New variants of glowworm swarm optimization based on step size

Abstract

In 2005, Krishnanand and Ghose (Multimodal function optimization using a glowworm metaphor with applications to collective robotics, 2005a), presented the idea of glowworm metaphor to determine multiple minima in the optimization problem arising in robotics applications. That research paper highlights the glowworm swarm behavior for determining multiple local minima for multimodal functions with application to robotics. Since then, a number of research papers have appeared to improve the performance of glowworm swarm optimization (GSO). In this paper, two major contributions are made. Firstly, a mathematical result is proved which shows that the step size of GSO has a significant influence on the convergence of GSO. Secondly, three variants of GSO are proposed which depend on different step size. Based on the implementation of the proposed variants and the original GSO on 15 benchmark problems, it is concluded that one of the proposed variants is a definite improvement over the original GSO and the remaining variants.

Appendix A: List of test problems

Problem 1: Easom 2D problem

$$ Min\,\,f_{1} (x) = - \cos (x_{1} )\cos (x_{2} )\exp \left( { - (x_{1} - \pi )^{2} - (x_{2} - \pi )^{2} } \right) $$

Subject to $- 10 \le x_{1} ,x_{2} \le 10$. It has minimum value $f(x^{*} ) = - 1$ at $x^{*} = (\pi ,\pi ).$

Problem 2: Becker and Lago problem

$$ Min\,\,f_{2} (x) = \left( {\left| {x_{1} } \right| - 5} \right)^{2} + \left( {\left| {x_{2} } \right| - 5} \right)^{2} $$

Subject to $- 10 \le x_{1} ,x_{2} \le 10.$ it has four minima located at $x^{*} = \left( { \pm 5, \pm 5} \right)$ all with $f(x^{*} ) = 0.$

Problem 3: Aluffi-Pentini’s problem

$$ Min\,\,f_{3} (x) = 0.25x_{1}^{4} - 0.5x_{1}^{2} + 0.1x_{1} + 0.5x_{2}^{2} $$

Subject to $- 10 \le x_{1} ,x_{2} \le 10.$ it has two local minima and global minima is located at $x^{*} = \left( { - 1.0465,\,0} \right)$ with $f(x^{*} ) \approx - 0.3523.$

Problem 4: Wood’s problem

$$ \begin{aligned} Min\,\,f_{4} (x) = 100(x_{2} - x_{1}^{2} )^{2} + (1 - x_{1} )^{2} + 90(x_{4} - x_{3}^{2} ) + (1 - x_{3} )^{2} + 10.1[(x_{2} - 1)^{2} + (x_{4} - 1)^{2} ] + 19.8(x_{2} - 1)(x_{4} - 1) \hfill \\ \end{aligned} $$

Subject to $- 10 \le x_{1} ,x_{2} ,x_{3} ,x_{4} \le 10.$ minima is located at $x^{*} = \left( {1,1,1,1} \right)$ with $f(x^{*} ) = 0.$

Problem 5: Miele and Cantrell’s problem$Min\,\,f_{5} (x) = (\exp (x_{1} ) - x_{2} )^{4} + 100(x_{2} - x_{3} )^{6} + (\tan (x_{3} - x_{4} ))^{4} + x_{1}^{8}$ Subject to $- 1 \le x_{1} ,x_{2} ,x_{3} ,x_{4} \le 1.$ minima is located at $x^{*} = \left( {0,1,1,1} \right)$ with $f(x^{*} ) = 0.$

Problem 6: Bohachevsky 1 problem

$$ Min\,\,f_{6} (x) = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos (3\pi x_{1} ) - 0.4\cos (4\pi x_{2} ) + 0.7 $$

Subject to $- 50 \le x_{1} ,x_{2} \le 50.$ the number of local minima is unknown but it has global minima at $x^{*} = \left( {0,\,0} \right)$ with $f(x^{*} ) = 0.$

Problem 7: Bohachevsky 2 problem

$$ Min\,\,f_{7} (x) = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos (3\pi x_{1} )\cos (4\pi x_{2} ) + 0.3 $$

Subject to $- 50 \le x_{1} ,x_{2} \le 50.$ the number of local minima is unknown but it has global minima at $x^{*} = \left( {0,\,0} \right)$ with $f(x^{*} ) = 0.$

