Bolstering efficient SSGAs based on an ensemble of probabilistic variable-wise crossover strategies

Abstract

A crossover operator in genetic algorithms (GAs) plays an essential role as the main search operator to breed offspring by exchanging information between individuals. Although different types of crossover operators have been developed for real-coded GAs (RCGAs), there has been very little research on combining different crossover operators to build more effective and efficient RCGAs. In this work, we propose new steady-state generation alternation-based RCGAs (SSGAs) ameliorated with (i) an ensemble of different probabilistic variable-wise crossover strategies, which is realized by the corresponding parallel populations, to utilize synergetic and complementary effect with their efficient operations, and (ii) efficient operation at each evolution step to obtain further performance enhancement. To investigate the performance of this ensemble with respect to search abilities and computation time, we compare the proposed algorithms against various SSGAs when running 27 benchmark functions. Empirical studies showed that the proposed algorithms exhibit better performance than the contestant SSGAs on these functions. Moreover, a comparison with the state-of-the-art evolutionary algorithms on eight difficult benchmark functions clearly demonstrated outperformance of the proposed algorithms.

Appendix: Test function suite

$$\begin{aligned}&f_1 (x)=\sum \limits _{i=1}^D {z_i^2 } +450,\,z=x-o.\\&f_2 (x)=\sum \limits _{i=1}^D {\left( {\sum \limits _{j=1}^i {z_j } }\right) }^2,\,z=x-o. \\&f_3 (x)= \sum \limits _{i=1}^D {z_i^2 } \left( +2z_{i+1}^2 -0.3\cos (3\pi z_i)\right. \\&\qquad \qquad \left. -0.4\cos (4\pi z_{i+1})+0.7\right) ,\,z=x-o. \\&f_4 (x)\!=\!\sum \limits _{i=1}^D \left( {z_i^2} \!+\!z_{i\!+\!1}^2\right) ^{0.25}\left( {\sin ^2\left( {50\left( z_i^2 \!+\!z_{i+1}^2\right) ^{0.1}}\right) \!+\!1}\right) ,\\&\qquad z=x-o. \\&f_5 (x)=\sum \limits _{i=1}^D {\left| {z_i}\right| } +\mathop \prod \limits _{i=1}^D \left| {z_i}\right| ,\,z=x-o. \\&f_6 (x)=\mathop {\max }\limits _i\left\{ {\left| {z_i}\right| ,\,1\le i\le D}\right\} +450,\,z=x-o. \\&f_7 (x)\!=\!\sum \limits _{i=1}^D \left( {100}(z_i^2 \!+\! z_{i+1})^{2} \!+\! (z_i-1)^2\right) +390,\,z\!=\!x\!-\!o \\&f_8 (x)=\sum \limits _{i=1}^{D-1} \left( {100}(z_i^2 + z_{i+1}\right) ^{2}+ \left( z_i-1)^2\right) -900,\\&\qquad z={\mathbf {M}}_1 \left( {\frac{2.048(x-o)}{100}}\right) +1.\\&f_9 (x)=-20\exp \left( {-0.2\sqrt{\frac{1}{D}\sum \limits _{i=1}^D {z_i^2 } }}\right) \\&\quad -\exp \left( {\frac{1}{D}\sum \limits _{i=1}^D {\cos (2\pi z_i)}}\right) +20+e-140,\,z=x-o.\\&f_{10} (x)=-20\exp \left( {-0.2\sqrt{\frac{1}{D}\sum \limits _{i=1}^D {z_i^2 } }}\right) \\&\quad -\exp \left( {\frac{1}{D}\sum \limits _{i=1}^D {\cos (2\pi z_i)}}\right) +20+e-700,\\&z=\Lambda ^{10}{\mathbf {M}}_2 T_{asy}^{0.5} \left( {\mathbf {M}}_1 (x-o)\right) . \end{aligned}$$

