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Benchmarking and comparison of nature-inspired population-based continuous optimisation algorithms

  • Methodologies and Application
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Abstract

This paper describes an experimental investigation into four nature-inspired population-based continuous optimisation methods: the Bees Algorithm, Evolutionary Algorithms, Particle Swarm Optimisation, and the Artificial Bee Colony algorithm. The aim of the proposed study is to understand and compare the specific capabilities of each optimisation algorithm. For each algorithm, thirty-two configurations covering different combinations of operators and learning parameters were examined. In order to evaluate the optimisation procedures, twenty-five function minimisation benchmarks were designed by the authors. The proposed set of benchmarks includes many diverse fitness landscapes, and constitutes a contribution to the systematic study of optimisation techniques and operators. The experimental results highlight the strengths and weaknesses of the algorithms and configurations tested. The existence and extent of origin and alignment search biases related to the use of different recombination operators are highlighted. The analysis of the results reveals interesting regularities that help to identify some of the crucial issues in the choice and configuration of the search algorithms.

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Birmingham, Birmingham, B15 2TT, UK

    D. T. Pham

  2. Department of Biology, University of Bergen, Postboks 7803, 5020 , Bergen, Norway

    M. Castellani

Authors
  1. D. T. Pham
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  2. M. Castellani
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Corresponding author

Correspondence to M. Castellani.

Additional information

Communicated by R. John.

Appendices

Appendix A: Optimisation benchmarks

1.1 General features

All functions are defined within the interval:

\(I =\{x\in \mathrm{R}^\mathrm{n}; -100<x_{i}<100, i=1,2,\ldots ,n\}.\)

All but functions 22, 23, and 24 are two-dimensional.

For all functions, \(f(x^{\mu })=0\) where \(x^{\mu }=(x^{\mu }_{1},\ldots ,x^{\mu }_{n})\) is the global minimum point.

1.2 Multimodal functions

1.2.1 Function 1

This function maps two “holes”. The global optimum is located at the origin of the search space.

$$\begin{aligned} f( {x_1 ,x_2 })&= 1.0-0.8\cdot e^{-\left( {\frac{^{\left( {( {x_1 -50})^2+( {x_2 -50})^2}\right) }}{200}}\right) }\\&-1.0\cdot e^{-\left( {\frac{^{\left( {( {x_1 -0})^2+( {x_2 -0})^2}\right) }}{200}}\right) }\\ x^\mu&= (0,0) \end{aligned}$$

1.2.2 Function 2

This function maps two “holes”. The global optimum is located far from the origin of the search space.

$$\begin{aligned} f( {x_1 ,x_2 })&= 1.0-1.0\cdot e^{-\left( {\frac{^{\left( {( {x_1 -50})^2+( {x_2 -50})^2}\right) }}{200}}\right) }\\&-0.8\cdot e^{-\left( {\frac{^{\left( {( {x_1 -0})^2+( {x_2 -0})^2}\right) }}{200}}\right) }\\&x^\mu =(50,50) \end{aligned}$$

1.2.3 Function 3

This function maps nine basins of equal size arranged on a grid. The global minimum corresponds to one of the lateral basins of the grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.8\cdot e^{-\left( {\frac{^{( {x_1 -60})^2+( {x_2 -60})^2}}{100}}\right) }\\&\quad -1.0\cdot e^{-\left( {\frac{^{( {x_1 -0})^2+( {x_2 -60})^2}}{100}}\right) }-0.8\cdot e^{-\left( {\frac{^{( {x_1 +60})^2+( {x_2 -60})^2}}{100}}\right) } \\&\quad -0.8\cdot e^{-\left( {\frac{^{( {x_1 -60})^2+( {x_2 -0})^2}}{100}}\right) }-0.8\cdot e^{-\left( {\frac{^{( {x_1 -0})^2+( {x_2 -0})^2}}{100}}\right) }\\&\quad -0.8\cdot e^{-\left( {\frac{^{( {x_1 +60})^2+( {x_2 -0})^2}}{100}}\right) } \\&\quad -0.8\cdot e^{-\left( {\frac{^{( {x_1 -60})^2+( {x_2 +60})^2}}{100}}\right) }-0.8\cdot e^{-\left( {\frac{^{( {x_1 -0})^2+( {x_2 +60})^2}}{100}}\right) }\\&\quad -0.8\cdot e^{-\left( {\frac{^{( {x_1 +60})^2+( {x_2 +60})^2}}{100}}\right) } \\&x^\mu =(0,60) \end{aligned}$$

