  This paper presents two inexact composite gradient methods, one inner
accelerated and another doubly accelerated, for solving a class of nonconvex
spectral composite optimization problems. More specifically, the objective
function for these problems is of the form $f_1 + f_2 + h$ where $f_1$ and
$f_2$ are differentiable nonconvex matrix functions with Lipschitz continuous
gradients, $h$ is a proper closed convex matrix function, and both $f_2$ and
$h$ can be expressed as functions that operate on the singular values of their
inputs. The methods essentially use an accelerated composite gradient method to
solve a sequence of proximal subproblems involving the linear approximation of
$f_1$ and the singular value functions underlying $f_2$ and $h$. Unlike other
composite gradient-based methods, the proposed methods take advantage of both
the composite and spectral structure underlying the objective function in order
to efficiently generate their solutions. Numerical experiments are presented to
demonstrate the practicality of these methods on a set of real-world and
randomly generated spectral optimization problems.
