Table 1 Detailed information on the 12 benchmarks
From: A modified algorithm for computation issues in UAV-enabled wireless communications
F | Formula | Range | Optimal value |
|---|---|---|---|
f1 | \(\sum \limits _{i = 1}^{n} {{x_{i}}^{2}} \) | [ − 100, 100] | 0 |
f2 | \( {10^{6}}*{x_{1}}^{2} + \sum \limits _{i = 2}^{n} {{x_{i}}^{2}} \) | [ − 100, 100] | 0 |
f3 | \(\sum \limits _{i = 1}^{n} {i*{x_{i}}^{2}} \) | [ −100, 100] | 0 |
f4 | \( \sum \limits _{i = 1}^{n} {{{\left \lfloor {{x_{i}} + 0.5} \right \rfloor }^{2}}} \) | [ −100, 100] | 0 |
f5 | \( \sum \limits _{i = 1}^{n} {{x_{i}}^{2}} + \prod \limits _{i = 1}^{n} {{x_{i}}^{2}} \) | [ − 100, 100] | 0 |
f6 | \( \sum \limits _{i = 1}^{n} {\left [ {{x_{i}}^{2} - 10\cos (2\pi {x_{i}}) + 10} \right ]} \) | [ − 5.12, 5.12] | 0 |
f7 | \( \frac {1}{{4000}}\sum \limits _{i = 1}^{n} {{x_{i}}^{2}} - \prod \limits _{i = 1}^{n} {\cos (\frac {{{x_{i}}}}{{\sqrt i }})} + 1 \) | [ − 600, 600] | 0 |
f8 | \( \begin {aligned} & - 20\exp (- 0.2\sqrt {\frac {1}{n}\sum \limits _{i = 1}^{n} {{x_{i}}^{2}}}) \\ & + \exp (\frac {1}{n}\sum \limits _{i = 1}^{n} {\cos (2\pi {x_{i}})}) + 20 + {\mathrm {e}} \\ \end {aligned} \) | [ − 32, 32] | 0 |
f9 | \( \begin {aligned} &\frac {\pi }{{30}}\left \{ \begin {aligned} 10 \cdot \sin \left [ {1 + 0.25{{({x_{1}} + 1)}^{2}}} \right ] + \\ \sum \limits _{i = 1}^{n - 1} {\left \{ \begin {aligned} {\left [ {0.25\left ({{x_{i}} + 1} \right)} \right ]^{2}} \\ \left [ {1 + 10\left ({\sin {{\left ({\pi \left ({1 + 0.25\left ({{x_{i + 1}} + 1} \right)} \right)} \right)}^{2}}} \right)} \right ] \\ \end {aligned} \right \}} \\ \end {aligned} \right \} \\ & + \sum \limits _{i = 1}^{n} {{y_{i}}} \\ \end {aligned} \)\(\begin {array}{l} {y_{i}} = \left \{ {\begin {array}{*{20}{c}} {100{{({x_{i}} - 10)}^{4}}}&{{x_{i}} > 10}&{}\\ 0&{ - 10 \le {x_{i}} \le 10}&{}\\ {100{{(- {x_{i}} - 10)}^{4}}}&{{x_{i}} < - 10}&{} \end {array}} \right. \end {array}\) | [ − 50, 50] | 0 |
f10 | \( 0.1\left \{ \begin {aligned} {\sin ^{2}}(3\pi {x_{1}}) + \\ \sum \limits _{i = 1}^{n - 1} {{{({x_{i}} - 1)}^{2}}} [1 + {\sin ^{2}}(3\pi {x_{i + 1}})] \\ + ({x_{n}} - 1)[1 + {\sin ^{2}}(2\pi {x_{n}})] \\ \end {aligned} \right \} + \sum \limits _{i = 1}^{n} {{y_{i}}} \)\(\begin {array}{l} {y_{i}} = \left \{ {\begin {array}{*{20}{c}} {100{{({x_{i}} - 10)}^{4}}}&{{x_{i}} > 5}&{}\\ 0&{ - 5 \le {x_{i}} \le 5}&{}\\ {100{{(- {x_{i}} - 10)}^{4}}}&{{x_{i}} < - 5}&{} \end {array}} \right. \end {array}\) | [ − 50, 50] | 0 |
f11 | \(\begin {aligned} {\left ({{1}{\text {.5}} - {x_{1}} + {x_{1}}{x_{2}}} \right)^{2}} + {\left ({2.25 - {x_{1}} + {x_{1}}{x_{2}}^{2}} \right)^{2}} \\ + {\left ({2.625 - {x_{1}} + {x_{1}}{x_{2}}^{3}} \right)^{2}} \\ \end {aligned} \) | [ − 4.5, 4.5] | 0 |
f12 | (x12+x22)0.25〈{sin[50(3x12+x22)0.1]}2+1〉 | [ − 100, 1000] | 0 |