1.2.4 Function 4

This function describes the same fitness landscape of function 3 rotated of 4/9\(\cdot \pi \).

$$\begin{aligned}&f( {r_1 ,r_2 })=1.0-0.8\cdot e^{-\left( {\frac{^{( {r_1 -60})^2+( {r_2 -60})^2}}{100}}\right) }\\&\quad -1.0\cdot e^{-\left( {\frac{^{( {r_1 -0})^2+( {r_2 -60})^2}}{100}}\right) }-0.8\cdot e^{-\left( {\frac{^{( {r_1 +60})^2+( {r_2 -60})^2}}{100}}\right) } \\&\quad -0.8\cdot e^{-\left( {\frac{^{( {r_1 -60})^2+( {r_2 -0})^2}}{100}}\right) }-0.8\cdot e^{-\left( {\frac{^{( {r_1 -0})^2+( {r_2 -0})^2}}{100}}\right) }\\&\quad -0.8\cdot e^{-\left( {\frac{^{( {r_1 +60})^2+( {r_2 -0})^2}}{100}}\right) } \\&\quad -0.8\cdot e^{-\left( {\frac{^{( {r_1 -60})^2+( {r_2 +60})^2}}{100}}\right) }\\&\quad -0.8\cdot e^{-\left( {\frac{^{( {r_1 -0})^2+( {r_2 +60})^2}}{100}}\right) }\\&\quad -0.8\cdot e^{-\left( {\frac{^{( {r_1 +60})^2+( {r_2 +60})^2}}{100}}\right) } \\&r_1 =x_1 \cdot \cos \left( {\frac{4}{9}\cdot \pi }\right) -x_2 \cdot \sin \left( {\frac{4}{9}\cdot \pi }\right) \\&r_2 =x_1 \cdot \sin \left( {\frac{4}{9}\cdot \pi }\right) +x_2 \cdot \cos \left( {\frac{4}{9}\cdot \pi }\right) \\&x^\mu =(-25.7,-54.2) \end{aligned}$$

1.2.5 Function 5

This function maps eleven large secondary basins, and a narrow and steep hole near the top-right corner of the search space where the global minimum is located. The secondary minima are arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-1.0\cdot e^{-\left( {\frac{^{( {x_1 -80})^2+( {x_2 -80})^2}}{100}}\right) ^4}\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 +50})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 -50})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 +50})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +50})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 -50})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 +0})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 +0})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +100})^2+( {x_2 +100})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 -100})^2+( {x_2 +100})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +100})^2+( {x_2 -100})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{400}}\right) } \\&x^\mu =(80,80) \end{aligned}$$

1.2.6 Function 6

This function maps eleven large secondary basins, and a narrow and steep hole at the origin of the search space where the global minimum is located. The secondary minima are arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-1.0\cdot e^{-\left( {\frac{^{( {x_1 -0})^2+( {x_2 -0})^2}}{100}}\right) ^4}\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 +50})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 -50})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 +50})^2}}{400}}\right) }\\&\quad -0.75\cdot e^ {-\left( {\frac{^{( {x_1 -50})^2+( {x_2 -50})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +50})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 -50})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 +0})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 +0})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 +100})^2+( {x_2 +100})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 -100})^2+( {x_2 +100})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +100})^2+( {x_2 -100})^2}}{400}}\right) } \\&x^\mu =(0,0) \end{aligned}$$

1.2.7 Function 7

This function maps eleven large secondary basins, and a narrow and steep hole near the top-right corner of the search space where the global minimum is located. The secondary minima are not arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-1.0\cdot e^{-\left( {\frac{^{( {x_1 -85})^2+( {x_2 -80})^2}}{100}}\right) ^4}\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +65})^2+( {x_2 +45})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -40})^2+( {x_2 +70})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 -80})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +55})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +15})^2+( {x_2 -60})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +65})^2+( {x_2 -10})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 -80})^2+( {x_2 +5})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{400}}\right) } -0.75\cdot e^{-\left( {\frac{^{( {x_1 -70})^2+( {x_2 +90})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +95})^2+( {x_2 -90})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +85})^2+( {x_2 +80})^2}}{400}}\right) } \\&\quad x^\mu =(85,80) \end{aligned}$$

