# A Corrected Inexact Proximal Augmented Lagrangian Method with a Relative Error Criterion for a Class of Group-quadratic Regularized Optimal Transport Problems

arXiv: https://arxiv.org/abs/2311.01976 · HTML: https://mlsanity.com/paper/2311.01976/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2023.

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

- [PINS: Proximal Iterations with Sparse Newton and Sinkhorn for Optimal Transport](https://mlsanity.com/paper/2502.03749/) (2025)
- [An Inexact Halpern Iteration with Application to Distributionally Robust Optimization](https://mlsanity.com/paper/2402.06033/) (2024)
- [Accelerating Multi-Block Constrained Optimization Through Learning to Optimize](https://mlsanity.com/paper/2409.17320/) (2024)
- [A Provably Convergent and Practical Algorithm for Gromov--Wasserstein Optimal Transport](https://mlsanity.com/paper/2605.04175/) (2026)

## Abstract (arXiv)

The optimal transport (OT) problem and its related problems have attracted significant attention and have been extensively studied in various applications. In this paper, we focus on a class of group-quadratic regularized OT problems which aim to find solutions with specialized structures that are advantageous in practical scenarios. To solve this class of problems, we propose a corrected inexact proximal augmented Lagrangian method (ciPALM), with the subproblems being solved by the semi-smooth Newton (sc Ssn) method. We establish that the proposed method exhibits appealing convergence properties under mild conditions. Moreover, our ciPALM distinguishes itself from the recently developed semismooth Newton-based inexact proximal augmented Lagrangian (sc Snipal) method for linear programming. Specifically, sc Snipal uses an absolute error criterion for the approximate minimization of the subproblem for which a summable sequence of tolerance parameters needs to be pre-specified for practical implementations. In contrast, our ciPALM adopts a relative error criterion with a textitsingle tolerance parameter, which would be more friendly to tune from computational and implementation perspectives. These favorable properties position our ciPALM as a promising candidate for tackling large-scale problems. Various numerical studies validate the effectiveness of employing a relative error criterion for the inexact proximal augmented Lagrangian method, and also demonstrate that our ciPALM is competitive for solving large-scale group-quadratic regularized OT problems.