Convergence of the kinetic annealing for general potentials. (English) Zbl 1517.60100
The paper is concerned with the kinetic Langevin simulated annealing. The authors study the Markov process \((Z_t)_{t\geq 0} = (X_t, Y_t)_{t\geq0}\) on \(\mathbb{R}^{2d}\) that solves
\(dX_t=dY_t \)
\(dY_t=-\nabla U(X_t)dt-\gamma_t Y_t dt +\sqrt{2\gamma_t \beta_t^{-1}}dB_t \)
where \(\gamma_t:\mathbb{R}_+\rightarrow\mathbb{R}_+\) is a friction parameter. The convergence of the kinetic langevin simulated annealing is proven under mild assumptions on the potential \(U\) for slow logarithmic cooling schedules, which widely extends the scope of the previous results of P. Monmarché [Probab. Theory Relat. Fields 172, No. 3–4, 1215–1248 (2018; Zbl 1404.60120)]. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established.
\(dX_t=dY_t \)
\(dY_t=-\nabla U(X_t)dt-\gamma_t Y_t dt +\sqrt{2\gamma_t \beta_t^{-1}}dB_t \)
where \(\gamma_t:\mathbb{R}_+\rightarrow\mathbb{R}_+\) is a friction parameter. The convergence of the kinetic langevin simulated annealing is proven under mild assumptions on the potential \(U\) for slow logarithmic cooling schedules, which widely extends the scope of the previous results of P. Monmarché [Probab. Theory Relat. Fields 172, No. 3–4, 1215–1248 (2018; Zbl 1404.60120)]. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)
MSC:
| 60J60 | Diffusion processes |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 90C15 | Stochastic programming |
Keywords:
hypocoercivity; Langevin diffusion; metastability; simulated annealing; stochastic optimizationCitations:
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