QPMSO-Type 3b

Attractor $\boldsymbol{\Omega}^{t}_{i,s}$

Type-3 particles differ from Type-1 and Type-2 particles in that they only periodically orient towards the global best solution $\boldsymbol{x}_{\text{g-best}}^{t}$:

\[\boldsymbol{\Omega}^{t}_{i,s} = \begin{cases} \boldsymbol{\varphi}^{t} _{i,s} \boldsymbol{x}^{t}_{\text{g-best}} + \left( 1-\boldsymbol{\varphi}^{t} _{i,s} \right) \boldsymbol{x}^{t}_{\text{ms-best}} & \text{if update-iteration} \\ \boldsymbol{\varphi}^{t} _{i,s} \boldsymbol{x}_{i,s,\text{p-best}} + \left( 1-\boldsymbol{\varphi}^{t} _{i,s} \right) \boldsymbol{x}^{t}_{\text{ms-best}} & \text{otherwise} \end{cases}\]

Characteristic length $\boldsymbol{L}^{t}_i$

\[\boldsymbol{L}^{t}_{i,s} = \begin{cases} 2 \alpha \left| \boldsymbol{x}_{\text{g-best}}^{t} - \boldsymbol{x}^{t}_{\text{mg-best}} \right| & \text{if update-iteration} \\ 2 \alpha \left| \boldsymbol{x}^{t}_{i,s} - \boldsymbol{x}^{t}_{\text{ms-best}} \right| & \text{otherwise} \end{cases}\]

$\boldsymbol{x}^{t}_{\text{mg-best}}$ is the global mean best position and defined as follows:

\[\boldsymbol{x}^{t}_{\text{mg-best}} = \frac{1}{ns} \sum^{ns}_{i=1} \boldsymbol{x}^{t}_{\text{s-best}}\]

Above equations imply, that each particle periodically loses its memory and reorients towards the global best particle pulled back by the mean best position of the swarm.

In contrast to the Type-1 and Type-2 particles, only one parameter is required for the calculation of the Type-3 particles: the frequency with which an update iteration is performed.

Finding a suitable range for $ni^{up}$

To evaluate a suitable range for the parameter $ni^{up}$ controlling how strong the best particles of each swarm are attracted to the global best particle different functions from the CEC 2017 are optimised using different settings for $ni^{up}$. For more details on the target functions see Benchmark. The functions have been evaluated for $N=10$ and $N=30$ dimensions, respectively. The number of function evaluates was limited to $10000 \cdot N$. The optimisation of each function was repeated 100 times (per variation of $ni^{up}$). The mean value of the function values at the end of the iteration and the standard deviation are given in the two following tables for $N=10$ and $N=30$, respectively.

Results for $N=10$

References

[1] J. Sun, W. Fang, X. Wu, V. Palade, and W. Xu, ‘Quantum-Behaved Particle Swarm Optimization: Analysis of Individual Particle Behavior and Parameter Selection’, Evolutionary Computation, vol. 20, no. 3, pp. 349–393, Sep. 2012, doi: 10.1162/EVCO_a_00049.

---