On the mean‐field limit for the consensus‐based optimization

摘要

This paper is concerned with the large particle limit for the consensus‐based optimization (CBO), which was postulated in the pioneering works by Carrillo, Pinnau, Totzeck and many others. In order to solve this open problem, we adapt a compactness argument by first proving the tightness of the empirical measures {μN}N≥2$$ {left{{mu}amp;amp;#x0005E;Nright}}_{Nge 2} $$ associated to the particle system and then verifying that the time marginal of the limit measure μ$$ mu $$ is the unique weak solution to the mean‐field CBO equation. Such results are further extended to the model of particle swarm optimization (PSO).