# The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates

arXiv: https://arxiv.org/abs/1801.01994 · HTML: https://mlsanity.com/paper/1801.01994/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2018.

## Built on by (most cited first, 13 of 13)

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- [ADMM for Structured Fractional Minimization](https://mlsanity.com/paper/2411.07496/) (2024)
- [A Linearized Alternating Direction Multiplier Method for Federated Matrix Completion Problems](https://mlsanity.com/paper/2503.12733/) (2025)
- [A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints](https://mlsanity.com/paper/2509.02894/) (2025)
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- [Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties](https://mlsanity.com/paper/2603.01304/) (2026)

## Abstract (arXiv)

We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit of the proximal alternating direction method of multipliers and its linearized variant, respectively. The proximal terms are introduced via variable metrics, a fact which allows us to derive new proximal splitting algorithms for nonconvex structured optimization problems, as particular instances of the general schemes. Under mild conditions on the sequence of variable metrics and by assuming that a regularization of the associated augmented Lagrangian has the Kurdyka-Lojasiewicz property, we prove that the iterates converge to a KKT point of the objective function. By assuming that the augmented Lagrangian has the Lojasiewicz property, we also derive convergence rates for both the augmented Lagrangian and the iterates.