  In this paper, we describe and establish iteration-complexity of two
accelerated composite gradient (ACG) variants to solve a smooth nonconvex
composite optimization problem whose objective function is the sum of a
nonconvex differentiable function $ f $ with a Lipschitz continuous gradient
and a simple nonsmooth closed convex function $ h $. When $f$ is convex, the
first ACG variant reduces to the well-known FISTA for a specific choice of the
input, and hence the first one can be viewed as a natural extension of the
latter one to the nonconvex setting. The first variant requires an input pair
$(M,m)$ such that $f$ is $m$-weakly convex, $\nabla f$ is $M$-Lipschitz
continuous, and $m \le M$ (possibly $m<M$), which is usually hard to obtain or
poorly estimated. The second variant on the other hand can start from an
arbitrary input pair $(M,m)$ of positive scalars and its complexity is shown to
be not worse, and better in some cases, than that of the first variant for a
large range of the input pairs. Finally, numerical results are provided to
illustrate the efficiency of the two ACG variants.
