Attractive and Repulsive Fully Informed Particle Swarm Optimization based on the modified Fitness Model

Abstract

A novel Attractive and Repulsive Fully Informed Particle Swarm Optimization based on the modified Fitness Model (ARFIPSOMF) is presented. In ARFIPSOMF, a modified fitness model is used as a self-organizing population structure construction mechanism. The population structure is gradually generated as the construction and the optimization processes progress asynchronously. An attractive and repulsive interacting mechanism is also introduced. The cognitive and the social effects on each particle are distributed by its ‘contextual fitness’ value $F$. Two kinds of experiments are conducted. Results focusing on the optimization performance show that the proposed algorithm maintains stronger diversity of the population during the convergent process, resulting in good solution quality on a wide range of test functions, and converge faster. Moreover, the results concerning on topologic characteristics of the population structure indicate that (1) the final population structures developed by optimizing different test functions differ, which is an important for improving ARFIPSOMF performance, and (2) the final structures developed by optimizing some test functions exhibit scale-free property approximately.

Appendix A

1.1 A.1 Convergence analysis of ARFIPSO

To analyze the convergence of ARFIPSO, the theory of stability of linear system is used in this section. For analysis purpose, consider the situation that $P_g$ keeps constant during a period of time. Due to particle $i$ chosen randomly, results can be applied to all other particles. In addition, due to each dimension updated independently from others, without loss of generality, the one-dimensional case is used for convergence analysis of ARFIPSO. The Eqs. (8) and (9) can be transformed as:

$$\begin{aligned}&v_i (t+1)=wv_i (t)+(1-F_i (t)) \times c_i \nonumber \\&\times \left( \sum \limits _{\begin{array}{c} j\in n(B(i)) j\ne g j\ne i \end{array}} {r_j (p_j (t)-x_i (t))} -\sum \limits _{j\in n(W(i))} {r_j (p_j (t)-x_i (t))}\right) \, \nonumber \\&\,+F_i (t) \times \beta \times (r_g (p_g (t)-x_i (t))+r_i (p_i (t)-x_i (t))) \end{aligned}$$

(19)

$$\begin{aligned} x_i (t+1)=x_i (t)+v_i (t+1) \end{aligned}$$

(20)

Then, Eq. (19) becomes:

$$\begin{aligned} v_i (t+1)&=wv_i (t)-\left( \left( 1-F_i (t)c_i \right) \right. \nonumber \\&\quad \left. \left( \sum \limits _{j\in B(i)} {r_j }-\sum \limits _{j\in {W}(i)} {r_j } \right) \right) \nonumber \\&\quad +F_i (t)\beta r_g +F_i (t)\beta r_j )x_i (t) \nonumber \\&\quad \,+((1-F_i (t)c_i )\left( \sum \limits _{j\in {B}(i)} {r_j } -\sum \limits _{j\in {W}(i)} {r_j } )\right) p_j \nonumber \\&\quad +F_i (t)\beta r_g p_g +F_i (t)\beta r_j p_i \, \end{aligned}$$

(21)

Substituting Eqs. (20) and (21) into Eq. (22), the iterative process is obtained as follows:

$$\begin{aligned}&x_i (t\!+\!1)\!-\!\left( 1\!+\!w\!-\!(((1\!-\!F_i (t)c_i )\left( \sum \limits _{j\in {B}(i)} {r_j } \!-\!\sum \limits _{j\in {W}(i)} {r_j } \right) \right) \nonumber \\&\qquad + F_i (t)\beta r_g +F_i (t)\beta r_j))x_i (t)+wx_i (t-1) \nonumber \\&\quad = \left( (1-F_i (t)c_i )\left( \sum \limits _{j\in B(i)} {r_j } -\sum \limits _{j\in W(i)} {r_j } \right) \right) p_j \nonumber \\&\qquad + F_i (t)\beta r_g p_g +F_i (t)\beta r_j p_i \end{aligned}$$

(22)

Equation (22) can be viewed as a two-order discrete system with $((1 - F_{i} (t)c_{i} )(\sum \nolimits _{{j \in {{B}}(i)}} {r_{j} } - \sum \nolimits _{{j \in {{W}}(i)}} {r_{j} } ){\text {)}}p_{j} { + {F}}_{{{i}}} (t)\beta r_{g} p_{g} + {{F}}_{{{i}}} (t)\beta r_{j} p_{i}$ as the input.

