Table 2 Basic unimodal functions F1–F8.

\(F1 = \sum\limits_{i = 1}^{n} {ix_{i}^{4} } + random[0,1)\)

30

[-128,128]

0

\(F2 = - \cos \left( {x_{1} } \right)\cos \left( {x_{2} } \right)e^{{ - \left( {x_{1} - \pi } \right)^{2} - \left( {x_{2} - \pi } \right)^{2} }}\)

2

[-100,100]

−1

\(F3 = \sum\nolimits_{i = 1}^{n - 1} {\left[ {\left( {100\left( {x_{i + 1} - x_{i} } \right)^{2} } \right) + \left( {x_{i} - 1} \right)^{2} } \right]}\)

30

[-30,30]

0

\(F4 = \left( {x_{i} - 1} \right)^{2} + \sum\nolimits_{i = 2}^{n} {i\left( {2x_{i}^{2} - x_{i - 1} } \right)^{2} }\)

30

[-10,10]

0

\(F5 = x_{i}^{2} + 2x_{2}^{2} - 0.3\cos \left( {3\pi x_{1} } \right)\left( {4\pi x_{3} } \right) + 0.3\)

2

[-100,100]

0

\(F6 = \sum\nolimits_{k = 1}^{n} {\left[ {\sum\nolimits_{i = 1}^{n} {\left( {i^{k} + \beta } \right)\left( {\left( {\frac{{x_{i} }}{i}} \right)^{k} - 1} \right)} } \right]^{2} }\)

4

[-4,4]

0

\(F7 = \sum\nolimits_{k = 1}^{n} {\left[ {\left( {\sum\nolimits_{i = 1}^{n} {x_{i}^{k} } } \right) - b_{k} } \right]^{2} }\)

4

[0,4]

0

\(F8 = - \sum\nolimits_{i = 1}^{4} {\exp \left[ { - \sum\nolimits_{j = 1}^{6} {a_{ij} \left( {x_{j} - p_{ij} } \right)^{2} } } \right]}\)

6

[0,1]

−3.32