State-Dependent Temperature Control for Langevin Diffusions

Xuefeng Gao, Zuo Quan Xu, Xunyu Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review


We study the temperature control problem for Langevin diffusions in the context of nonconvex optimization. The classical optimal control of such a problem is of the bang-bang type, which is overly sensitive to errors. A remedy is to allow the diffusions to explore other temperature values and hence smooth out the bang-bang control. We accomplish this by a stochastic relaxed control formulation incorporating randomization of the temperature control and regularizing its entropy. We derive a state-dependent, truncated exponential distribution, which can be used to sample temperatures in a Langevin algorithm, in terms of the solution to an Hamilton-Jacobi- Bellman partial differential equation. We carry out a numerical experiment on a one-dimensional baseline example, in which the Hamilton-Jacobi-Bellman equation can be easily solved, to compare the performance of the algorithm with three other available algorithms in search of a global optimum.

Original languageEnglish
Pages (from-to)1250-1268
Number of pages19
JournalSIAM Journal on Control and Optimization
Issue number3
Publication statusPublished - Jun 2022


  • Boltzmann exploration
  • HJB equation
  • Langevin diffusion
  • entropy regularization
  • nonconvex optimization
  • stochastic relaxed control

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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