Constriction PSO

Early PSO studies showed that particle velocities may grow without a bound, oscillate around the optimum, and/or fail to converge.

To combat, a constriction factor was introduced to ensure stable particle trajectories. The key idea here is that the we scale the velocity update so that the particle motion is damped and converges.

Standard PSO:

$$v_i(t+1)=w{v}_i(t)+c_1{r}_1(p_i-x_i)+c_2{r}_2(g-x_i)$$

Constriction PSO:

$$v_i(t+1)=\mathcal{X}[v_i(t)+c_1{r}_1(p_i-x_i)+c_2{r}_2(g-x_i)]$$

where $\mathcal{X}$ is the constriction coefficient. It reduces the velocity magnitude and stabilizes swarm dynamics.

Let $\varphi =c_1+c_2$. The constriction factor is given as:

$$\mathcal{X}=\frac{2}{\left|2-\varphi -\sqrt{{\varphi }^2-4\varphi }\right|}$$

with the stability condition of $\varphi >4$.

A common parameter configuration is $c_1=c_2=2.05$, which gives $\varphi =4.1$ and $\mathcal{X}\approx 0.729$.

The velocity update is then:

$$v_i(t+1)=0.729[v_i(t)+2.05r_1(p_i-x_i)+2.05r_2(g-x_i)]$$