$$\begin{aligned}&f_{11} (x)\!=\!\sum \limits _{i=1}^D {\frac{z_i^2 }{4000}} \!+\!\mathop \prod \limits _{i=1}^D \cos \left( {\frac{z_i }{\sqrt{i} }}\right) \!+\!1\!-\!180,\,z\!=\!x-o.\\&f_{12} (x)=\sum \limits _{i=1}^D {\frac{z_i^2 }{4000}} +\mathop \prod \limits _{i=1}^D \cos \left( {\frac{z_i }{\sqrt{i} }}\right) +1-500,\\&z=\Lambda ^{100}{\mathbf {M}}_1 \frac{600 (x-o)}{100}.\\&f_{13} (x)\!=\!\sum \limits _{i=1}^D {\left( {z_i^2 \!-\! 10\cos (2\pi z_i)\!+\!10}\right) -330},\,z\!=\!x-o.\\&f_{14} (x)=\sum \limits _{i=1}^D {\left( {z_i^2 -10\cos (2\pi z_i)+10}\right) -300,}\\&z={\mathbf {M}}_1 \Lambda ^{10}{\mathbf {M}}_2 T_{asy}^{0.2} \left( {T_{osz} \left( {{\mathbf {M}}_1 \left( {\frac{5.12(x-o)}{100}}\right) }\right) }\right) .\\&f_{15} (x)=\sum \limits _{i=1}^D {\left( {z_i^2 -10\cos (2\pi z_i)+10}\right) -200,}\,\\ {\mathop {x}\limits ^{\frown }}&={\mathbf {M}}_1 \frac{5.12(x-o)}{100},\\&y_i \!=\! {\left\{ \begin{array}{ll} {{\mathop {x}\limits ^{\frown }}}_i &{} \text{ if } \left| {{\mathop {x}\limits ^{\frown }}}_i\right| \ge 0.5\\ { round}\left( 2{{\mathop {x}\limits ^{\frown }}}_i\right) /2 &{} \text{ if } \left| {{\mathop {x}\limits ^{\frown }}}_i\right| >0.5\\ \end{array}\right. } \quad \!\text{ for } i\!=\!1,2,\ldots , D\\&z={\mathbf {M}}_1 \Lambda ^{10}{\mathbf {M}}_2 T_{asy}^{0.2} ({T_{osz} (y)}). \end{aligned}$$

$$\begin{aligned}&f_{16} (x)=\sum \limits _{i=1}^D \left( {\sum \limits _{k=0}^{kmax} {\left[ {a^k\cos \left( 2\pi b^k(z_i +0.5)\right) }\right] } }\right) \\&\quad -D\sum \limits _{k=0}^{kmax} {\left[ {a^k\cos (2\pi b^k0.5)}\right] } -600,\\&\quad a=0.5, b = 3,\,kmax=20,\\&\quad z=\Lambda ^{10}{\mathbf {M}}_2 T_{asy}^{0.5} \left( {{\mathbf {M}}_1 \frac{0.5(x-o)}{100}}\right) .\\&f_{17} (x)=g(z_1, z_2)+g(z_2, z_3)+\cdots +g(z_{D-1}, z_D)\\&\quad +g(z_D, z_1)+600,\\&g(m,n)=0.5+\frac{\sin ^2\left( \sqrt{m^2+n^2}\right) -0.5}{\left( {1+0.001(m^2+n^2)}\right) ^{2}},\\&z={\mathbf {M}}_2 T_{asy}^{0.5} \left( {{\mathbf {M}}_1 (x-o)}\right) .\\&f_{18} (x)=\frac{\pi }{D}\left\{ 10\sin ^2(\pi y_1)+\sum \limits _{i=1}^{D-1} (y_i -1)^2\right. \\&\quad \left. \times \left[ {1+10\sin ^2(\pi y_{i+1})}\right] + (y_D -1)^{2}\right\} \\&\quad +\sum \limits _{i=1}^D {u(x_i, 10,100,4)},\\&\quad \hbox {where }y_i =1+\frac{1}{4}(x_i +1)\hbox { and }\\&\quad u(x_i, a,k,m)= {\left\{ \begin{array}{ll} k(x_i -a)^m,&{}{x_i >a}\\ 0,&{}{-a\le x_i \le a}\\ k(-x_i -a)^m,&{}{x_i <-a}\\ \end{array}\right. }. \end{aligned}$$

$$\begin{aligned}&f_{19} (x)=0.1\left\{ 10\sin ^2(3\pi x_1)+\sum \limits _{i=1}^{D-1} (x_i-1)^2\right. \\&\quad \left. \left[ {1+\sin ^2(3\pi x_{i+1})}\right] + (x_D -1)[1+\sin ^2(2\pi x_D)]\right\} \\&\quad +\sum \limits _{i=1}^D {u(x_i, 5,100,4)}.\\&f_{20} (x)=\sum \limits _{i=1}^{11} {\left[ {a_i -\frac{x_1 (b_i^2 +b_i x_2)}{b_i^2 +b_i x_3 +x_4 }}\right] }^2.\\&f_{21} (x)=4x_1^2 -2.1x_1^4 +\frac{1}{3}x_1^6 +x_1 x_2 -4x_2^2 +4x_2^4 .\\&f_{22} (x)=\left( {x_2 -\frac{5.1}{4\pi ^2}x_1^2 +\frac{5}{\pi }x_1 -6}\right) ^{2}\\&\quad +10\left( {1-\frac{1}{8\pi }}\right) \cos (x_1)+10.\\&f_{23-24} (x)=-\sum \limits _{i=1}^4 {c_i } \exp \left[ {-\sum \limits _{j=1}^D{a_{ij} (x_j -p_{ij})^2} }\right] ,\\&\quad \hbox {with }D=3,6\hbox { for }f_{22} (x)\hbox { and }f_{23} (x),\hbox { respectively.}\\&f_{25-27} (x)=-\sum \limits _{i=1}^D {\left[ {(x-a_i)(x-a_i)^T+c_i}\right] },\\&\quad \hbox {with }D=5,7,\hbox { and }10\hbox { for }f_{24} (x),f_{25} (x),\\&\quad \hbox {and }f_{26} (x),\hbox { respectively.} \end{aligned}$$

