QPMSO-Type 2
Attractor $\boldsymbol{\Omega}^{t}_{i,s}$
Type-1 particles are closely related to Type-2 particles of the single-swarm QPSO algorithm (see ). The stochastic attractor takes the following form:
\[\boldsymbol{\Omega}^{t}_{i,s} = \beta \boldsymbol{A}^{t}_{i,s}+ (1-\beta)\boldsymbol{\Gamma}^{t} _{i,s}\]$\boldsymbol{A}^{t} _{i,s}$ is the stochastic attractor of the single phase QPSO, now defined in terms of the best candidate solution ever explored by the $s$-th swarm:
\[\boldsymbol{A}^{t}_{i,s} = \boldsymbol{\varphi}^{t} _{i,s} \boldsymbol{x}_{i,s,\text{p-best}} + \left( 1-\boldsymbol{\varphi}^{t} _{i,s} \right) \boldsymbol{x}_\text{s-best}^{t} ~~~\text{where:}~~~ \boldsymbol{\varphi}^{t} _{i,s}\sim \boldsymbol{U}(0,1).\]Therein, $\alpha$ is the contraction–expansion (CE) coefficient controlling the convergence speed of the algorithm. Note that by introducing multiple swarms, the CE parameter $\alpha$ now controls the “inner-swarm” behaviour.
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.
In order to let the different swarms benefit from the experience of the other swarms, we define a second attractor $\boldsymbol{\Gamma}^{t} _{i,s}$, which directs the less successful swarms towards the most successful swarm.
\[\boldsymbol{\Gamma}^{t} _{i,s} = \boldsymbol{\varphi}^{t} _{i,s} \boldsymbol{x}_{i,s,\text{p-best}} + \left( 1-\boldsymbol{\varphi}^{t} _{i,s} \right) \boldsymbol{x}_\text{g-best}^{t} ~~~\text{where:}~~~ \boldsymbol{\varphi}^{t} _{i,s} \sim \boldsymbol{U}(0,1).\]$\beta$ is a contraction–expansion (CE) coefficient similar to $\alpha$ controlling the convergence speed towards the so-far found global minimum. $0 \leq \beta \leq 1$ holds and for $\beta=1$ the single-swarm QPSO-Type2 algorithm is recovered. Assuming the same $\boldsymbol{\varphi}^{t} _{i,s}$ for both attractors, above the function of motion of the particles can be simplified to:
\[\begin{aligned} \boldsymbol{\Omega}^{t}_{i,s} &= \beta \boldsymbol{A}^{t}_{i,s} + (1-\beta)\boldsymbol{\Gamma}^{t} _{i,s} \\ &= \boldsymbol{\varphi}^{t} _{i,s} \boldsymbol{x}_{i,s,\text{p-best}} + \left( 1-\boldsymbol{\varphi}^{t} _{i,s} \right) \left( \beta \boldsymbol{x}_\text{s-best}^{t} + (1-\beta) \boldsymbol{x}_\text{g-best}^{t} \right) \end{aligned}\]Characteristic length $\boldsymbol{L}^{t}_i$
\[\boldsymbol{L}^{t}_i = 2 \alpha \left( \beta \left| \boldsymbol{x}^{t}_i - \boldsymbol{x}^{t}_{\text{ms-best}} \right| + (1-\beta) \left| \boldsymbol{x}^{t}_i - \boldsymbol{x}^{t}_{\text{g-best}} \right| \right) ~~~ \text{where}~~~ \boldsymbol{x}^{t}_{\text{ms-best}} = \frac{1}{np} \sum^{np}_{i=1} \boldsymbol{x}^{t}_{\text{i,s,p-best}}\]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.