1.2.8 Function 8

This function maps eleven large secondary basins, and a narrow and steep hole at the origin of the search space where the global minimum is located. The secondary minima are not arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-1.0\cdot e^{-\left( {\frac{^{( {x_1 -0})^2+( {x_2 -0})^2}}{100}}\right) ^4}\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 -65})^2+( {x_2 -45})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -40})^2+( {x_2 +70})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 +35})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +50})^2+( {x_2 -80})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +55})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 +15})^2+( {x_2 -60})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +65})^2+( {x_2 -10})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -80})^2+( {x_2 +5})^2}}{400}}\right) }-0.75\cdot e^{-\left( {\frac{^{( {x_1 -70})^2+( {x_2 +90})^2}}{400}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{( {x_1 +85})^2+( {x_2 +80})^2}}{400}}\right) } \\&\quad +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +90})^2+( {x_2 -70})^2}}{400}}\right) } \\&x^\mu =(0,0) \end{aligned}$$

1.3 Minimum surrounded by flat surface

1.3.1 Function 9

This function maps a multimodal search surface. The global minimum lies at the origin of the search space, and its basin is characterised by a large flat step and a steep and narrow ending. Four secondary minima are arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })= \left\{ \begin{array}{l} ( {( {x_1 +0})^2+( {x_2 +0})^2\le 625})\Rightarrow 0.4-0.4\cdot e^{-\left( {\frac{( {x_1 +0})^2+( {x_2 +0})^2}{10}}\right) } \\ else\Rightarrow 1.0-0.75\cdot e^{-\left( {\frac{^{\left( {x_1 +75}\right) ^2+\left( {x_2 +75}\right) ^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 -75})^2+( {x_2-75})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -75})^2+( {x_2 +75})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 -75})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{1000}}\right) } \\ \end{array}\right. \\&x^\mu =(0,0) \end{aligned}$$

1.3.2 Function 10

This function maps a multimodal search surface. The global minimum lies far from the origin of the search space, and its basin is characterised by a large flat step and a steep and narrow ending. Four secondary minima are arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })= \left\{ \begin{array}{l} ( {( {x_1 +75})^2+( {x_2 +75})^2\le 625})\Rightarrow 0.4-0.4\cdot e^{-\left( {\frac{( {x_1 +75})^2+( {x_2 +75})^2}{10}}\right) } \\ else\Rightarrow 1.0-0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 +75})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 -75})^2+( {x_2 -75})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -75})^2+( {x_2 +75})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 -75})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{1000}}\right) } \\ \end{array}\right. \\&x^\mu =(-75,-75) \end{aligned}$$

1.3.3 Function 11

This function maps a multimodal search surface. The global minimum lies at the origin of the search space, and its basin is characterised by a large flat step and a steep and narrow ending. Four secondary minima are not arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=\left\{ {\begin{array}{l} ( {( {x_1 +0})^2+( {x_2 +0})^2\le 625})\Rightarrow 0.4-0.4\cdot e^{-\left( {\frac{( {x_1 +0})^2+( {x_2 +0})^2}{10}}\right) } \\ else\Rightarrow 1.0-0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 +75})^2}}{1000}}\right) }\\ -0.75\cdot e^{-( {\frac{^{( {x_1 -50})^2+( {x_2 -60})^2}}{1000}})} \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -70})^2+( {x_2 +55})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 +45})^2+( {x_2 -85})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{1000}}\right) } \\ \end{array}} \right. \\&x^\mu =(0,0) \end{aligned}$$

1.3.4 Function 12

This function maps a multimodal search surface. The global minimum lies far from the origin of the search space, and its basin is characterised by a large flat step and a steep and narrow ending. Four secondary minima are not arranged on a grid.

