In this paper, we consider the generalized (higher order) Langevin equation for the purpose of simulated annealing and optimization of nonconvex functions. Our approach modifies the underdamped Langevin equation by replacing the Brownian noise with an appropriate Ornstein–Uhlenbeck process to account for memory in the system. Under reasonable conditions on the loss function and the annealing schedule, we establish convergence of the continuous time dynamics to a global minimum. In addition, we investigate the performance numerically and show better performance and higher exploration of the state space compared to the underdamped Langevin dynamics with the same annealing schedule.


  1. nonconvex optimization
  2. generalized Langevin equation
  3. simulated annealing

MSC codes

  1. 60J25
  2. 46N10
  3. 60J60

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The authors would like to thank Tony Lelievre, Gabriel Stoltz, Urbain Vaes and the anonymous referees for their helpful remarks.

Supplementary Materials

PLEASE NOTE: These supplementary files have not been peer-reviewed.
Index of Supplementary Materials
Title of paper: On the Generalized Langevin Equation for Simulated Annealing
Authors: Martin Chak, Nikolas Kantas, and Grigorios A. Pavliotis
File: supplement.pdf
Type: PDF
Contents: This file includes background/additional results, a description of the numerical methodology used and figures/tables from additional numerical experiments. The background results provide full justification of the main results in the manuscript; additional results provide a peripheral perspectve on the main results; the numerical methodology allows reproduction of the numerical experiments; the additional numerical experiments provide a wider variety of examples.


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Information & Authors


Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 139 - 167
ISSN (online): 2166-2525


Submitted: 2 December 2021
Accepted: 22 August 2022
Published online: 1 March 2023


  1. nonconvex optimization
  2. generalized Langevin equation
  3. simulated annealing

MSC codes

  1. 60J25
  2. 46N10
  3. 60J60



Martin Chak Contact the author
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
Nikolas Kantas
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.

Funding Information

Engineering and Physical Sciences Research Council (EPSRC): EP/P031587/1, EP/L024926/1, EP/L020564/1
Funding: The first author was supported by an EPSRC studentship. The second and third authors were partially supported by JPMorgan Chase & Co. under J.P. Morgan A.I. Research Awards 2019. The third author was also partially supported by the EPSRC through grants EP/P031587/1, EP/L024926/1, and EP/L020564/1.

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