  The leaky ReLU network with a group sparse regularization term has been
widely used in the recent years. However, training such a network yields a
nonsmooth nonconvex optimization problem and there exists a lack of approaches
to compute a stationary point deterministically. In this paper, we first
resolve the multi-layer composite term in the original optimization problem by
introducing auxiliary variables and additional constraints. We show the new
model has a nonempty and bounded solution set and its feasible set satisfies
the Mangasarian-Fromovitz constraint qualification. Moreover, we show the
relationship between the new model and the original problem. Remarkably, we
propose an inexact augmented Lagrangian algorithm for solving the new model and
show the convergence of the algorithm to a KKT point. Numerical experiments
demonstrate that our algorithm is more efficient for training sparse leaky ReLU
neural networks than some well-known algorithms.