Problem 8: Branin problem

$$ \begin{aligned} Min\,\,f_{8} (x) = a(x_{2} - bx_{1}^{2} + cx_{1} - d)^{2} + g(1 - h)\cos (x_{1} ) + g, \hfill \\ a = 1,b = \frac{5.1}{{4\pi^{2} }},c = \frac{5}{\pi },d = 6,g = 10,h = \frac{1}{8\pi } \hfill \\ \end{aligned} $$

Subject to $- 5 \le x_{1} \le 10,\,0 \le x_{1} \le 15.$ it have three minima located at $x^{*} = \left( { - \pi ,\,\,12.275} \right),\,\left( {\pi ,\,2.275} \right),\,\,\left( {3\pi ,\,2.475} \right)$ all with $f(x^{*} ) = {5 \mathord{\left/ {\vphantom {5 {(4\pi }}} \right. \kern-0pt} {(4\pi }}).$

Problem 9: Eggcrate problem

$$ Min\,\,f_{9} (x) = x_{1}^{2} + x_{2}^{2} + 25\left( {\sin^{2} x_{1} + \sin^{2} x_{2} } \right) $$

Subject to $- 2\pi \le x_{1} ,\,x_{2} \le 2\pi .$ it is highly nonlinear and has many local optima. The global minima is located at $x^{*} = (0,0)$ with $f(x^{*} ) = 0.$

Problem 10: Modified Rosenbrock problem

$$ Min\,\,f_{10} (x) = 100\left( {x_{2} - x_{1}^{2} } \right)^{2} + [6.4(x_{2} - 0.5)^{2} - x_{1} - 0.6]^{2} $$

Subject to $- 5 \le x_{1} ,x_{2} \le 5.$ It has two global minima located at $x^{*} \approx \left( {0.3412,\,0.1164} \right),\,(1,\,1)$ with $f(x^{*} ) = 0.$

Problem 11: Periodic problem

$$ Min\,\,f_{11} (x) = 1 + \sin^{2} x_{1} + \sin^{2} x_{2} - 0.1\exp ( - x_{1}^{2} - x_{2}^{2} ) $$

Subject to $- 10 \le x_{1} ,x_{2} \le 10.$ it has a global minimum at $x^{*} = (0,0)$ with $f(x^{*} ) = 0.9$ and also have 49 local minima all with minimum value 1.

Problem 12: Powell problem

$$ Min\,\,f_{12} (x) = (x_{1} + 10x_{2} )^{2} + 5(x_{3} - x_{4} )^{2} + (x_{2} - 2x_{3} )^{4} + 10(x_{1} - x_{4} )^{4}$$

Subject to $- 10 \le x_{1} ,x_{2} ,x_{3} ,x_{4} \le 10.$ it has a global minimum at $x^{*} = (0,0,0,0)$ with $f(x^{*} ) = 0.$

Problem 13: Camel Back-3 Hump problem

$$ Min\,\,f_{13} (x) = 2x_{1}^{2} - 1.05x_{1}^{4} + \frac{1}{6}x_{1}^{6} + x_{1} x_{2} + x_{2}^{2} $$

Subject to $- 5 \le x_{1} ,x_{2} \le 5.$ It has three local minima, one of them is global at $x^{*} = (0,0)$ with $f(x^{*} ) = 0.$

Problem 14: Camel Back-6 Hump problem

$$ Min\,\,f_{14} (x) = 4x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4} $$

Subject to $- 5 \le x_{1} ,x_{2} \le 5.$ It has three local minima with values $f \approx - 1.0316,\, - 0.2154,\,2.1042,$ and two global minima at $x^{*} \approx (0.089842, - 0.712656),\,( - 0.089846,0.712656)$ with $f(x^{*} ) \approx - 1.0316.$

Problem 15: McCormick problem

$$ Min\,\,f_{15} (x) = \sin (x_{1} + x_{2} ) + (x_{1} - x_{2} )^{2} - \frac{3}{2}x_{1} + \frac{5}{2}x_{2} + 1 $$

Subject to $- 1.5 \le x_{1} \le 4$ and $- 3 \le x_{2} \le 3.$ It has local minima at $(2.59,1.59)$ and a global minima at $x^{*} = ( - 0.547, - 1.547)$ with $f(x^{*} ) \approx - 1.9133.$