# Frank-Wolfe Splitting via Augmented Lagrangian Method

arXiv: https://arxiv.org/abs/1804.03176 · HTML: https://mlsanity.com/paper/1804.03176/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2018.

## Built on by (most cited first, 7 of 7)

- [Diversifying AI: Towards Creative Chess with AlphaZero](https://mlsanity.com/paper/2308.09175/) (2023)
- [LeanVec: Searching vectors faster by making them fit](https://mlsanity.com/paper/2312.16335/) (2023)
- [Faster Projection-Free Augmented Lagrangian Methods via Weak Proximal Oracle](https://mlsanity.com/paper/2210.13968/) (2022)
- [SAIF: Sparse Adversarial and Imperceptible Attack Framework](https://mlsanity.com/paper/2212.07495/) (2022)
- [Federated Frank-Wolfe Algorithm](https://mlsanity.com/paper/2408.10090/) (2024)
- [A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron](https://mlsanity.com/paper/2503.01441/) (2025)
- [Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes](https://mlsanity.com/paper/2505.11259/) (2025)

## Abstract (arXiv)

Minimizing a function over an intersection of convex sets is an important task in optimization that is often much more challenging than minimizing it over each individual constraint set. While traditional methods such as Frank-Wolfe (FW) or proximal gradient descent assume access to a linear or quadratic oracle on the intersection, splitting techniques take advantage of the structure of each sets, and only require access to the oracle on the individual constraints. In this work, we develop and analyze the Frank-Wolfe Augmented Lagrangian (FW-AL) algorithm, a method for minimizing a smooth function over convex compact sets related by a "linear consistency" constraint that only requires access to a linear minimization oracle over the individual constraints. It is based on the Augmented Lagrangian Method (ALM), also known as Method of Multipliers, but unlike most existing splitting methods, it only requires access to linear (instead of quadratic) minimization oracles. We use recent advances in the analysis of Frank-Wolfe and the alternating direction method of multipliers algorithms to prove a sublinear convergence rate for FW-AL over general convex compact sets and a linear convergence rate for polytopes.