Pigeon-inspired optimization (PIO) is a swarm intelligence optimizer inspired by the homing behavior of pigeons. PIO consists of two optimization stages which employ the map and compass operator, and the landmark operator, respectively. In canonical PIO, these two operators treat every bird equally, which deviates from the fact that birds usually act heterogenous roles in nature. In this paper, we propose a new variant of PIO algorithm considering bird heterogeneity — HPIO. Both of the two operators are improved through dividing the birds into hub and non-hub roles. By dividing the birds into two groups, these two groups of birds are respectively assigned with different functions of “exploitation” and “exploration”, so that they can closely interact with each other to locate the best promising solution. Extensive experimental studies illustrate that the bird heterogeneity produced by our algorithm can benefit the information exchange between birds so that the proposed PIO variant significantly outperforms the canonical PIO.
This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB1200100), National Natural Science Foundation of China (Grant Nos. 61425014, 61521091, 91538204, 61671031, 61722102).
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Figure 1
(Color online) Networks associating with the bird topology. (a) A fully-connected network, where each two nodes are linked by an edge; (b) a scale-free network, where the node degree follows the power-law distribution, which means most nodes (non-hub nodes) have low degrees, while a few nodes (hub nodes) have high degrees; (c) a random network
Figure 2
(Color online) Variation of the average fitness of birds as the optimization proceeds.
Figure 3
(Color online) Variation of hub/non-hub birds' characteristics as optimization proceeds. (a) Average fitness of the hub/non-hub birds; (b) average times of hub/non-hub birds being learned from other birds.
| Canonical PIO | HPIO |
| Input | |
| $N_p$: number of individuals in pigeon swarm; $D$: dimension of the search space; $R$: the map and compass factor; | |
| $t_1$: the number of iterations that the map and compass operator is executed; | |
| $t_2$: the maximum number of iterations that the landmark operator is executed; | |
| Search range: the borders of the search space. | |
| $k_i$: degree of $i$th bird; | |
| $k_c$: minimum of hub birds' degrees (viz. the threshold which splits the birds into hub and non-hub groups). | |
| Initialization | |
| Set initial values for $N_p,~D,~R,~t_1,~t_2$ and the search range. | |
| Set initial position $\boldsymbol{X}_i^0\in\mathbb{R}^D$ and velocity $\boldsymbol{V}_i^0\in\mathbb{R}^D$ in the search range for each pigeon individual $f\left(\boldsymbol{X}_i^0\right)~(1\le~i\le~N_p)$. | |
| Calculate fitness values of different pigeon individuals $f\left(\boldsymbol{X}_i^0\right)~(1\le~i\le~N_p)$. | |