To analyze the convergent condition of sequence $\{Ex_i (t)\}$ $(Ex_i (t)$ is the expectation of random variable $x_i (t))$, Eq. (23) is obtained from Eq. (22):

$$\begin{aligned}&Ex_i (t+1)-(1+w-(((1-F_i (t)c_i )(\frac{1}{2}\left| {n(B(i))} \right| \nonumber \\&\qquad -\frac{1}{2} \left| {n(W(i))} \right| ))+\frac{1}{2}F_i (t)\beta \nonumber \\&\qquad +\frac{1}{2}{F}_i (t)\beta ))Ex_i (t)+wEx_i (t-1) \nonumber \\&\quad =((1-F_i (t)c_i )\left( \frac{1}{2}\left| {n(B(i))} \right| -\frac{1}{2}\left| {n(W(i))} \right| )\right) p_j \nonumber \\&\qquad + \frac{1}{2}{F}_i (t)\beta p_g+\frac{1}{2}{F}_i (t)\beta p_i \end{aligned}$$

(23)

Let $\phi =\frac{1}{2}((1-F_i (t)c_i )(\left| {n(B(i))} \right| -\left| {n(W(i))} \right| ))+\frac{1}{2}{F}_i (t)\beta +\frac{1}{2}{F}_i (t)\beta )$.

The characteristic equation of the iterative process shown in Eq. (23) is

$$\begin{aligned} \lambda ^2-(1+w-\phi )\lambda +w=0 \end{aligned}$$

(24)

According to the theory of stability of linear system, the convergent condition of iterative process $\{Ex_i (t)\}$ is that absolute values of eigenvalues $\lambda _{1}$ and $\lambda _{2}$ are both less than 1. That is,

$$\begin{aligned} \left| {\frac{1+w-\phi \pm \sqrt{(1+w-\phi )^2-4w} }{2}} \right| <1 \end{aligned}$$

(25)

Thus, the convergent condition of iterative process $\{Ex_i (t)\}$ is written as:

$$\begin{aligned}&0 \le w {<} 1\;{\text {and}}\;0 {<} ((1 - F_{i} (t)c_{i} )(\left| {n(B(i)} \right| ) - \left| {n(W(i))} \right| ))\nonumber \\&\quad + F_{{{i}}} (t)\beta + {{F}}_{{{i}}} (t)\beta ) < 4(1 + w) \end{aligned}$$

(26)

Next, the Eq. (26) is analyzed according the value of $F_i$ in detail:

• 1. If $F_i =1$, then $0<F_i \beta <2(1+w)$. So, particle $i$ is convergent when $0<\beta <2(1+w)$.

• 2. If $F_i =0$, then $0<c_i \times (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )<4(1+w)$. Further analysis is given as follows:

  • 1) If particle $i$ satisfies $\left| {n(B(i))} \right| -\left| {n(W(i))} \right| \le 0$, then it is divergent.

  • 2) If particle $i$ satisfies $\left| {n(B(i))} \right| -\left| {n(W(i))} \right| >0$, then $0<c_i <\frac{4(1+w)}{(\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )}$. Moreover, $0\le w<1, \min (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )=1, \min \,4(1+w)=4$ and $\max \,4(1+w)=8$. So,

  • [1 ] When $c_i \ge 8$, $c_i =k_i \ge 8\ge \frac{\max 4(1+w)}{\min (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )}$. Thus, particle $i$ is divergent;

  • [2 ] When $c_i \le 2$, i.e., $k_i \le 2$, then $\max (\left| {n(B(i))} \right| \!-\!\left| {n(W(i))} \right| )\!=\!2$ and $c_i \!=\!k_i \!\le \! 2\!\le \! \frac{\min \,4(1\!+\!w)}{\max \,(\left| {n(B(i))} \right| \!-\!\left| {n(W(i))} \right| )}$. Thus, the particle $i$ is convergent;

  • [3 ] $2<c_i =k_i <8$. Thus, particle $i$ may be convergent or divergent.

• 3. If $0<F_i <1$ then $0<(1-F_i ) \times c_i \times (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )+2F_i \beta <4(1+\omega )$.

