Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 54, Issue 3, Pages 3091-3121Publisher
SIAM PUBLICATIONS
DOI: 10.1137/21M1412323
Keywords
swarm optimization; consensus based optimization; Laplace's principle; tightness
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This study proposes a continuous description of particle swarm optimization (PSO) based on stochastic differential equations, and formally demonstrates the connection between PSO and consensus-based optimization (CBO) through the zero-inertia limit. By rigorously deriving CBO from PSO through the limit of zero inertia, a quantified convergence rate is obtained. The results are illustrated with numerical examples.
Recently a continuous description of particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in [Math. Models Methods Appl. Sci., 31 (2021), pp. 1625--1657] where the authors formally showed the link between PSO and the consensus-based optimization (CBO) through the zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in [S. Grassi and L. Pareschi, Math. Methods Appl. Sci., 31 (2021), pp. 1625--1657] by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating both the weak and strong convergence of the corresponding stochastic differential equations of Mckean type in the continuous path space and the results are illustrated with some numerical examples.
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