# Particle Swarm Optimisation: A Historical Review Up to the Current Developments

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## Abstract

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## 1. Introduction

## 2. Particle Swarm Optimisation

#### The Gbest and Lbest Models

## 3. Modifications to the Particle Swarm Optimisation

#### 3.1. Algorithm Convergence Improvements

#### 3.1.1. The Inertia Weight Parameter

#### 3.1.2. The Constriction Factor

#### 3.2. Neighbourhoods

#### 3.2.1. Static Neighbourhood

#### 3.2.2. Dynamic Neighbourhood

#### 3.2.3. Near Neighbour Interactions

#### 3.3. The Stagnation Problem

## 4. Particle Swarm Optimisation Variants

#### 4.1. Cooperative Particle Swarm Optimisation

#### Two Steps Forward, One Step Back

#### 4.2. Adaptive Particle Swarm Optimisation

#### 4.3. Constrained Optimisation Problems

#### 4.4. Multi-Objective Optimisation

#### 4.5. Multimodal Function Optimisation

#### 4.5.1. Objective Function Stretching

#### 4.5.2. Nbest Technique

#### 4.5.3. Subpopulations and Multi-Swarm

#### 4.6. The Fully Informed Particle Swarm Optimisation

#### 4.7. Parallel Implementations of Particle Swarm Optimisation

## 5. Connections to Other Artificial Intelligence Tools

#### 5.1. Hybrid Variants of Particle Swarm Optimisation

#### 5.1.1. Evolutionary Computation Operators

#### 5.1.2. PSO with Genetic Algorithms

#### 5.1.3. PSO With Differential Evolution

#### 5.1.4. PSO with Simulated Annealing

#### 5.1.5. PSO With Other Evolutionary Algorithms

#### 5.2. Artificial Neural Networks with Particle Swarm Optimisation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PSO | Particle Swarm Optimisation |

GA | Genetic Algorithms |

ANN | Artificial Neural Network |

FSM | Preservation of Feasible Solutions Method |

GCPSO | Guaranteed Convergence PSO |

FIPS | Fully Informed Particle Swarm |

STPSO | Stretched PSO |

APSO | Adaptive PSO |

DE | Differential Evolution |

EA | Evolutionary Algorithms |

SA | Simulated Annealing |

PPSO | Parallelized PSO |

CPU | Central Processing Unit |

GPU | Graphics Processing Unit |

COP | Constrained Optimisation Problem |

FDR | Fitness-Distance-Ratio |

MPPSO | Multi-Phase PSO |

SPSO | Standard PSO |

CPSO | Cooperative PSO |

MOPSO | Multi-Objective PSO |

SCD | Special Crowding Distance |

EPSO | Evolutionary PSO |

DEEPSO | Differential Evolutionary PSO |

ACO | Ant Colony Optimisation |

DNSPSO | Diversity Enhanced PSO with Neighborhood Search |

ABC | Artificial Bee Colony |

## Mathematical notation

D | Mean distance of each particle to other particles |

$F(\xb7)$ | Penalty function to be minimised or maximised |

$H(\xb7)$ | Function stretching for multimodal function optimisation |

${H}_{p}(\xb7)$ | Penalty factor in a penalty function |

$K(\xb7)$ | Constriction factor |

${N}_{i}$ | Neighbourhood of the particle i |

${R}_{1}$ | Cognitive uniformly distributed random vector used to compute the particle’s velocity |

${R}_{2}$ | Social uniformly distributed random vector used to compute the particle’s velocity |

S | Search space, defined by the domain of the function to be optimised, |

that contains all the feasible solutions for the problem | |

$\alpha $ | Diagonal matrix whose diagonal values are within the range of $[0,1]$ |

$\u03f5$ | Absolute different between the last and the current best fitness value, or the algorithm accuracy |

$\gamma (\xb7)$ | Power of a penalty function |

$\widehat{y}$ | Position of the best particle in the swarm or in the neighbourhood (target particle) |

$\omega (\xb7)$ | Inertia weight parameter used to compute the velocity of each particle |

$\rho $ | Diagonal matrix that represents the architecture of the swarm |

$\sigma $ | Scale parameter of Cauchy mutation |

$\tau $ | Index of the global best particle in the swarm |

$\theta (\xb7)$ | Multi-stage assignment function in a penalty function |

${\phi}_{1}$ | Cognitive real acceleration coefficient used to compute the particle’s velocity |

${\phi}_{2}$ | Social real acceleration coefficient used to compute the particle’s velocity |

${\phi}_{3}$ | Deviation real acceleration coefficient used to compute the particle’s velocity |

${\overrightarrow{P}}_{\phantom{\rule{0.222222em}{0ex}}t}^{\phantom{\rule{0.222222em}{0ex}}j}$ | Prior best position that maximises the FDR measure |

$\overrightarrow{V}$ | Particle’s velocity |

${\overrightarrow{c}}^{\phantom{\rule{0.222222em}{0ex}}j}$ | Position of the centroid of the group j |

$\overrightarrow{g}$ | Global best position of a particle in the swarm |

${\overrightarrow{p}}_{\phantom{\rule{0.222222em}{0ex}}t}^{\phantom{\rule{0.222222em}{0ex}}i}$ | Personal best position of particle i |

$\overrightarrow{x}$ | Position vector of a solution found in the search space |

${\overrightarrow{x}}_{max}$ | Upper limit of the dimension d in the search space |

${\overrightarrow{x}}_{min}$ | Lower limit of the dimension d in the search space |

$\overrightarrow{{y}^{*}}$ | Set of feasible solutions that forms the Pareto front |

d | Number of dimensions of the search space |

${e}_{f}$ | Evolutionary factor used in the APSO |

$f(\xb7)$ | Objective function to be minimised or maximised |

g | Set of inequality function constraints |

h | Set of equality function constraints |

${h}_{p}(\xb7)$ | Dynamic modified penalty value in a penalty function |

l | Number of particles in the swarm or in the neighbourhood |

m | Number of inequality constraints |

n | Non-linear modulation index |

p | Number of equality constraints |

${q}_{i}(\xb7)$ | Relative violated function of the constraints in a penalty function |

s | Number of particles in the swarm |

t | The number of the current iteration |

${w}_{g}$ | Parameter, in the form of a diagonal matrix, to add variability to the best position in the swarm |

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**MDPI and ACS Style**

Freitas, D.; Lopes, L.G.; Morgado-Dias, F. Particle Swarm Optimisation: A Historical Review Up to the Current Developments. *Entropy* **2020**, *22*, 362.
https://doi.org/10.3390/e22030362

**AMA Style**

Freitas D, Lopes LG, Morgado-Dias F. Particle Swarm Optimisation: A Historical Review Up to the Current Developments. *Entropy*. 2020; 22(3):362.
https://doi.org/10.3390/e22030362

**Chicago/Turabian Style**

Freitas, Diogo, Luiz Guerreiro Lopes, and Fernando Morgado-Dias. 2020. "Particle Swarm Optimisation: A Historical Review Up to the Current Developments" *Entropy* 22, no. 3: 362.
https://doi.org/10.3390/e22030362