  We study the temperature control problem for Langevin diffusions in the
context of non-convex 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 HJB partial differential equation. We carry out
a numerical experiment on a one-dimensional baseline example, in which the HJB
equation can be easily solved, to compare the performance of the algorithm with
three other available algorithms in search of a global optimum.
