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

Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing