# ADMM for Multiaffine Constrained Optimization

arXiv: https://arxiv.org/abs/1802.09592 · HTML: https://mlsanity.com/paper/1802.09592/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2018.

## Built on by (most cited first, 6 of 6)

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- [Imitation Learning of Stabilizing Policies for Nonlinear Systems](https://mlsanity.com/paper/2109.10854/) (2021)
- [MOSIC: Model-Agnostic Optimal Subgroup Identification with Multi-Constraint for Improved Reliability](https://mlsanity.com/paper/2504.20908/) (2025)
- [Learning DNN networks using un-rectifying ReLU with compressed sensing application](https://mlsanity.com/paper/2101.06940/) (2021)
- [LRTuckerRep: Low-rank Tucker Representation Model for Multi-dimensional Data Completion](https://mlsanity.com/paper/2508.03755/) (2025)

## Abstract (arXiv)

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-Lojasiewicz (K-L) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions.