  We study the convergence of Langevin-Simulated Annealing type algorithms with
multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential
function to minimize, we consider the stochastic equation $dY_t = - \sigma
\sigma^\top \nabla V(Y_t) dt + a(t)\sigma(Y_t)dW_t + a(t)^2\Upsilon(Y_t)dt$,
where $(W_t)$ is a Brownian motion, where $\sigma : \mathbb{R}^d \to
\mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a :
\mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where
$\Upsilon$ is a correction term. This setting can be applied to optimization
problems arising in Machine Learning. The case where $\sigma$ is a constant
matrix has been extensively studied however little attention has been paid to
the general case. We prove the convergence for the $L^1$-Wasserstein distance
of $Y_t$ and of the associated Euler-scheme $\bar{Y}_t$ to some measure
$\nu^\star$ which is supported by $\text{argmin}(V)$ and give rates of
convergence to the instantaneous Gibbs measure $\nu_{a(t)}$ of density $\propto
\exp(-2V(x)/a(t)^2)$. To do so, we first consider the case where $a$ is a
piecewise constant function. We find again the classical schedule $a(t) =
A\log^{-1/2}(t)$. We then prove the convergence for the general case by giving
bounds for the Wasserstein distance to the stepwise constant case using
ergodicity properties.
