Table 5 Benchmark functions for our proposed algorithms.

\(f_{1} \left( {x,y} \right) = x^{2} + y^{2}\)

FOPID-based HESPSOALO

2

20

[− 5, 5]

\(f_{2} \left( {x,y} \right) = 20 + \left( {x^{2} - 10 \cdot \cos \left( {2\pi x} \right)} \right) + \left( {y^{2} - 10 \cdot \cos \left( {2\pi y} \right)} \right)\)

2

20

[− 5, 5]

\(\begin{aligned} f_{3} \left( {x,y} \right) & = - 20\exp \left( { - 0.2\sqrt {0.5\left( {x^{2} + y^{2} } \right)} } \right) \\ & \quad - \exp \left( {0.5\left( {\cos \left( {2\pi x} \right) + \cos \left( {2\pi y} \right)} \right)} \right) + 20 + e \\ \end{aligned}\)

2

20

[− 5, 5]

\(f_{4} \left( {x,y} \right) = 1 + \frac{{x^{2} + y^{2} }}{4000} - \cos \left( x \right) \cdot \cos \left( {\frac{y}{\sqrt 2 }} \right) + 20 + e\)

FOPID-based HPSOGWO

2

20

[− 5, 5]

\(\begin{aligned} f_{5} \left( {x,y} \right) & = \sin ^{2} \left( {3\pi x} \right) + (x - 1)^{2} \left( {1 + \sin ^{2} \left( {3\pi y} \right)} \right) \\ & \quad + (y - 1)^{2} \left( {1 + \sin ^{2} \left( {2\pi y} \right)} \right) \\ \end{aligned}\)

FOPID-based HGWOALO

2

20

[− 10, 10]

\(\begin{aligned} f_{6} \left( {x,y} \right) & = (1.5 - x + xy)^{2} + (2.25 - x + xy^{2} )^{2} \\ & \quad + (2.625 - x + xy^{3} )^{2} \\ \end{aligned}\)

2

20

[− 5, 5]

\(f_{7} \left( {x,y} \right) = (x + 2y - 7)^{2} + (2x + y - 5)^{2}\)

2

20

[-5, 5]

\(f_{8} \left( {x,y} \right) = (x^{2} + y - 11)^{2} + (x + y^{2} - 7)^{2}\)

General (All algorithms)

2

20

[-5, 5]

\(f_{9} \left( {x,y} \right) = 2x^{2} - 1.05x^{4} + \frac{{x^{6} }}{6} + xy + y^{2}\)

2

20

[-5, 5]