  We introduce the Asynchronous PALM algorithm, a new extension of the Proximal
Alternating Linearized Minimization (PALM) algorithm for solving nonsmooth,
nonconvex optimization problems. Like the PALM algorithm, each step of the
Asynchronous PALM algorithm updates a single block of coordinates; but unlike
the PALM algorithm, the Asynchronous PALM algorithm eliminates the need for
sequential updates that occur one after the other. Instead, our new algorithm
allows each of the coordinate blocks to be updated asynchronously and in any
order, which means that any number of computing cores can compute updates in
parallel without synchronizing their computations. In practice, this
asynchronization strategy often leads to speedups that increase linearly with
the number of computing cores.
  We introduce two variants of the Asynchronous PALM algorithm, one stochastic
and one deterministic. In the stochastic \textit{and} deterministic cases, we
show that cluster points of the algorithm are stationary points. In the
deterministic case, we show that the algorithm converges globally whenever the
Kurdyka-{\L}ojasiewicz property holds for a function closely related to the
objective function, and we derive its convergence rate in a common special
case. Finally, we provide a concrete case in which our assumptions hold.