$$\begin{aligned}&f( {x_1 ,x_2 })=\left\{ {\begin{array}{l} ( {( {x_1 +75})^2+( {x_2 +75})^2\le 625})\Rightarrow 0.4-0.4\cdot e^{-\left( {\frac{( {x_1 +75})^2+( {x_2 +75})^2}{10}}\right) } \\ else\Rightarrow 1.0-0.75\cdot e^{-\left( {\frac{^{( {x_1 +75})^2+( {x_2 +75})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 -60})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 -70})^2+( {x_2 +55})^2}}{1000}}\right) }\\ -0.75\cdot e^{-\left( {\frac{^{( {x_1 +45})^2+( {x_2 -85})^2}}{1000}}\right) } \\ +-0.75\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{1000}}\right) } \\ \end{array}} \right. \\&x^\mu =(-75,-75) \end{aligned}$$

1.4 Narrow valleys

1.4.1 Function 13

This function maps two valleys on opposite sides of the search space.

$$ \begin{aligned}&f( {x_1 ,x_2 })=\left\{ {\begin{array}{l} ( {x_1 \ge 0 \& x_2 \ge 0}) \Rightarrow 1.0-0.75\cdot e^{-\left( {\frac{^{( {x_2 -50})^2}}{50}}\right) } \\ ( {x_1 <0 \& x_2 <0}) \Rightarrow 1.0-1.0\cdot \left( {\frac{x_2 +100}{70}}\right) \cdot e^{-\left( {\frac{^{( {x_1 +x_2 +130})^2}}{25}}\right) } \\ \end{array}} \right. \\&x^\mu =(-100,-30) \end{aligned}$$

1.4.2 Function 14

This function maps two pairs of valleys of opposite slopes that represent competing basins of attraction. The minimum is located near the borders of the search surface at the end of the narrowest valley. The four valleys join at the origin, where a further basin is located.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-1.0\cdot e^{-\left( {\frac{^{x_1 -90}}{25}}\right) ^2}\cdot e^{-\left( {\frac{( {x_1 +2\cdot x_2 })^2}{2}}\right) }\\&\quad -0.75\cdot e^{-\left( {\frac{^{x_1 +100}}{25}}\right) ^2}\cdot e^{-\left( {\frac{( {x_1 +2\cdot x_2 })^2}{1000}}\right) } \\&\quad +-0.80\cdot e^{-\left( {\frac{^{( {x_1 +0})^2+( {x_2 +0})^2}}{150}}\right) }\\&\quad -0.75\cdot \left( {\frac{x_2 }{100}}\right) ^2\cdot \\&\quad e^{-\left( {\frac{^{( {3\cdot x_1 -x_2 })^2}}{1000}}\right) }\\&x^\mu =(90,-45) \end{aligned}$$

1.4.3 Function 15

This function maps a narrow parabolic valley surrounded by a large flat surface. The valley is located in the half plane of positive x1 values (x1 \(>\) 0). The other half of the fitness landscape is covered by a sliding plane.

$$\begin{aligned}&f( {x_1, x_2 })=\left\{ {\begin{array}{l} ( {x_1 \ge 0})\Rightarrow 1.0-\frac{x_1^4 }{90^4}\cdot e^{-\left( {\frac{^{\left( {x_1^2 +4\cdot x_2^2 -8100}\right) ^2}}{250}}\right) } \\ else\Rightarrow 1.0-\frac{x_1 }{125} \\ \end{array}} \right. \\&x^\mu =( {90,0}) \end{aligned}$$

1.5 Wavelike

1.5.1 Function 16

This function combines two cosinusoidal functions. Each function depends on one of the two input variables, and its amplitude increases linearly with the associated variable. The global minimum is in a narrow hole that is added to one of the “pockets” of the search surface. Function 16 has an overall unimodal characteristic corresponding to a plane slanted toward the positive values of the two input variables. The optimum is far from the origin and does not correspond to the minimum of the unimodal characteristic (i.e. the slanted plane).

$$\begin{aligned} f( {x_1, x_2 })&= 1.0-\left( {0.4+0.4\cdot \cos \left( {2\cdot \pi \cdot \frac{x_1 -75}{25}}\right) }\right) \\&\cdot \frac{x_1 +100}{400} \\&-\left( {0.4+0.4\cdot \cos \left( {2\cdot \pi \cdot \frac{x_2 -75}{25}}\right) }\right) \\&\cdot \frac{x_2 +100}{400}-0.30\cdot e^{-\left( {\frac{^{( {x_1 -75})^2+( {x_2 -75})^2}}{10}}\right) } \\ x^\mu&= (75,75) \end{aligned}$$

