  We present a comprehensive study of the resolution and stability properties
of sparse promoting optimization theories applied to narrow band array imaging
of localized scatterers. We consider homogeneous and heterogeneous media, and
multiple and single scattering situations. When the media is homogeneous with
strong multiple scattering between scatterers, we give a non-iterative
formulation to find the locations and reflectivities of the scatterers from a
nonlinear inverse problem in two steps, using either single or multiple
illuminations. We further introduce an approach that uses the top singular
vectors of the response matrix as optimal illuminations, which improves the
robustness of sparse promoting optimization with respect to additive noise.
When multiple scattering is negligible, the optimization problem becomes linear
and can be reduced to a hybrid-$\ell_1$ method when optimal illuminations are
used. When the media is random, and the interaction with the unknown
inhomogeneities can be primarily modeled by wavefront distortions, we address
the statistical stability of these methods. We analyze the fluctuations of the
images obtained with the hybrid-$\ell_1$ method, and we show that it is stable
with respect to different realizations of the random medium provided the
imaging array is large enough. We compare the performance of the
hybrid-$\ell_1$ method in random media to the widely used Kirchhoff migration
and the multiple signal classification methods.
