  In this paper we consider minimization of a difference-of-convex (DC)
function with and without linear constraints. We first study a smooth
approximation of a generic DC function, termed difference-of-Moreau-envelopes
(DME) smoothing, where both components of the DC function are replaced by their
respective Moreau envelopes. The resulting smooth approximation is shown to be
Lipschitz differentiable, capture stationary points, local, and global minima
of the original DC function, and enjoy some growth conditions, such as
level-boundedness and coercivity, for broad classes of DC functions. We then
develop four algorithms for solving DC programs with and without linear
constraints based on the DME smoothing. In particular, for a smoothed DC
program without linear constraints, we show that the classic gradient descent
method as well as an inexact variant can obtain a stationary solution in the
limit with a convergence rate of $\mathcal{O}(K^{-1/2})$, where $K$ is the
number of proximal evaluations of both components. Furthermore, when the DC
program is explicitly constrained in an affine subspace, we combine the
smoothing technique with the augmented Lagrangian function and derive two
variants of the augmented Lagrangian method (ALM), named LCDC-ALM and composite
LCDC-ALM, focusing on different structures of the DC objective function. We
show that both algorithms find an $\epsilon$-approximate stationary solution of
the original DC program in $\mathcal{O}(\epsilon^{-2})$ iterations. Comparing
to existing methods designed for linearly constrained weakly convex
minimization, the proposed ALM-based algorithms can be applied to a broader
class of problems, where the objective contains a nonsmooth concave component.
Finally, numerical experiments are presented to demonstrate the performance of
the proposed algorithms.
