  Coupling a many-body-localized system to a dissipative bath necessarily leads
to delocalization. Here, we investigate the nature of the ensuing relaxation
dynamics and the information it holds on the many-body-localized state. We
formulate the relevant Lindblad equation in terms of the local integrals of
motion of the underlying localized Hamiltonian. This allows to map the quantum
evolution deep in the localized state to tractable classical rate equations. We
consider two different types of dissipation relevant to systems of ultra-cold
atoms: dephasing due to inelastic scattering on the lattice lasers and particle
loss. Only the latter mechanism shows a pronounced effects of interactions on
the relaxation of observables.
