  In this paper, we further study the forward-backward envelope first
introduced in [28] and [30] for problems whose objective is the sum of a proper
closed convex function and a twice continuously differentiable possibly
nonconvex function with Lipschitz continuous gradient. We derive sufficient
conditions on the original problem for the corresponding forward-backward
envelope to be a level-bounded and Kurdyka-{\L}ojasiewicz function with an
exponent of $\frac12$; these results are important for the efficient
minimization of the forward-backward envelope by classical optimization
algorithms. In addition, we demonstrate how to minimize some
difference-of-convex regularized least squares problems by minimizing a
suitably constructed forward-backward envelope. Our preliminary numerical
results on randomly generated instances of large-scale $\ell_{1-2}$ regularized
least squares problems [37] illustrate that an implementation of this approach
with a limited-memory BFGS scheme usually outperforms standard first-order
methods such as the nonmonotone proximal gradient method in [35].
