  A precise characterization of the extremal points of sublevel sets of
nonsmooth penalties provides both detailed information about minimizers, and
optimality conditions in general classes of minimization problems involving
them. Moreover, it enables the application of accelerated generalized
conditional gradient methods for their efficient solution. In this manuscript,
this program is adapted to the minimization of a smooth convex fidelity term
which is augmented with an unbalanced transport regularization term given in
the form of a generalized Kantorovich-Rubinstein norm for Radon measures. More
precisely, we show that the extremal points associated to the latter are given
by all Dirac delta functionals supported in the spatial domain as well as
certain dipoles, i.e., pairs of Diracs with the same mass but with different
signs. Subsequently, this characterization is used to derive precise
first-order optimality conditions as well as an efficient solution algorithm
for which linear convergence is proved under natural assumptions. This
behaviour is also reflected in numerical examples for a model problem.
