QPMSO-Type 4

For type-4 particles, we introduce the concept of “elite particles” into the QPSMO. Elite particles denote the particles with the best position of a swarm. Unlike the rest of the particles in the swarm, the elite particles have the ability to communicate with elite particles from the other swarms and thus learn from their successes/failures.

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

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

where $\boldsymbol{x}^{t}_{\text{me-best}}$ is the mean best position of all elite particles. With the elite particle being defined as the best particle of each swarm, the mean best position is calculated as follows:

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

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

\[\boldsymbol{L}^{t}_{i,s} = \begin{cases} 2 \beta \left| \boldsymbol{x}^{t}_{i,s} - \boldsymbol{x}^{t}_{\text{me-best}} \right| & \text{if elite particle} \\ 2 \alpha \left| \boldsymbol{x}^{t}_{i,s} - \boldsymbol{x}^{t}_{\text{ms-best}} \right| & \text{otherwise} \end{cases}\]

Therein, $\alpha$ and $\beta$ are the contraction–expansion (CE) coefficients controlling the convergence speed of the algorithm. In fact, $\alpha$ is the identical to the CE parameter of the single swarm QPSO.

In [4] it has been shown that linearly decreasing $\alpha$ from $1.0$ to $0.5$ during the course of the optimisation leads to a good performance of the algorithm in general.

$\beta$ is a new parameter controlling the contraction-expansion behaviour of the elite particles. To find a suitable range for $\beta$, an experimental study on different functions of the CEC 2017 event is conducted in the subsequent section.

Type-4b Particles

Type-4b particles are a variant of Type-4 particles.

Finding a suitable range for $\beta$

To evaluate a suitable range for the parameter $\beta$ 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 $\beta$. For more details on the target functions see Section 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 $\beta$). 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.

Type-4 Particles

Type-4b Particles

The investigation performed here suggests that the calibration success is insensitive to the choice of the $\beta$ parameter. If at all, only slight advantages can be identified. For example, if a linear decrease of $\beta$, starting at 1 towards 0.5 is chosen, then this corresponds to the recommendation for the choice of the parameter $\alpha$ in [1]. This would allow the substitution $\beta = \alpha$, which allows the extension of the single swarm algorithm to a multi-swarm algorithm based on type-4 particles without the introduction of a new parameter. However, for the sake of generality we keep the parameter $\beta$ and recommend to use the same linear decrease as for the parameter $\alpha$.

Influence of swarm size

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.

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