  This paper develops and analyzes an accelerated proximal descent method for
finding stationary points of nonconvex composite optimization problems. The
objective function is of the form $f+h$ where $h$ is a proper closed convex
function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$
is Lipschitz continuous on the domain of $h$. The main advantage of this method
is that it is "parameter-free" in the sense that it does not require knowledge
of the Lipschitz constant of $\nabla f$ or of any global topological properties
of $f$. It is shown that the proposed method can obtain an
$\varepsilon$-approximate stationary point with iteration complexity bounds
that are optimal, up to logarithmic terms over $\varepsilon$, in both the
convex and nonconvex settings. Some discussion is also given about how the
proposed method can be leveraged in other existing optimization frameworks,
such as min-max smoothing and penalty frameworks for constrained programming,
to create more specialized parameter-free methods. Finally, numerical
experiments are presented to support the practical viability of the method.
