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## Exact analysis of the sampling distribution for the canonical particle swarm optimiser and its convergence during stagnation

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Venue: | In Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation |

Citations: | 7 - 3 self |

### Citations

852 | The particle swarm-explosion, stability, and convergence in a multidimensional complex space,”
- Clerc, Kennedy
- 2002
(Show Context)
Citation Context ...the case for the best particle in a neighbourhood. The work was extended in [4] where mul134tiple multi-dimensional particles were covered. Similar assumptions were used by Clerc and Kennedy’s model =-=[5]-=-: one particle, one dimension, deterministic behaviour and stagnation. Under these conditions the swarm is a discrete-time linear dynamical system. The dynamics of the state (position and velocity) of... |

189 |
Fundamentals of computational swarm intelligence,
- Engelbrecht
- 2005
(Show Context)
Citation Context ...matrix. The model, therefore, predicts that the particle will converge to equilibrium if the magnitude of the eigenvalues is smaller than 1. A similar approach was used by van den Bergh [6] (see also =-=[2]-=-), who, again, modelled one particle, with no randomness and during stagnation. As in previous work, van den Bergh provided an explicit solution for the trajectory of the particle. He showed that the ... |

188 | Population structure and particle swarm performance,” - Kennedy, Mendes - 2002 |

183 |
The Particle Swarm Optimization Algorithm: Convergence Analysis and Parameter Selection,”
- Trelea
- 2003
(Show Context)
Citation Context ...ibrational frequency. Like the original model, this model assumes one particle, one dimension, no randomness and stagnation. Under the same assumptions as [5] and following a similar approach, Trelea =-=[11]-=- performed a lucid analysis of a 4parameter family of particle models and identified regions in the parameter space where the model exhibits qualitatively different behaviours (either stability, harmo... |

170 | An Analysis of Particle Swarm Optimizers,
- Bergh
- 2002
(Show Context)
Citation Context ...ormation on StdDev[xt], previous research has effectively assumed that limt→∞ E[xt] = p would eventually drive StdDev[xt] to zero. This assumption has, for example, been used in the proof provided in =-=[6]-=- and [2] that the PSO is not guaranteed to be an optimiser. However, as we have shown in this work, limt→∞ StdDev[xt] =0onlyify=ˆy, andso whether or not the PSO is an optimiser is still effectively a ... |

69 |
Bare bones particle swarms, in:
- Kennedy
- 2003
(Show Context)
Citation Context ...n finally rewrite as r psd = 2 (ν − μ2 ) · ˆy − y Δ ˛ 2 ˛ (26) Hence the search continues unless y =ˆy. It is interesting to note that the observation that led to the definition of the bare-bones PSO =-=[18]-=- that the standard deviation of the search distribution is proportional to ˛ ˆy−y ˛ 2 was fundamentally correct. There is, however, a multiplicative factor, p 2(ν − μ2 )/Δ, in Equation (26) which depe... |

57 | Particle swarm optimization: surfing the waves, in:
- Ozcan, Mohan
- 1999
(Show Context)
Citation Context ...on, in one dimension, in the absence of stochasticity and during stagnation. Also, y and ˆy were assumed to coincide, as is the case for the best particle in a neighbourhood. The work was extended in =-=[4]-=- where mul134tiple multi-dimensional particles were covered. Similar assumptions were used by Clerc and Kennedy’s model [5]: one particle, one dimension, deterministic behaviour and stagnation. Under... |

55 | The behavior of particles, in: - Kennedy - 1998 |

36 |
Stability analysis of the particle dynamics in particle swarm optimizer,
- Kadirkamanathan, Selvarajah, et al.
- 2006
(Show Context)
Citation Context ...he was able to show that a particle’s new velocity is the sum of three components: a forward force, a backward force and noise. Clerc studied the distributions of these forces. Kadirkamanathan et al. =-=[17]-=- were able to study the stability of particles in the presence of stochasticity by using Lyapunov stability analysis. They considered the behaviour of a single particle – the swarm best – with inertia... |

34 | Analysis of a simple particle swarm optimization system,
- Ozcan, Mohan
- 1998
(Show Context)
Citation Context ... considering simplifying assumptions such as isolated single individuals, search stagnation (i.e., no improved solutions are found) and, crucially, absence of randomness. For example, Ozcan and Mohan =-=[3]-=- studied the behaviour of one particle, in isolation, in one dimension, in the absence of stochasticity and during stagnation. Also, y and ˆy were assumed to coincide, as is the case for the best part... |

29 | Population Topologies and Their Influence in Particle Swarm Performance - Mendes - 2004 |

25 |
Adaptive particle swarm optimization, in
- Yasuda, Ide, et al.
(Show Context)
Citation Context ...n a point that is neither the global optimum nor indeed a local optimum. This implies that a PSO is not guaranteed to be an optimiser. A simplified model of particle was also studied by Yasuda et al. =-=[7]-=-. The assumptions were: one one-dimensional particle, stagnation and absence of stochasticity. Inertia was included in the model. Again an eigenvalue analysis of the resulting dynamical system was per... |

17 | Stagnation analysis in particle swarm optimization or what happens when nothing happens,”
- Clerc
- 2006
(Show Context)
Citation Context ...nextricably on the specific details of the fitness function, they were able to study in detail only the free response. To better understand the behaviour of the PSO during phases of stagnation, Clerc =-=[16]-=- analysed the distribution of velocities of one particle controlled by the standard PSO update rule with inertia and stochastic forces. Inparticular, he was able to show that a particle’s new velocity... |

15 |
Particle swarms and population diversity,”
- Blackwell
- 2005
(Show Context)
Citation Context ...the aim of determining for what parameter settings the systems is stable and what classes of behaviours are possible for a particle. Conditions for cyclic behaviour were analysed in detail. Blackwell =-=[19]-=- investigated how the spatial extent of a particle swarm varies over time. A simplified swarm model was adopted which is an extension of the one by Clerc and Kennedy where more than one particle and m... |

13 |
Particle swarm optimization–mass-spring system analogon,
- Brandstatter, Baumgartner
- 2002
(Show Context)
Citation Context ...e sense that they could change their personal best. Constriction was included but not stochasticity. [19] suggested that spatial extent decreases exponentially with time. Brandstätter and Baumgartner =-=[10]-=- drew an analogy between Clerc and Kennedy’s model [5] and a damped massspring oscillator, making it possible to rewrite the model using the notions of damping factor and natural vibrational frequency... |

12 | Dynamic system analysis and initial particles position in particle swarm optimization
- Campana, Fasano, et al.
- 2006
(Show Context)
Citation Context ...tively different behaviours (either stability, harmonic oscillations or zigzagging behaviour). The dynamical system approach proposed by Clerc and Kennedy has recently been extended by Campana et al. =-=[12, 13]-=- who studied an extended PSO. Under the assumption that no randomness is present, the resulting model is a discrete, linear and stationary dynamical system, for which [12, 13] formally expressed the f... |

9 | Adaptive Particle Swarm Optimization Using Velocity Feedback - Iwasaki, Yasuda |

9 | Particle swarms and population diversity I : Analysis - Blackwell - 2003 |

8 | Particle Swarm Optimization: Efficient Globally Convergent Modifications,”
- Campana, Fasano, et al.
(Show Context)
Citation Context ...tively different behaviours (either stability, harmonic oscillations or zigzagging behaviour). The dynamical system approach proposed by Clerc and Kennedy has recently been extended by Campana et al. =-=[12, 13]-=- who studied an extended PSO. Under the assumption that no randomness is present, the resulting model is a discrete, linear and stationary dynamical system, for which [12, 13] formally expressed the f... |