Further analysis is given as follows:

• 1. When $\max (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )=2$ and $c_i =\beta$, because $\min \,4(1+\omega )=4$, then $0<c_i \times \max (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )<4$. Thus, when $0<c_i =k_i \le 2$, particle $i$ is convergent

• 2. When $\min (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )=1$ and $c_i =\beta$, because $\max \,4(1+\omega )=8$ then $0<c_i \times \min (\left| {n(B(i))} \right| -\left| {n(W(i))} \right| )<8$. Thus, when $c_i =k_i \ge 8$, particle $i$ is divergent

• 3. When $2<c_i =k_i <8$ particle $i$ may be convergent or divergent.

To summarize, when the acceleration coefficient $c$ is set as the smaller value, nodes with fewer connections are more likely to converge; when the acceleration coefficient $c$ is set as the larger value, the nodes with more connections are more likely to diverge. Thus, the convergence analysis of ARFIPSO can be used to determine the parameters setting in next experiments.

1.2 A.2 Global convergence analysis of ARFIPSOMF

Solis and Wets (1981) provide the conditions which the stochastic optimization algorithm converges at the global optimum with probability 1. Major conclusions are summarized as follows:

Hypothesis 1

If $f(D(z,\xi ))\le f(z)$ and $\xi \in \Omega$, then $f(D(z,\xi ))\le f(\xi )$

Here, $D$ is the function for generating problem solutions, $\xi$ is a random vector from probability space $(R^n,B,u_k )$, $f$ is an objective function, $\Omega$ is the constraint solution space of the problem and $\Omega \subseteq R^n$, $u_k$ is the probability measure on $B$ and $B$ is the $\sigma$ field of $R^n$ subset.

Hypothesis 2

If for $\forall A(A$is Borel set of$\Omega )$, $v(A)>0$, then

$$\begin{aligned} \prod \limits _{k=0}^\infty {(1-\mu _k [A])} =0 \end{aligned}$$

(27)

$v(A)$ is $n$-dimension closure of $A$,$u_k (A)$ is the probability of $u_k$ generating $A$.

Theorem 1

If $f$ is a measurable function, $\Omega$ is a measurable subset of $R^n$ and $\left\{ {z_k } \right\} _0^\infty$ is the solution sequence generated by stochastic optimization algorithm, then the following formula (28) is found when Hypotheses 1 and 2 are satisfied:

$$\begin{aligned} \mathop {\lim }\limits _{k\rightarrow +\infty } \,P[z_k \in R_\varepsilon ]=1 \end{aligned}$$

(28)

$R_\varepsilon$ is the global optimum set.

Function $D$ is defined as:

$$\begin{aligned} D(P_g (t),x_i (t))=\left\{ \begin{array}{l} P_g (t),\,f(P_g (t))\le f(x_i (t)) \\ x_i (t),\,f(P_g (t))>f(x_i (t)) \\ \end{array} \right. \end{aligned}$$

(29)

We can prove that the formula (29) satisfies Hypothesis 1. Furthermore, when ARFIPSOMF searches stagnantly, for new particle $i$, $M_{i,t} =\Omega$ and for other particle $l$,

$$\begin{aligned} M_{l,t}&=x_{lk} (t)+wv_{lk} (t)+(1-F_l ) \times c_l\\&\quad \times \left( \sum \limits _{\begin{array}{l} j\in n({B}(l)) j\ne g j\ne l \end{array}} {r_{jk} (p_{jk} (t)-x_{lk} (t))}\right. \\&\quad \left. -\sum \limits _{j\in n({W}(l))} {r_{jk} (p_{jk} (t)-x_{lk} (t))} \right) \\&\quad +F_l \times \beta \times (r_{gk} (p_g (t){-}x_{lk} (t)){+}r_{lk} (p_l (t)-x_{lk} (t))) \end{aligned}$$

Thus, $\Omega \subseteq \mathop {{U}}\nolimits _{\begin{array}{c} l = 1 \\ l \ne i \end{array}}^{{N(t)}} \;M_{{l,t}} \;{{U}}\;M_{{i,t}}$.

Set $A=M_{i,t}$, where $A$ is the Borel subset of $\Omega$. Thus, $\nu [A]>0$ and $\mu _t [A]=\sum \nolimits _{i=0}^{N(t)} {\mu _{i,t} [A]=1}$. Therefore, Hypothesis 2 is satisfied. Because Hypotheses 1 and 2 and Theorem 1 are satisfied, ARFIPSOMF can converge at the global optimum with probability 1.