# A Proximal Alternating Direction Method of Multiplier for Linearly Constrained Nonconvex Minimization

arXiv: https://arxiv.org/abs/1812.10229 · HTML: https://mlsanity.com/paper/1812.10229/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Dec 2018.

## Built on by (most cited first, 14 of 14)

- [Complexity of Single Loop Algorithms for Nonlinear Programming with Stochastic Objective and Constraints](https://mlsanity.com/paper/2311.00678/) (2023)
- [A Single-Loop Gradient Descent and Perturbed Ascent Algorithm for Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/2207.05650/) (2022)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Stochastic Smoothed Primal-Dual Algorithms for Nonconvex Optimization with Linear Inequality Constraints](https://mlsanity.com/paper/2504.07607/) (2025)
- [Stochastic Smoothed Gradient Descent Ascent for Federated Minimax Optimization](https://mlsanity.com/paper/2311.00944/) (2023)
- [Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization](https://mlsanity.com/paper/2403.09090/) (2024)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions](https://mlsanity.com/paper/2502.19764/) (2025)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)
- [First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods](https://mlsanity.com/paper/2307.07605/) (2023)
- [Differentiable Projection-based Learn to Optimize in Wireless Network-Part I: Convex Constrained (Non-)Convex Programming](https://mlsanity.com/paper/2502.00053/) (2025)
- [A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints](https://mlsanity.com/paper/2509.02894/) (2025)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)

## Abstract (arXiv)

Consider the minimization of a nonconvex differentiable function over a polyhedron. A popular primal-dual first-order method for this problem is to perform a gradient projection iteration for the augmented Lagrangian function and then update the dual multiplier vector using the constraint residual. However, numerical examples show that this approach can exhibit "oscillation" and may not converge. In this paper, we propose a proximal alternating direction method of multipliers for the multi-block version of this problem. A distinctive feature of this method is the introduction of a "smoothed" (i.e., exponentially weighted) sequence of primal iterates, and the inclusion, at each iteration, to the augmented Lagrangian function a quadratic proximal term centered at the current smoothed primal iterate. The resulting proximal augmented Lagrangian function is inexactly minimized (via a gradient projection step) at each iteration while the dual multiplier vector is updated using the residual of the linear constraints. When the primal and dual stepsizes are chosen sufficiently small, we show that suitable "smoothing" can stabilize the "oscillation", and the iterates of the new proximal ADMM algorithm converge to a stationary point under some mild regularity conditions. Furthermore, when the objective function is quadratic, we establish the linear convergence of the algorithm. Our proof is based on a new potential function and a novel use of error bounds.