  In this paper, we generalize the well-known Nesterov's accelerated gradient
(AG) method, originally designed for convex smooth optimization, to solve
nonconvex and possibly stochastic optimization problems. We demonstrate that by
properly specifying the stepsize policy, the AG method exhibits the best known
rate of convergence for solving general nonconvex smooth optimization problems
by using first-order information, similarly to the gradient descent method. We
then consider an important class of composite optimization problems and show
that the AG method can solve them uniformly, i.e., by using the same aggressive
stepsize policy as in the convex case, even if the problem turns out to be
nonconvex. We demonstrate that the AG method exhibits an optimal rate of
convergence if the composite problem is convex, and improves the best known
rate of convergence if the problem is nonconvex. Based on the AG method, we
also present new nonconvex stochastic approximation methods and show that they
can improve a few existing rates of convergence for nonconvex stochastic
optimization. To the best of our knowledge, this is the first time that the
convergence of the AG method has been established for solving nonconvex
nonlinear programming in the literature.
