Simulated annealing acceptance probability puzzle My understanding of simulated annealing (SA) is that at any iteration $t$, a new sample $Y_t$ is generated, which, if the objective function $E$ is improved, i.e., $E(Y_t)<E(X_{t-1})$, then $Y_t$ is accepted, that is, $X_t=Y_t$, and the iteration moves on. Otherwise, the new sample is accepted by the following probability: $P(X_t=Y_t)=\exp(E(X_{t-1})-E(Y_t))\le1$. This setting is the same as in the Hastings-Metropolis (HM) algorithm, where detailed balance equation is indeed satisfied for the underlying Boltzmann distribution $\exp(-E(X)/T)$, assuming the proposal density is symmetric, e.g., a random walk.
However, in a number of papers, including this and this, which are popular papers, the acceptance probability seems to be set as $\frac{1}{1+\exp(E(Y_t)-E(X_{t-1}))}$, which is $\le\frac{1}{2}$ for the case $E(Y_t)\ge E(X_{t-1})$. Under this acceptance probability, the Metropolis' detailed balance equation no longer holds. I believe this is also the default setting for Matlab's implementation of SA.
Can somebody explain to me why do these authors use such an acceptance rate? I spent an entire weekend surveying a number of papers, none of them explain with any clarity. 
 A: This alternative acceptance probability is Barker’s formula which got published in the Australian Journal of Physics at the beginning of Barker’s PhD at the University of Adelaide.

As shown in the above screenshot, the basis  of Barker’s algorithm is indeed Barker’s acceptance probability, albeit written in a somewhat confusing way since the current value of the chain is kept if a Uniform variate is smaller than what is actually the rejection probability. 
As in Metropolis et al. (1953), the analysis is made on a discretised (finite) space, building the Markov transition matrix, stating the detailed balance equation (called microscopic reversibility). Interestingly, while Barker acknowledges that there are other ways of assigning the transition probability, his is the “most rapid” in terms of mixing. And equally interestingly, he discusses the scale of the random walk in the [not-yet-called] Metropolis-within-Gibbs move as major, targetting 0.5 as the right acceptance rate, and suggesting to adapt this scale on the go.
A: Notice that 
$$
\frac{1}{1 + e^{E(Y_t)-E(X_{t-1})}} =  \frac{e^{E(X_{t-1})-E(Y_t)}}{1 + e^{E(X_{t-1})-E(Y_t)}} \propto e^{E(X_{t-1})-E(Y_t)}
$$
so what you are proposing as a acceptance probability is really the proportional component of a normalized logistic probabilty.
I believe in the detailed balance equation the normalizing factor disappears leaving only the proportional component which you might have seen in some treatement.
