  We consider a class of difference-of-convex (DC) optimization problems whose
objective is level-bounded and is the sum of a smooth convex function with
Lipschitz gradient, a proper closed convex function and a continuous concave
function. While this kind of problems can be solved by the classical
difference-of-convex algorithm (DCA) [26], the difficulty of the subproblems of
this algorithm depends heavily on the choice of DC decomposition. Simpler
subproblems can be obtained by using a specific DC decomposition described in
[27]. This decomposition has been proposed in numerous work such as [18], and
we refer to the resulting DCA as the proximal DCA. Although the subproblems are
simpler, the proximal DCA is the same as the proximal gradient algorithm when
the concave part of the objective is void, and hence is potentially slow in
practice. In this paper, motivated by the extrapolation techniques for
accelerating the proximal gradient algorithm in the convex settings, we
consider a proximal difference-of-convex algorithm with extrapolation to
possibly accelerate the proximal DCA. We show that any cluster point of the
sequence generated by our algorithm is a stationary point of the DC
optimization problem for a fairly general choice of extrapolation parameters:
in particular, the parameters can be chosen as in FISTA with fixed restart
[15]. In addition, by assuming the Kurdyka-{\L}ojasiewicz property of the
objective and the differentiability of the concave part, we establish global
convergence of the sequence generated by our algorithm and analyze its
convergence rate. Our numerical experiments on two difference-of-convex
regularized least squares models show that our algorithm usually outperforms
the proximal DCA and the general iterative shrinkage and thresholding algorithm
proposed in [17].
