\right]}^{2}\end{array}\)

4

[−5,5]

0.0003075

\({{\varvec{F}}}_{16}\left({\varvec{x}}\right)={\left({{\varvec{x}}}_{2}-\frac{5.1}{4{{\varvec{\pi}}}^{2}}{{\varvec{x}}}_{1}^{2}+\frac{5}{{\varvec{\pi}}}{{\varvec{x}}}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8{\varvec{\pi}}}\right){\varvec{c}}{\varvec{o}}{\varvec{s}}{{\varvec{x}}}_{1}+10\)

2

[−5,10] x [0,15]

0.398

\({{\varvec{F}}}_{17}\left({\varvec{x}}\right)=-{\sum }_{{\varvec{j}}=1}^{4}{{\varvec{c}}}_{{\varvec{i}}{\varvec{j}}}{\varvec{e}}{\varvec{x}}{\varvec{p}}\left[-{\sum }_{{\varvec{j}}=1}^{4}{{\varvec{a}}}_{{\varvec{i}}{\varvec{j}}}{\left({{\varvec{x}}}_{{\varvec{j}}}-{{\varvec{p}}}_{{\varvec{i}}{\varvec{j}}}\right)}^{2}\right]\)

4

[0,1]

−3.86

\({{\varvec{F}}}_{18}\left({\varvec{x}}\right)=-{\sum }_{{\varvec{j}}=1}^{4}{{\varvec{c}}}_{{\varvec{i}}{\varvec{j}}}{\varvec{e}}{\varvec{x}}{\varvec{p}}\left[-{\sum }_{{\varvec{j}}=1}^{6}{{\varvec{a}}}_{{\varvec{i}}{\varvec{j}}}{\left({{\varvec{x}}}_{{\varvec{j}}}-{{\varvec{p}}}_{{\varvec{i}}{\varvec{j}}}\right)}^{2}\right]\)

6

[0,1]

−3.32

\({{\varvec{F}}}_{19}\left({\varvec{x}}\right)=-{\sum }_{{\varvec{i}}=1}^{5}[\left({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}\right)({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}{)}^{{\varvec{T}}}+{{\varvec{c}}}_{{\varvec{i}}}{]}^{-1}\)

4

[0,10]

−10

\({{\varvec{F}}}_{20}\left({\varvec{x}}\right)=-{\sum }_{{\varvec{i}}=1}^{10}[\left({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}\right)({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}{)}^{{\varvec{T}}}+{{\varvec{c}}}_{{\varvec{i}}}{]}^{-1}\)

4

[0,10]

−10

\({{\varvec{F}}}_{21}\left({\varvec{x}}\right)=-{\sum }_{{\varvec{i}}=1}^{7}{\left[\left({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}\right){\left({\varvec{x}}-{{\varvec{a}}}_{{\varvec{i}}}\right)}^{{\varvec{T}}}+{{\varvec{c}}}_{{\varvec{i}}}\right]}^{-1}\)

4

[0,10]

−10

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