A continuous-time limit for particle swarm optimization

Abstract

Particle swarm optimization (PSO) is a popular class of gradient-free methods for global optimization. We study a continuous-time model for momentum-free PSO with personal-best memory and global-best interaction. Unlike previous models which rely on smooth approximations, we keep the original discontinuous personal- and global-best processes. We study the small-step limit and show that weak limit points satisfy a stochastic differential inclusion: personal-best variables obey a relaxed record condition, while fast switching of the global-best selector leads to a convexified interaction. We then give conditions under which the true original selectors are recovered. Under suitable regularity and nondegeneracy assumptions, Brownian noise allows us to identify the personal-best processes with the latest record positions of the limiting trajectories and to prove that the particle attaining the global best is unique at almost every time. The resulting continuous-time dynamics retain both selection rules without smoothing them.

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BibTeX RIS

BibTeX

@misc{bianchi2026continuoustime,
  title = {{A continuous-time limit for particle swarm optimization}},
  author = {Pascal Bianchi and Radu-Alexandru Dragomir and Yerkin Yesbay},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07373},
  eprint = {2610.07373},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07373},
}

RIS

TY  - UNPB
TI  - A continuous-time limit for particle swarm optimization
AU  - Pascal Bianchi
AU  - Radu-Alexandru Dragomir
AU  - Yerkin Yesbay
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.07373
UR  - https://arxiv.org/abs/2610.07373
AB  - Particle swarm optimization (PSO) is a popular class of gradient-free methods for global optimization. We study a continuous-time model for momentum-free PSO with personal-best memory and global-best interaction. Unlike previous models which rely on smooth approximations, we keep the original discontinuous personal- and global-best processes. We study the small-step limit and show that weak limit points satisfy a stochastic differential inclusion: personal-best variables obey a relaxed record condition, while fast switching of the global-best selector leads to a convexified interaction. We then give conditions under which the true original selectors are recovered. Under suitable regularity and nondegeneracy assumptions, Brownian noise allows us to identify the personal-best processes with the latest record positions of the limiting trajectories and to prove that the particle attaining the global best is unique at almost every time. The resulting continuous-time dynamics retain both selection rules without smoothing them.
ER  -

Abstract

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