Order-2 Stability Analysis of Particle Swarm Optimization

@article{Liu2015Order2SA,
  title={Order-2 Stability Analysis of Particle Swarm Optimization},
  author={Qunfeng Liu},
  journal={Evolutionary Computation},
  year={2015},
  volume={23},
  pages={187-216}
}
  • Qunfeng Liu
  • Published 1 June 2015
  • Mathematics
  • Evolutionary Computation
Several stability analyses and stable regions of particle swarm optimization (PSO) have been proposed before. The assumption of stagnation and different definitions of stability are adopted in these analyses. In this paper, the order-2 stability of PSO is analyzed based on a weak stagnation assumption. A new definition of stability is proposed and an order-2 stable region is obtained. Several existing stable analyses for canonical PSO are compared, especially their definitions of stability and… 
Particle Swarm Optimization: Understanding Order-2 Stability Guarantees
TLDR
It is shown that the definition of order-2 stability which accurately reflects PSO behavior is that of convergence in second order moment to a constant, and not to zero.
Stability Analysis of the Particle Swarm Optimization Without Stagnation Assumption
TLDR
It is proved that the convergence of expectation and variance of the positions generated by the stochastic recurrence relation is independent of the mean and varianceof the distribution of the personal and global best vectors.
Particle swarm stability: a theoretical extension using the non-stagnate distribution assumption
TLDR
An extension of the state of the art theoretical model utilized for understanding the stability criteria of the particles in particle swarm optimization algorithms is presented, showing the condition that the expected value and variance of a particle’s personal and neighborhood best positions are convergent sequences is shown to be a necessary condition for order-1 and order-2 stability.
Analysis of Stability, Local Convergence, and Transformation Sensitivity of a Variant of the Particle Swarm Optimization Algorithm
TLDR
This paper identifies boundaries of coefficients for this algorithm that ensure particles converge to their equilibrium and investigates the local convergence property of this algorithm and proves that the original standard PSO algorithm is not sensitive to rotation, scaling, and translation of the search space.
Fitness-distance-ratio particle swarm optimization: stability analysis
TLDR
This paper theoretically derives the conditions necessary for order-1 and order-2 stability under the well known stagnation assumption and validates its theoretical findings against an assumption free FDR-PSO algorithm.
Particle swarm variants: standardized convergence analysis
TLDR
It was found that using a specially designed objective function for convergence analysis is both a simple and valid method for performing assumption free convergence analysis.
Particle swarm optimizer: The impact of unstable particles on performance
TLDR
It is shown empirically that a majority of PSO parameters that are theoretically unstable perform worse than a trivial random search across 28 objective functions, and across various dimensionalities.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 33 REFERENCES
Particle swarm variants: standardized convergence analysis
TLDR
It was found that using a specially designed objective function for convergence analysis is both a simple and valid method for performing assumption free convergence analysis.
The generalized PSO: a new door to PSO evolution
A generalized form of the particle swarm optimization (PSO) algorithm is presented. Generalized PSO (GPSO) is derived from a continuous version of PSO adopting a time step different than the unit.
Stability analysis of the particle dynamics in particle swarm optimizer
TLDR
Simulation results confirm the prediction from theory that stability of the particle dynamics requires increasing the maximum value of the random parameter when the inertia factor is reduced.
A study of particle swarm optimization particle trajectories
A Study of Collapse in Bare Bones Particle Swarm Optimization
  • T. Blackwell
  • Computer Science
    IEEE Transactions on Evolutionary Computation
  • 2012
TLDR
It is conjectured that, subject to spread, stability and no-collapse, there is a single encompassing particle swarm paradigm, and that an important aspect of parameter tuning within any particular manifestation is to remove any deleterious behavior that ensues from the dynamics.
The particle swarm - explosion, stability, and convergence in a multidimensional complex space
TLDR
This paper analyzes a particle's trajectory as it moves in discrete time, then progresses to the view of it in continuous time, leading to a generalized model of the algorithm, containing a set of coefficients to control the system's convergence tendencies.
Mean and Variance of the Sampling Distribution of Particle Swarm Optimizers During Stagnation
  • R. Poli
  • Computer Science, Mathematics
    IEEE Transactions on Evolutionary Computation
  • 2009
TLDR
A novel method is introduced that allows us to exactly determine all the characteristics of a PSO sampling distribution and explain how it changes over any number of generations, in the presence stochasticity.
Particle swarm optimization: surfing the waves
  • E. Ozcan, C. Mohan
  • Computer Science
    Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406)
  • 1999
TLDR
This paper takes the next step, generalizing to obtain closed form equations for trajectories of particles in a multi-dimensional search space.
...
1
2
3
4
...