  Standard approaches to difference-of-convex (DC) programs require exact
solution to a convex subproblem at each iteration, which generally requires
noiseless computation and infinite iterations of an inner iterative algorithm.
To tackle these difficulties, inexact DC algorithms have been proposed, mostly
by relaxing the convex subproblem to an approximate monotone inclusion problem.
However, there is no guarantee that such relaxation can lead to a finitely
terminating inner loop. In this paper, we point out the termination issue of
existing inexact DC algorithms by presenting concrete counterexamples.
Exploiting the notion of $\epsilon$-subdifferential, we propose a novel inexact
proximal linearized DC algorithm termed tPLDCA. Despite permission to a great
extent of inexactness in computation, tPLDCA enjoys the same convergence
guarantees as exact DC algorithms. Most noticeably, the inner loop of tPLDCA is
guaranteed to terminate in finite iterations as long as the inner iterative
algorithm converges to a solution of the proximal point subproblem, which makes
an essential difference from prior arts. In addition, by assuming the first
convex component of the DC function to be pointwise maximum of finitely many
convex smooth functions, we propose a computational friendly surrogate of
$\epsilon$-subdifferential, whereby we develop a feasible implementation of
tPLDCA. Numerical results demonstrate the effectiveness of the proposed
implementation of tPLDCA.
