# Strong Convergence of FISTA Iterates under H"olderian and Quadratic Growth Conditions

arXiv: https://arxiv.org/abs/2407.17063 · HTML: https://mlsanity.com/paper/2407.17063/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [On the Effectiveness of the z-Transform Method in Quadratic Optimization](https://mlsanity.com/paper/2507.03404/) (2025)

## Abstract (arXiv)

Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of Oleft(n-2right) in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions.