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Weak convergence of particle swarm optimization

Vianney Bruned, André Mas, Sylvain Wlodarczyk

Abstract

Particle swarm optimization algorithm is a stochastic meta-heuristic solving global optimization problems appreciated for its efficacity and simplicity. It consists in a swarm of particles interacting among themselves and searching the global optimum. The trajectory of the particles has been well-studied in a deterministic case and more recently in a stochastic context. Assuming the convergence of PSO, we proposed here two CLT for the particles corresponding to two kinds of convergence behavior. These results can lead to build confidence intervals around the local minimum found by the swarm or to the evaluation of the risk. A simulation study confirms these properties.

Weak convergence of particle swarm optimization

Abstract

Particle swarm optimization algorithm is a stochastic meta-heuristic solving global optimization problems appreciated for its efficacity and simplicity. It consists in a swarm of particles interacting among themselves and searching the global optimum. The trajectory of the particles has been well-studied in a deterministic case and more recently in a stochastic context. Assuming the convergence of PSO, we proposed here two CLT for the particles corresponding to two kinds of convergence behavior. These results can lead to build confidence intervals around the local minimum found by the swarm or to the evaluation of the risk. A simulation study confirms these properties.

Paper Structure

This paper contains 14 sections, 10 theorems, 116 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. First case: oscillatory ($p\neq g$)
  4. Second case: non-oscillatory and stagnant ($p=g$)
  5. The swarm at a fixed step
  6. Simulation and numerical results
  7. Oscillatory case
  8. Non-oscillatory case
  9. Swarm at a fixed step
  10. Conclusion
  11. Derivations of the results
  12. First case: oscillatory
  13. Second case: non-oscillatory and stagnant
  14. Proof of Lemma \ref{['drift']}

Key Result

Theorem 1

Set $\mathfrak{L}=c\left( \frac{1-\omega}{1+\omega}\right) \left( 1+\omega-\frac{c}{2}\right)$ and Denote finally $\Gamma=\mathfrak{C}\cdot\mathrm{diag}\left( p-g\right)^{\odot2}$ then : where $\mathcal{N}\left( 0,\Gamma\right)$ denotes the Gaussian centered random vector of $\mathbb{R}^{d}$ with covariance matrix $\Gamma$.

Figures (13)

  • Figure 1: Display of constraint $\mathbf{A}_{3}$ in the plane $\left(\omega,c\right)$.
  • Figure 2: Display of constraint $\mathbf{B}_{1}$ in the plane $\left(c,\omega\right)$.
  • Figure 3: Contour of the Himmelblau's function in the space $\left[-6,6\right]^{2}$.
  • Figure 4: Normal probability plot of $H^{s}_{1}(N)$ on the first coordinate.
  • Figure 5: Normal probability plot of $H^{s}_{1}(N)$ on the second coordinate.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Corollary 4
  • Remark 3
  • Theorem 5
  • Remark 4
  • Corollary 6
  • ...and 4 more