| Set $\boldsymbol{X}_{\rm~gbest}$ is the position of the bird having the best fitness among all the birds. | |
| Set $\boldsymbol{X}_{{\rm~pbest}_i}=\boldsymbol{X}_i^0,~1\le~i\le~N_p$. | |
| Set $\boldsymbol{X}_{{\rm~cbest}_i}=\boldsymbol{X}_j^0,~j\in\left\{i,\mathcal{N}\left(i\right)\right\}$ and $j$'s fitness is the best, $\mathcal{N}\left(i\right)$ is the set of the neighbors of $i$. | |
| Randomly generate a scale-free network using Barabási-Albert model. This network has $N_p$ nodes, each representing an individual bird. Regard bird $j~(j\neq~i)$ as bird $i$'s neighbor if $j$ is linked to $i$ in this network. | |
| Map and compass operator execution | |
| For $N_c=1$ to $t_1$ do | For $N_c=1$ to $t_1$ do |
| For $i=1$ to $N_p$ do | For $i=1$ to $N_p$ do |
| $\boldsymbol{V}_i^{N_c}=\boldsymbol{V}_i^{N_c-1}{\rm~exp}\left(-R\cdot~N_c\right)+{\rm~rand}\left(0,1\right)$ | $\boldsymbol{V}_i^{N_c}=\boldsymbol{V}_i^{N_c}\cdot~{\rm~exp}{\left(-R\cdot~t\right)}$ |
| $\cdot(\boldsymbol{X}_{\rm~gbest}^{N_c-1}-\boldsymbol{X}_i^{N_c-1}),$ | |
| (${\rm~rand}\left(0,1\right)$ is a random number among 0,1)) | $+\left\{ ~~\begin{aligned} ~~\frac{1}{k_i}\sum_{j\in\mathcal{N}\left(i\right)}{{{\rm~rand}\left(0,1\right)}\cdot\left(\boldsymbol{X}_{{\rm~pbest}_j}-x_i\right)},&~~k_i>k_c,\\ ~~{\rm~rand}\left(0,1\right)\cdot(\boldsymbol{X}_{{\rm~cbest}_i}-x_i),&~~k_i\leq~k_c, ~~\end{aligned} ~~\right.$ ($\boldsymbol{X}_{{\rm~pbest}_j}$ is defined as $j$'s best position through all the iterations before $N_c$ (not included), $\boldsymbol{X}_{{\rm~cbest}_i}$ is defined as the best position of $i$'s neighbors and $i$ through all iterations before $N_c$ (not included); (${\rm~rand}\left(0,1\right)$ is a random number among 0,1)) |
| $\boldsymbol{X}_i^{N_c}=\boldsymbol{X}_i^{N_c-1}+\boldsymbol{V}_i^{N_c}$, | $\boldsymbol{X}_i^{N_c}=\boldsymbol{X}_i^{N_c-1}+\boldsymbol{V}_i^{N_c}$, |
| Update $\boldsymbol{X}_{\rm~gbest}$. | Update $\boldsymbol{X}_{{\rm~pbest}_j},~\boldsymbol{X}_{{\rm~cbest}_i}$. |
| End for | End for |
| End for | End for |
| Landmark operator execution | |
| For $N_c=t_1+1$ to $t_1+t_2$ do | For $N_c=t_1+1$ to $t_1+t_2$ do |
| Rank all the available birds individuals according to their | Rank all the available birds individuals according to |
| fitness values $f(\boldsymbol{X}_i^{N_c})~(1\le~i\le~N_p)$. | their degrees in the scale-free network. |
| Abandon half of the birds having relatively worse fitness | Abandon the birds whose degrees are the lowest in |
| in the swarm ($N_p=N_p/2$). | the swarmp (update $N_p$). |
| $F(\boldsymbol{X}_i^{N_c})=\left\{ ~~\begin{aligned} ~~\frac{1}{f(\boldsymbol{X}_i^{N_c})+\varepsilon},~&{\text{for}}~~f~~~{\text{minimization}},\\ ~~f(\boldsymbol{X}_i^{N_c}),~~~~~~&{\text{for}}~~f~~{\text{maximization}},\\ ~~\end{aligned} ~~\right.$ | $F(\boldsymbol{X}_i^{N_c})=\left\{ ~~\begin{aligned} ~~\frac{1}{f(\boldsymbol{X}_i^{N_c})+\varepsilon},~&{\text{for}}~~f~~{\text{minimization}},\\ ~~f(\boldsymbol{X}_i^{N_c}),~~~~~~&{\text{for}}~~f~~{\text{maximization}},\\ ~~\end{aligned} ~~\right.$ |
