An analytical framework for consensus-based global optimization method

Math. Models Methods Appl. Sci.

Tse³

et al. 2017

Self Cite

130

208

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

Section: A Consensus-based Optimization (Cbo) Algorithmmentioning

confidence: 99%

“…Despite its simplicity, we will argue that the algorithm performs surprisingly well and allows for an analytical treatment as the system possesses a mean-field limit towards a partial differential equation (cf. [13]).…”

Section: A Consensus-based Optimization (Cbo) Algorithmmentioning

confidence: 99%

A consensus-based model for global optimization and its mean-field limit

Math. Models Methods Appl. Sci.

Tse³

et al. 2017

Self Cite

130

208

“…The main advantages of the scheme are: it allows for a formal passage to the mean-field limit and hence a rigorous analysis on the PDE level. Indeed, there is a convergence analysis, that indicates a good behavior of the particle scheme, available [3]. The updates of the particle states are based on a comparison to an averaged state.…”

Section: Consensus-based Global Optimizationmentioning

confidence: 99%

“…CBO allows for passage to the mean-field limit resulting in a nonlocal, degenerate, parabolic PDE. Exploiting tools from PDE analysis, it is possible to rigorously prove convergence results for the algorithm (see [3]). In the present article we discuss numerical results obtained with the Particle Swarm Optimization (PSO) [4], Wind-Driven Optimization (WDO) [6] and CBO and show that CBO leads to very competitive results.…”

mentioning

confidence: 99%

A Numerical Comparison of Consensus‐Based Global Optimization to other Particle‐based Global Optimization Schemes

Proc Appl Math and Mech

Blauth

et al. 2018

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We compare a first-order stochastic swarm intelligence model called consensus-based optimization (CBO), which may be used for the global optimization of a function in multiple dimensions, to other particle swarm algorithms for global optimization. CBO allows for passage to the mean-field limit resulting in a nonlocal, degenerate, parabolic PDE. Exploiting tools from PDE analysis, it is possible to rigorously prove convergence results for the algorithm (see [3]). In the present article we discuss numerical results obtained with the Particle Swarm Optimization (PSO) [4], Wind-Driven Optimization (WDO) [6] and CBO and show that CBO leads to very competitive results.

“…Later, it was even shown that special classes of singular interaction potentials allow passing to the mean-field limit [5,26]. Nowadays, particle games are also employed in the field of global optimization [16,34].…”

Section: Introductionmentioning

confidence: 99%

Instantaneous control of interacting particle systems in the mean-field limit

Burger

Journal of Computational Physics

et al. 2020

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Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit,i.e., for an increasing number of particles.