$f_{28} (x){-}f_{35} (x)$ is based on the following composition function model:

$$\begin{aligned} f_c (x)=\sum \limits _{i=1}^4 { \{ \omega _i \cdot \left[ \lambda _i g_i (x)+bias_i\right] \}+f^{*}} \end{aligned}$$

$g_i (x):$ the $i$th basic function to build the composition function $f_c (x)$;

$n$: the number of basic functions;

$o_i:$ new shifted optimum position for each $g_i (x)$, which is used to define the positions of the global and local optima;

$bias_i:$ to define which one is the global optimum;

$\lambda _i:$ to control the height of $g_i (x)$;

$\omega _i:$ the normalized weight of the weight $w_i$ for $g_i (x)$, which is defined as

$$\begin{aligned}&\omega _i \!=\!\frac{w_i }{\sum \nolimits _{i=1}^n {w_i}}\\&\hbox { where }w_i \!=\!\frac{1}{\sqrt{\sum \nolimits _{j=1}^D {(x_j \!-\! o_{ij})^2}}}\exp \left( -\frac{\sum \nolimits _{j=1}^D {(x_j -o_{ij})^2} }{2D\sigma _i^2 }\right) , \end{aligned}$$

in which $\sigma _i$ is to control the coverage range of $g_i (x)$, i.e., a small value of $g_i (x)$ gives a narrow range and vice versa. If $x=o_i$, then $\omega _j = {\left\{ \begin{array}{ll} 1 &{} j=i \\ 0 &{} j \ne i\\ \end{array}\right. }$ for $j$=1,2,...,$n$, and thus $f_c (x)=bias_i +f^*$; this indicates that global optimum has the smallest bias.

	$n$	$\sigma$	$\lambda$	bias	$g_i$
$f_{28} (x)$	5	${[10,20,30,40,50]}$	[1,1e-6,1e-26,1e-6,0.1]	${[0,100,200,300,400]}$	$g_1$: rotated Rosenbrock’s function
					$g_2$: rotated Different Powers function
					$g_3$: rotated Bent Cigar function
					$g_4$: rotated Discus function
					$g_5$: Sphere function
$f_{29} (x)$	3	${[20,20,20]}$	${[1,1,1]}$	${[0,100,200]}$	$g_1 -g_3$: Schwefel’s function
$f_{30} (x)$	3	${[20,20,20]}$	${[1,1,1]}$	${[0,100,200]}$	$g_1 -g_3$: Rotated Schwefel’s function
$f_{31} (x)$	3	${[20,20,20]}$	[0.25,1,2.5]	${[0,100,200]}$	$g_1$: rotated Schwefel’s function
					$g_2$:rotated Rastrigin’s function
					$g_3$: rotated Weierstrass’s function
$f_{32} (x)$	3	${[10,30,50]}$	[0.25,1,2.5]	${[0,100,200]}$	$g_1$:rotated Schwefel’s function
					$g_2$: rotated Rastrigin’s function
					$g_3$: rotated Weierstrass’s function
$f_{33} (x)$	5	${[10,10,10,10,10]}$	[0.25,1,1e-7,2.5,10]	${[0,100,200,300,400]}$	$g_1$: rotated Schwefel’s function
					$g_2$: rotated Rastrigin’s function
					$g_3$: rotated high conditional elliptic function
					$g_4$: rotated Weierstrass’s function
					$g_5$: rotated Griewank’s function
$f_{34} (x)$	5	${[10,10,10,20,20]}$	[100,10,2.5,25,0.1]	${[0,100,200,300,400]}$	$g_1$: rotated Griewank’s function
					$g_2$: rotated Rastrigin’s function
					$g_3$: rotated Schwefel’s function
					$g_4$: rotated Weierstrass’s function
					$g_5$: Sphere function
$f_{35} (x)$	5	${[10,20,30,40,50]}$	[2.5,2,5e-3,2.5,5e-4,0.1]	${[0,100,200,300,400]}$	$g_1$: rotated expanded Griewank’s function + Rosenbrock’s function
					$g_2$: rotated Schaffers F7 function
					$g_3$: rotated Schwefel’s function
					$g_4$: rotated Expanded Schaffer’s F6 function
					$g_5$: Sphere function