  A regularized minimization model with $l_1$-norm penalty (RP) is introduced
for training the autoencoders that belong to a class of two-layer neural
networks. We show that the RP can act as an exact penalty model which shares
the same global minimizers, local minimizers, and d(irectional)-stationary
points with the original regularized model under mild conditions. We construct
a bounded box region that contains at least one global minimizer of the RP, and
propose a linearly constrained regularized minimization model with $l_1$-norm
penalty (LRP) for training autoencoders. A smoothing proximal gradient
algorithm is designed to solve the LRP. Convergence of the algorithm to a
generalized d-stationary point of the RP and LRP is delivered. Comprehensive
numerical experiments convincingly illustrate the efficiency as well as the
robustness of the proposed algorithm.