1.5.2 Function 17

This function combines two sinusoidal functions. The two functions have constant amplitude and variable period. The global minimum is in a narrow hole that is added to one of the “pockets” of the search surface.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.2+0.2\cdot \\&\quad \sin \left( {2+\frac{x_2 +100}{100}\cdot \pi \cdot \frac{x_1 }{50}}\right) \\&\quad +-0.2+0.2\cdot \sin \left( {2+\frac{x_1 +100}{100}\cdot \pi \cdot \frac{x_2 }{50}}\right) -0.2\\&\quad \cdot e^{-\left( {\frac{^{( {x_1 -50})^2+( {x_2 -50})^2}}{10}}\right) } \\&x^\mu =(50,50) \end{aligned}$$

1.6 “Noisy” unimodal

1.6.1 Function 18

This function has overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 10 % of that of the unimodal curve. The peak lies far from the origin.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.3\cdot \left( {\left( {\frac{x_1 -50}{50}}\right) ^2+\left( {\frac{x_2 -50}{50}}\right) ^2+1}\right) \\&-0.3\cdot \left( {\left( {\frac{x_1 -50}{25}}\right) ^2+\left( {\frac{x_2 -50}{25}}\right) ^2+1}\right) \\&-0.3\cdot \left( {\left( {\frac{x_1 -50}{10}}\right) ^2+\left( {\frac{x_2 -50}{10}}\right) ^2+1}\right) \\&-0.05\cdot \left| {\cos ( {2\cdot \pi \cdot ( {x_1 -50})})+\cos ( {2\cdot \pi \cdot ( {x_2 -50})})} \right| \\&x^\mu =(50,50) \end{aligned}$$

1.6.2 Function 19

This function has an overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 25 % of that of the unimodal curve. The peak lies far from the origin.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.25\cdot \left( {\left( {\frac{x_1 -50}{50}}\right) ^2+\left( {\frac{x_2 -50}{50}}\right) ^2+1}\right) \\&-0.25\cdot \left( {\left( {\frac{x_1 -50}{25}}\right) ^2+\left( {\frac{x_2 -50}{25}}\right) ^2+1}\right) \\&-0.25\cdot \left( {\left( {\frac{x_1 -50}{10}}\right) ^2+\left( {\frac{x_2 -50}{10}}\right) ^2+1}\right) \\&-0.125\cdot \left| {\cos ( {2\cdot \pi \cdot ( {x_1 -50})})+\cos ( {2\cdot \pi \cdot ( {x_2 -50})})} \right| \\&x^\mu =(50,50) \end{aligned}$$

1.6.3 Function 20

This function has an overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 40 % of that of the unimodal curve. The peak lies far from the origin.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.2\cdot \left( {\left( {\frac{x_1 -50}{50}}\right) ^2+\left( {\frac{x_2 -50}{50}}\right) ^2+1}\right) \\&\quad -0.2\cdot \left( {\left( {\frac{x_1 -50}{25}}\right) ^2+\left( {\frac{x_2 -50}{25}}\right) ^2+1}\right) \\&\quad -0.2\cdot \left( {\left( {\frac{x_1 -50}{10}}\right) ^2+\left( {\frac{x_2 -50}{10}}\right) ^2+1}\right) \\&\quad -0.2\cdot \left| {\cos ( {2\cdot \pi \cdot ( {x_1 -50})})+\cos ( {2\cdot \pi \cdot ( {x_2 -50})})} \right| \\&x^\mu =(50,50) \end{aligned}$$

1.6.4 Function 21

This function is similar to function 19 but the period of the cosinusoidal noise component is multiplied by a factor 4.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.25\cdot \left( {\left( {\frac{x_1 -50}{50}}\right) ^2+\left( {\frac{x_2 -50}{50}}\right) ^2+1}\right) \\&\quad -0.25\cdot \left( {\left( {\frac{x_1 -50}{25}}\right) ^2+\left( {\frac{x_2 -50}{25}}\right) ^2+1}\right) \\&\quad -0.25\cdot \left( {\left( {\frac{x_1 -50}{10}}\right) ^2+\left( {\frac{x_2 -50}{10}}\right) ^2+1}\right) \\&\quad -0.125\cdot \left| {\cos \left( {2\cdot \pi \cdot \frac{( {x_1 -50})}{4}}\right) +\cos \left( {2\cdot \pi \cdot \frac{( {x_2 -50})}{4}}\right) } \right| \\&x^\mu =(50,50) \end{aligned}$$

