Enhanced shuffled frog-leaping algorithm for solving numerical function optimization problems

Abstract

The shuffled frog-leaping algorithm (SFLA) is a relatively new meta-heuristic optimization algorithm that can be applied to a wide range of problems. After analyzing the weakness of traditional SFLA, this paper presents an enhanced shuffled frog-leaping algorithm (MS-SFLA) for solving numerical function optimization problems. As the first extension, a new population initialization scheme based on chaotic opposition-based learning is employed to speed up the global convergence. In addition, to maintain efficiently the balance between exploration and exploitation, an adaptive nonlinear inertia weight is introduced into the SFLA algorithm. Further, a perturbation operator strategy based on Gaussian mutation is designed for local evolutionary, so as to help the best frog to jump out of any possible local optima and/or to refine its accuracy. In order to illustrate the efficiency of the proposed method (MS-SFLA), 23 well-known numerical function optimization problems and 25 benchmark functions of CEC2005 are selected as testing functions. The experimental results show that the enhanced SFLA has a faster convergence speed and better search ability than other relevant methods for almost all functions.

Appendices

Appendix 1

The theoretical minimum values and the optimum location for functions in Tables 1 and 2

Function	$X_{opt}$	$f_{opt}$
$f_{1\ldots } f_{5}$	[0]$^{n}$	0
$f_{6}$	$[-0.5]^{n}$	0
$f_{7}$	$[0]^{n}$	0
$f_{8}$	$[420.96]^{n}$	$-$418.9829*n
$f_{9\ldots } f_{13}$	$[0]^{n}$	0
$f_{14}$	($-$32, 32)	1
$f_{15}$	(0.1928, 0.1908, 0.1231, 0.1358)	0.00030
$f_{16}$	(0.089, $-$0.712), ($-$0.089, 0.712)	$-1.0316$
$f_{17}$	($-$3.14, 12.27), (3.14, 2.275), (9.42, 2.42)	0.398
$f_{18}$	(0, $-$1)	3
$f_{19}$	(0.114, 0.556, 0.852)	$-3.86$
$f_{20}$	(0.201, 0.15, 0.477, 0.275, 0.311, 0.657)	$-3.32$
$f_{21}$	5 local minima in $a_{ij}$, $j=1,2,{\ldots },5$	$-10.1532$
$f_{22}$	7 local minima in $a_{ij}$, $j=1,2,{\ldots },7$	$-10.4028$
$f_{23}$	10 local minima in $a_{ij}$, $j=1,2,{\ldots },10$	$-10.5363$

Appendix 2

See Tables 11, 12, 13, 14, 15, 16 and 17.

Table 11 $a_{ij}$ in $f_{14}$

Table 12 $a_{i}$ and $b_{i }$ in $f_{15}$

Table 13 $a_{i}$ and $c_{i }$ in $f_{19}$

Table 14 $p_{ij}$ in $f_{19}$

Table 15 $a_{ij}$ and $c_{i}$ in $f_{20}$

Table 16 $p_{ij}$ in $f_{20}$

Table 17 $a_{ij}$ and $c_{i}$ in $f_{21}$, $f_{22}$, $f_{23}$