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

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**A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems.** / Liang, Jiaming; Monteiro, Renato; Sim, Chee Khian.

Research output: Contribution to journal › Article › peer-review

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*Computational Optimization and Applications*.

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*Computational Optimization and Applications*.

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TY - JOUR

T1 - A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

AU - Liang, Jiaming

AU - Monteiro, Renato

AU - Sim, Chee Khian

N1 - 12 month embargo. Springer. This is a post-peer-review, pre-copyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/[insert DOI].

PY - 2021/4/29

Y1 - 2021/4/29

N2 - 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 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 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 ƒ is m-weakly convex, ∇ƒ is M-Lipschitz 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.

AB - 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 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 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 ƒ is m-weakly convex, ∇ƒ is M-Lipschitz 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.

KW - FISTA variants

KW - Nonconvex composite optimisation problem

KW - Accelerated gradient method

M3 - Article

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

ER -

ID: 27427389