| $\boldsymbol{X}_{\rm~center}^{N_c}=\frac{\sum_{i=1}^{N_p}{\boldsymbol{X}_i^{N_c}F(\boldsymbol{X}_i^{N_c})}}{N_p\sum_{i=1}^{N_p}{F(\boldsymbol{X}_i^{N_c})}}$, | $\boldsymbol{X}_{\rm~center}^{N_c}=\frac{\sum_{i=1}^{N_p}{\boldsymbol{X}_i^{N_c}F(\boldsymbol{X}_i^{N_c})}}{N_p\sum_{i=1}^{N_p}{F(\boldsymbol{X}_i^{N_c})}}$, |
| $\boldsymbol{X}_i^{N_c}=\boldsymbol{X}_i^{N_c-1}+{\rm~rand}\cdot(\boldsymbol{X}_{\rm~center}^{N_c}-\boldsymbol{X}_i^{N_c-1})$, | $\boldsymbol{X}_i^{N_c}=\boldsymbol{X}_i^{N_c-1}+{\rm~rand}\cdot(\boldsymbol{X}_{\rm~center}^{N_c}-\boldsymbol{X}_i^{N_c-1})$, |
| Update $\boldsymbol{X}_{\rm~gbest}$. | Update $\boldsymbol{X}_{\rm~gbest}$. |
| End for | End for |
| Note: if $N_p=1$ after a number of iterations executing the landmark operator, the iterative process will be forced to | |
| terminated, namely, the actual number of the landmark operator execution iteration may less than $t_2$. | |
| Output | |
| $\boldsymbol{X}_{\rm~gbest}$ is output as the best final solution according to the fitness function $f$. | |
| Method | Description |
| PIO | The canonical PIO |
| ERPIO | The PIO using the bird interaction topology of ErdH os-Rényi random network (refer to Figure |
| EHPIO | The HPIO whose bird interaction topology network is replaced as ErdH os-Rényi random network |
| SWPIO | The PIO using the bird interaction topology of small-world network (refer to Figure |
| SHPIO | The HPIO whose bird interaction topology network is replaced as small-world network |
| SFPIO | The PIO using the bird interaction topology of scale-free network (please refer to Figure |
| SIPIO | The SFPIO whose birds use a selective-informed learning strategy aiming at bringing in bird heterogeneity (The map and compass operator of SIPIO is the same with that of HPIO, while the landmark of SIPIO remains the same with that of canonical PIO) |
| Category | Number | Function | Expression |
| Unimodal | 1 | Sphere | $f_1\left(x\right)=\sum_{i=1}^{D}x_i^2$ |
| 2 | Rosenbrock | $f_2\left(x\right)=\sum_{i=1}^{D}{{(100(x}_i^2-x_{i+1}})^2+{(x_i-1)}^2)$ | |
| 3 | Schwefel P2.22 | $f_3\left(x\right)=\sum_{i=1}^{D}\left|x_i\right|+\prod_{i=1}^{D}\left|x_i\right|$ | |
| 4 | Quartic Noise | $f_4\left(x\right)=\sum_{i=1}^{D}{{ix}_i^2+{\rm~random}\left[0,\left.~1\right)\right.}$ | |
| 5 | Schwefel P1.22 | $f_5\left(x\right)={(\sum_{i=1}^{D}x_i)}^2$ | |
| Mutimodal | 6 | Ackley | $f_6\left(x\right)=-20\exp{\left(-0.2\sqrt{\frac{1}{D}\sum_{i=1}^{D}x_i^2}\right)}$ |
| $-\exp{\left(\frac{1}{D}\sum_{i=1}^{D}\cos{\left(2\pi~x_i\right)}\right)}+20+e$ | |||
| 7 | Rastrigin | $f_7\left(x\right)=\sum_{i=1}^{D}{(x_i^2-10\cos{\left(2\pi~x_i\right)}+10)}$ | |
| 8 | Rastrigin (discrete) | $f_8\left(x\right)=\sum_{i=1}^{D}{(y_i^2-10\cos{\left(2\pi~y_i\right)}+10)},$ | |
