A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

Abstract

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.

28 pages

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• A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems
• A quasi-Newton proximal splitting method
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Cited by in corpus (3)

• An Average Curvature Accelerated Composite Gradient Method for Nonconvex Smooth Composite Optimization Problems
• Accelerated Inexact First-Order Methods for Solving Nonconvex Composite Optimization Problems
• A FISTA-Type First Order Algorithm on Composite Optimization Problems that is Adaptable to the Convex Situation