# Do optimization techniques map to sampling techniques?

From any generic sampling algorithm, one can derive an optimization algorithm.

Indeed, to maximize an arbitrary function $f: \textbf{x} \rightarrow f(\textbf{x})$, it suffices to draw samples from $g \sim e^{f/T}$. For $T$ small enough, these samples will fall near the global maximum (or local maxima in practice) of the function $f$.

By “sampling” I mean, drawing a pseudo-random sample from a distribution given a log-likelihood function known up to a constant. For instance, MCMC sampling, Gibbs sampling, Beam Sampling, etc. By “optimization” I mean the attempt to find parameters maximizing the value of a given function.

Is the reverse possible?
Given a heuristic to find the maximum of a function or a combinatorial expression, can we extract an efficient sampling procedure?

HMC for instance seems to take advantage of gradient information. Can we construct a sampling procedure that takes advantage of a BFGS-like approximation of the Hessian?
(edit: apparently yes: http://papers.nips.cc/paper/4464-quasi-newton-methods-for-markov-chain-monte-carlo.pdf)
We can use MCTS in combinatorial problems, can we translate that into a sampling procedure?

Context: a difficulty in sampling is often that most of the mass of the probability distribution lies within a very small region. There are interesting techniques to find such regions, but they do not directly translate into unbiased sampling procedures.

Edit: I now have a lingering feeling that the answer to that question is somewhat equivalent to the equality of complexity classes #P and NP, making the answer a likely “no”. It does explain why every sampling technique yields an optimization technique but not vice versa.