| $y_i=\left\{ ~~\begin{aligned} ~~x_i,&~~|x_i|<\frac{1}{2},\\ ~~{\rm~round}(2x_i)/2,&~~|x_i|\ge\frac{1}{2}\\ ~~\end{aligned} \right.$ | |||
| 9 | Weierstrass | $f_9\left(x\right)=\sum_{i=1}^{D}\left(\sum_{k=0}^{k_{\rm~max}}\left[a^k\cos{\left(2\pi~b^k\left(x_i+0.5\right)\right)}\right]\right)$ | |
| $-D\sum_{k=0}^{k_{\rm~max}}\left[a^k\cos{\left(\pi~b^k\right)}\right]$ | |||
| 10 | Griewank | $f_{10}\left(x\right)=\frac{1}{4000}\sum_{i=1}^{D}x_i^2-\prod_{i=1}^{D}{\cos{\left(\frac{x_i}{\sqrt~i}\right)}+1}$ | |
| Rotated multimal$^{\rm~c)}$ | 11 | Ackley (rotated) | $f_{11}\left(x\right)=-20\exp{\left(-0.2\sqrt{\frac{1}{D}\sum_{i=1}^{D}y_i^2}\right)}$ |
| $-\exp{\left(\frac{1}{D}\sum_{i=1}^{D}\cos{\left(2\pi~y_i\right)}\right)}+20+e$ | |||
| 12 | Rastrigin (rotated) | $f_{12}\left(x\right)=\sum_{i=1}^{D}{(y_i^2-10\cos{\left(2\pi~y_i\right)}+10)}$ | |
| 13 | Rastrigin (discrete and rotated) | $f_{13}\left(x\right)=\sum_{i=1}^{D}{(z_i^2-10\cos{\left(2\pi~z_i\right)}+10)},$ | |
| $z_i=\left\{ ~~\begin{aligned} ~~y_i,&~~|y_i|<\frac{1}{2},\\ ~~{\rm~round}(2y_i)/2,&~~|y_i|\ge\frac{1}{2}\\ ~~\end{aligned} \right.$ | |||
| 14 | Weierstrass (rotated) | $f_{14}\left(x\right)=\sum_{i=1}^{D}\left(\sum_{k=0}^{k_{\rm~max}}\left[a^k\cos{\left(2\pi~b^k\left(y_i+0.5\right)\right)}\right]\right)$ | |
| $-D\sum_{k=0}^{k_{\rm~max}}\left[a^k\cos{\left(\pi~b^k\right)}\right]$ | |||
| 15 | Griewank (rotated) | $f_{15}\left(x\right)=\frac{1}{4000}\sum_{i=1}^{D}y_i^2-\prod_{i=1}^{D}{\cos{\left(\frac{y_i}{\sqrt~i}\right)}+1}$ | |
| Composite | 16 | Composite 1 | $f_{16}$ is composed by 10 Sphere functions |
| 17 | Composite 2 | $f_{17}$ is composed by 10 functions |
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| Category | Number | Function | Search range of $x_i$ ($1\leqslant~i\leqslant~D$) | Goal |
| 1 | Sphere | $\left[-100,100\right]$ | 0.01 | |
| 2 | Rosenbrock | $\left[-2.048,2.048\right]$ | 100 | |
| Unimodal | 3 | Schwefel P2.22 | $\left[-10,10\right]$ | 0.01 |
| 4 | Quartic noise | $\left[-1.28,1.28\right]$ | 0.05 | |
| 5 | Schwefel P1.22 | $\left[-10,10\right]$ | 100 | |
| 6 | Ackley | $\left[-32,32\right]$ | 0.01 | |
| 7 | Rastrigin | $\left[-5.12,5.12\right]$ | 100 | |
| Mutimodal | 8 | Rastrigin (discrete) | $\left[-0.5,0.5\right]$ | 0.01 |
| 9 | Weierstrass | $\left[-0.5,0.5\right]$ | 0.01 | |
| 10 | Griewank | $\left[-600,600\right]$ | 0.05 | |
| 11 | Ackley (rotated) | $\left[-32,32\right]$ | 0.01 | |
| 12 | Rastrigin (rotated) | $\left[-5.12,5.12\right]$ | 100 | |
| Rotated multimal | 13 | Rastrigin (discrete and rotated) | $\left[-5.12,5.12\right]$ | 100 |
| 14 | Weierstrass (rotated) | $\left[-0.5,0.5\right]$ | 1 | |
| 15 | Griewank (rotated) | $\left[-600,600\right]$ | 0.05 | |
| Composite | 16 | Composite 1 | $\left[-5,5\right]$ | 0.01 |
| 17 | Composite 2 | $\left[-5,5\right]$ | 10 |
| Benchmark | PIO | ERPIO | EHPIO | SWPIO | SHPIO | SFPIO | SIPIO | HPIO | HPIO |
| function | ($k_c=9$) | ($k_c=9$) | ($k_c=9$) | ($k_c=9$) | (best $k_c$) | ||||
| $f_1$ | 1.92${\rm~E}-$3 | 5.74${\rm~E}-$7 | 1.35${\rm~E}-$14 | 5.73${\rm~E}-$6 | 3.25${\rm~E}-$14 | 2.22${\rm~E}-$4 | 4.66${\rm~E}-$5 | 1.05${\rm~E}-$14 $(k_c=15)$ | |
| $f_2$ | 3.11${\rm~E}$1 | 2.82${\rm~E}$1 | 2.87E1 | 2.90${\rm~E}$1 | 2.91${\rm~E}$1 | 2.87${\rm~E}$1 | 2.88${\rm~E}$1 | 2.87${\rm~E}$1 $(k_c=17)$ | |
| $f_3$ | – | 3.42${\rm~E}-$6 | 2.25${\rm~E}-$9 | 5.20${\rm~E}-$4 | 5.25${\rm~E}-$9 | 3.35${\rm~E}-$3 | 9.08${\rm~E}-$4 | 9.62${\rm~E}-$10 $(k_c=17)$ | |
| $f_4$ | 6.99${\rm~E}-$4 | 4.99${\rm~E}-$5 | 5.28${\rm~E}-$5 | 5.13${\rm~E}-$5 | 9.87${\rm~E}-$5 | 5.48${\rm~E}-$5 | 5.45${\rm~E}-$5 | 4.81${\rm~E}-$5 $(k_c=5)$ | |
| $f_5$ | 1.77 | 6.96${\rm~E}-$2 | 7.35${\rm~E}-$11 | 1.35${\rm~E}-$1 | 6.13${\rm~E}-$11 | 2.55${\rm~E}-$1 | 1.46${\rm~E}-$2 | 5.36${\rm~E}-$11 $(k_c=9)$ | |
| $f_6$ | – | 2.82${\rm~E}-$5 | 4.31${\rm~E}-$9 | 4.11${\rm~E}-$4 | 5.09${\rm~E}-$9 | 4.33${\rm~E}-$3 | 2.18${\rm~E}-$3 | 4.27${\rm~E}-$9 $(k_c=9)$ | |
| $f_7$ | 7.42${\rm~E}-$1 | 3.66${\rm~E}-$1 | 5.28${\rm~E}-$12 | 3.87${\rm~E}-$1 | 6.02${\rm~E}-$12 | 3.92${\rm~E}-$1 | 5.27${\rm~E}-$2 | 1.25${\rm~E}-$12 $(k_c=5)$ | |
| $f_8$ | 9.12${\rm~E}-$1 | 4.58${\rm~E}-$1 | 9.88${\rm~E}-$11 | 4.89${\rm~E}-$1 | 9.98${\rm~E}-$11 | 5.47${\rm~E}-$1 | 9.87${\rm~E}-$2 | 2.84${\rm~E}-$11 $(k_c=1)$ | |
| $f_9$ | 6.41${\rm~E}-$3 | 3.59${\rm~E}-$3 | 3.29${\rm~E}-$9 | 4.36${\rm~E}-$3 | 9.17${\rm~E}-$9 | 4.30${\rm~E}-$3 | 6.55${\rm~E}-$3 | 2.03${\rm~E}-$9 $(k_c=1)$ | |
| $f_{10}$ | 3.43${\rm~E}-$3 | 3.60${\rm~E}-$6 | 4.68${\rm~E}-$14 | 1.11${\rm~E}-$4 | 4.17${\rm~E}-$14 | 7.26${\rm~E}-$4 | 5.34${\rm~E}-$4 | 4.98${\rm~E}-$15 $(k_c=15)$ | |
| $f_{11}$ | – | 5.01${\rm~E}-$5 | 6.25${\rm~E}-$8 | 7.68${\rm~E}-$4 | 8.30${\rm~E}-$8 | 4.43${\rm~E}-$3 | 4.31${\rm~E}-$3 | 4.21${\rm~E}-$8 $(k_c=9)$ | |
| $f_{12}$ | 1.12 | 5.22${\rm~E}-$1 | 5.00${\rm~E}-$12 | 5.82${\rm~E}-$1 | 5.82${\rm~E}-$12 | 6.45${\rm~E}-$1 | 8.64${\rm~E}-$2 | 4.02${\rm~E}-$12 $(k_c=5)$ | |
| $f_{13}$ | 1.34 | 6.08${\rm~E}-$1 | 6.49${\rm~E}-$11 | 6.78${\rm~E}-$1 | 7.68${\rm~E}-$11 | 7.21${\rm~E}-$1 | 1.42${\rm~E}-$1 | 2.25${\rm~E}-$11 $(k_c=1)$ | |
| $f_{14}$ | 1.06${\rm~E}-$2 | 6.66${\rm~E}-$3 | 5.93${\rm~E}-$9 | 7.21${\rm~E}-$3 | 3.84${\rm~E}-$9 | 7.25${\rm~E}-$3 | 3.02${\rm~E}-$3 | 8.65${\rm~E}-$10 $(k_c=1)$ | |
| $f_{15}$ | 4.94${\rm~E}-$3 | 2.14${\rm~E}-$5 | 8.63${\rm~E}-$13 | 1.14${\rm~E}-$4 | 9.51${\rm~E}-$13 | 6.32${\rm~E}-$4 | 5.04${\rm~E}-$4 | 1.06${\rm~E}-$13 $(k_c=9)$ | |
| $f_{16}$ | 1.34 | 8.99${\rm~E}-$1 | 5.26${\rm~E}-$1 | 9.00${\rm~E}-$1 | 6.00${\rm~E}-$1 | 9.21${\rm~E}-$1 | 6.55${\rm~E}-$1 | 1.68${\rm~E}-$1 $(k_c=17)$ | |
| $f_{17}$ | 2.66 | 1.32 | 8.12${\rm~E}-$1 | 9.57${\rm~E}-$1 | 8.62${\rm~E}-$1 | 9.68${\rm~E}-$1 | 9.59${\rm~E}-$1 | 1.01${\rm~E}-$1 $(k_c=17)$ |
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