  We theoretically investigate Bloch oscillations in a one-dimensional
Bose-Hubbard chain, with single-particle losses from the odd lattice sites
described by the Lindblad equation. For a single particle the time evolution of
the state is completely determined by a non-Hermitian effective Hamiltonian. We
analyse the spectral properties of this Hamiltonian for an infinite lattice and
link features of the spectrum to observable dynamical effects, such as
frequency doubling in breathing modes. We further consider the case of many
particles in the mean-field limit leading to complex nonlinear Schr\"odinger
dynamics. Analytic expressions are derived for the generalised nonlinear
stationary states and the nonlinear Bloch bands. The interplay of nonlinearity
and particle losses leads to peculiar features in the nonlinear Bloch bands,
such as the vanishing of solutions and the formation of additional exceptional
points. The stability of the stationary states is determined via the
Bogoliubov-de Gennes equation and is shown to strongly influence the mean-field
dynamics. Remarkably, even far from the mean-field limit, the stability of the
nonlinear Bloch bands appears to affect the quantum dynamics. This is
demonstrated numerically for a two-particle system.
