# The Primal-Dual Hybrid Gradient Method for Semiconvex Splittings

arXiv: https://arxiv.org/abs/1407.1723 · HTML: https://mlsanity.com/paper/1407.1723/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2014.

## Built on by (most cited first, 4 of 4)

- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Enhanced total variation minimization for stable image reconstruction](https://mlsanity.com/paper/2201.02979/) (2022)
- [Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity](https://mlsanity.com/paper/2312.06540/) (2023)

## Abstract (arXiv)

This paper deals with the analysis of a recent reformulation of the primal-dual hybrid gradient method [Zhu and Chan 2008, Pock, Cremers, Bischof and Chambolle 2009, Esser, Zhang and Chan 2010, Chambolle and Pock 2011], which allows to apply it to nonconvex regularizers as first proposed for truncated quadratic penalization in [Strekalovskiy and Cremers 2014]. Particularly, it investigates variational problems for which the energy to be minimized can be written as G(u) + F(Ku), where G is convex, F semiconvex, and K is a linear operator. We study the method and prove convergence in the case where the nonconvexity of F is compensated by the strong convexity of the G. The convergence proof yields an interesting requirement for the choice of algorithm parameters, which we show to not only be sufficient, but necessary. Additionally, we show boundedness of the iterates under much weaker conditions. Finally, we demonstrate effectiveness and convergence of the algorithm beyond the theoretical guarantees in several numerical experiments.