# A Multi-step Inertial Forward--Backward Splitting Method for Non-convex Optimization

arXiv: https://arxiv.org/abs/1606.02118 · HTML: https://mlsanity.com/paper/1606.02118/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2016.

## Built on by (most cited first, 3 of 3)

- [Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexity](https://mlsanity.com/paper/2201.11411/) (2022)
- [Beyond L1: Faster and Better Sparse Models with skglm](https://mlsanity.com/paper/2204.07826/) (2022)
- [Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality](https://mlsanity.com/paper/2410.16849/) (2024)

## Abstract (arXiv)

In this paper, we propose a multi-step inertial Forward--Backward splitting algorithm for minimizing the sum of two non-necessarily convex functions, one of which is proper lower semi-continuous while the other is differentiable with a Lipschitz continuous gradient. We first prove global convergence of the scheme with the help of the Kurdyka-Lojasiewicz property. Then, when the non-smooth part is also partly smooth relative to a smooth submanifold, we establish finite identification of the latter and provide sharp local linear convergence analysis. The proposed method is illustrated on a few problems arising from statistics and machine learning.