A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems
A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems
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DOI：
10.1007/s10589-021-00280-9
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发表时间：
2019-05
影响因子：
2.2
通讯作者：
Jiaming Liang;R. Monteiro;C. Sim
中科院分区：
文献类型：
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作者：
Jiaming Liang;R. Monteiro;C. Sim
本文描述并建立了求解光滑非凸复合优化问题的两个加速复合梯度（ACG）变量的迭代复杂性，该问题的目标函数是一个具有Lipschitz连续梯度的非凸可微函数f和一个简单的非光滑闭凸函数h之和.当f是凸的，第一个ACG变量减少到众所周知的FISTA的输入的特定选择，因此第一个可以被看作是一个自然的扩展，后者的非凸设置。第一个变体需要一个输入对（M，m）使得fism-弱凸，是M-Lipschitz连续的，并且（可能），这通常很难获得或估计不好。另一方面，第二种变体可以从任意输入对（M，m）的正标量和它的复杂性被证明是不是更糟，在某些情况下，比第一种变体的输入对的大范围。最后，数值结果来说明两个ACG变体的效率。
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 functionfwith a Lipschitz continuous gradient and a simple nonsmooth closed convex functionh. Whenfis 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 thatfism-weakly convex,isM-Lipschitz continuous, and(possibly), 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.