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Abstract
In this paper, we describe and establish iterationcomplexity 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 ƒ with a Lipschitz continuous
gradient and a simple nonsmooth closed convex function h. When ƒ is convex, the first ACG
variant reduces to the wellknown 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 ƒ is mweakly convex, ∇ƒ is MLipschitz continuous,
and m ≤ 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.
Original language  English 

Pages (fromto)  649679 
Number of pages  31 
Journal  Computational Optimization and Applications 
Volume  79 
DOIs  
Publication status  Published  13 May 2021 
Keywords
 FISTA variants
 Nonconvex composite optimisation problem
 Accelerated gradient method
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 1 Finished

Extension to the relaxed PeacemanRachford splitting method
Monteiro, R., Sim, C. K. & Liang, J.
23/07/18 → 30/04/21
Project: Research