  We consider a recently proposed class of MCMC methods which uses proximity
maps instead of gradients to build proposal mechanisms which can be employed
for both differentiable and non-differentiable targets. These methods have been
shown to be stable for a wide class of targets, making them a valuable
alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found
wide application in imaging contexts. The wider stability properties are
obtained by building the Moreau-Yosida envelope for the target of interest,
which depends on a parameter $\lambda$. In this work, we investigate the
optimal scaling problem for this class of algorithms, which encompasses MALA,
and provide practical guidelines for the implementation of these methods.
