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.

在本文中，我们描述并建立了两个加速复合梯度（ACG）变体的迭代复杂度，以解决光滑非凸复合优化问题，该问题的目标函数是具有Lipschitz连续梯度和简单非光滑闭合的非凸可微函数f的和凸函数h。当f为凸形时，对于特定的输入选择，第一个ACG变体简化为众所周知的FISTA，因此第一个可以看作是后一个对非凸设置的自然扩展。第一个变体要求输入对（M，  m）使f为m-弱凸\（\ nabla f \）是M -Lipschitz连续的，并且是\（m \ le M \）（可能是\（m <M \）），通常很难获得或估计不正确。另一方面，第二个变体可以从任意一个正标量输入对（M，  m）开始，对于大范围的输入，它的复杂度显示出比第一个变体更差，并且在某些情况下要好于第一个变体输入对。最后，提供了数值结果来说明这两种ACG变体的效率。