  We consider the problem of minimizing a difference-of-convex (DC) function,
which can be written as the sum of a smooth convex function with Lipschitz
gradient, a proper closed convex function and a continuous possibly nonsmooth
concave function. We refine the convergence analysis in [38] for the proximal
DC algorithm with extrapolation (pDCA$_e$) and show that the whole sequence
generated by the algorithm is convergent when the objective is level-bounded,
{\em without} imposing differentiability assumptions in the concave part. Our
analysis is based on a new potential function and we assume such a function is
a Kurdyka-{\L}ojasiewicz (KL) function. We also establish a relationship
between our KL assumption and the one used in [38]. Finally, we demonstrate how
the pDCA$_e$ can be applied to a class of simultaneous sparse recovery and
outlier detection problems arising from robust compressed sensing in signal
processing and least trimmed squares regression in statistics. Specifically, we
show that the objectives of these problems can be written as level-bounded DC
functions whose concave parts are {\em typically nonsmooth}. Moreover, for a
large class of loss functions and regularizers, the KL exponent of the
corresponding potential function are shown to be 1/2, which implies that the
pDCA$_e$ is locally linearly convergent when applied to these problems. Our
numerical experiments show that the pDCA$_e$ usually outperforms the proximal
DC algorithm with nonmonotone linesearch [24, Appendix A] in both CPU time and
solution quality for this particular application.