1.6.5 Function 22

This function is similar to function 19 but the period of the cosinusoidal noise component is divided by a factor 4.

$$\begin{aligned}&f( {x_1 ,x_2 })=1.0-0.25\cdot \left( {\left( {\frac{x_1 -50}{50}}\right) ^2+\left( {\frac{x_2 -50}{50}}\right) ^2+1}\right) \\&-0.25\cdot \left( {\left( {\frac{x_1 -50}{25}}\right) ^2+\left( {\frac{x_2 -50}{25}}\right) ^2+1}\right) \\&-0.25\cdot \left( {\left( {\frac{x_1 -50}{10}}\right) ^2+\left( {\frac{x_2 -50}{10}}\right) ^2+1}\right) \\&-0.125\cdot \left| {\cos ( {2\cdot \pi \cdot 4\cdot ( {x_1 -50})})+\cos ( {2\cdot \pi \cdot 4\cdot ( {x_2 -50})})} \right| \\&x^\mu =(50,50) \end{aligned}$$

1.7 Different dimensionality

1.7.1 Function 23

This function maps seven secondary basins, and the global minimum is located far from the origin. The secondary minima are not arranged on a grid. The function is ten-dimensional

$$\begin{aligned}&f( {x_1 ,\ldots ,x_{10} })=1.0-1.0\cdot e^{\sum \nolimits _{i=1}^{10} {( {x_i -75})^2} }\\&-\sum \limits _{j=0}^6 {0.8\cdot e^{\frac{\sum \nolimits _{i=1}^{10} {-( {x_i +sign_{ij} \cdot j\cdot 15})^2} }{2000}}} \\&sign_{ij} =2\cdot \min \left( {1,\frac{i+1}{j+1}}\right) -1 \\&x^\mu =(75,\ldots ,75) \end{aligned}$$

1.7.2 Function 24

This function maps seven secondary basins, and the global minimum is located far from the origin. The secondary minima are not arranged on a grid. The function is fifteen-dimensional

$$\begin{aligned}&f( {x_1 ,\ldots ,x_{15} })=1.0-1.0\cdot e^{\sum \nolimits _{i=1}^{15} {( {x_i -75})^2} }\\&-\sum \limits _{j=0}^6 {0.8\cdot e^{\frac{\sum \nolimits _{i=1}^{15} {-( {x_i +sign_{ij} \cdot j\cdot 15})^2} }{2000}}} \\&sign_{ij} =2\cdot \min \left( {1,\frac{i+1}{j+1}}\right) -1 \\&x^\mu =(75,\ldots ,75) \end{aligned}$$

1.7.3 Function 25

This function maps seven secondary basins, and the global minimum is located far from the origin. The secondary minima are not arranged on a grid. The function is twenty-dimensional

$$\begin{aligned}&f( {x_1 ,\ldots ,x_{20} })=1.0-1.0\cdot e^{\sum \nolimits _{i=1}^{20} {( {x_i -75})^2} }\\&\quad -\sum \limits _{j=0}^6 {0.8\cdot e^{\frac{\sum \nolimits _{i=1}^{20} {-( {x_i +sign_{ij} \cdot j\cdot 15})^2} }{2000}}} \\&\quad sign_{ij} =2\cdot \min \left( {1,\frac{i+1}{j+1}}\right) -1 \\&\quad x^\mu =(75,\ldots ,75) \end{aligned}$$

B: Algorithms configurations

See Tables 15, 16, 17 and 18.

Table 15 EA configurations
Full size table
Table 16 PSO configurations
Full size table
Table 17 ABC configurations
Full size table
Table 18 Bees Algorithm configurations
Full size table

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Pham, D.T., Castellani, M. Benchmarking and comparison of nature-inspired population-based continuous optimisation algorithms. Soft Comput 18, 871–903 (2014). https://doi.org/10.1007/s00500-013-1104-9

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