  In this note, we propose a FISTA-type first order algorithm, VAR-FISTA, to
solve a composite optimization problem. A distinctive feature of VAR-FISTA is
its ability to exploit the convexity of the function in the problem, resulting
in an improved iteration complexity when the function is convex compared to
when it is nonconvex. The iteration complexity result for the convex and
nonconvex case obtained in the note are compatible to the best known in the
literature so far.
