This paper discusses the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximately solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with the linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman’s kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.


  1. primal-dual method
  2. optimal transport
  3. Wasserstein barycenter
  4. saddle-point

MSC codes

  1. Primary: 65Y20
  2. 49Q22; Secondary: 90C05
  3. 90C06
  4. 90C08
  5. 90C47

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Information & Authors


Published In

cover image SIAM Journal on Mathematics of Data Science
SIAM Journal on Mathematics of Data Science
Pages: 1369 - 1395
ISSN (online): 2577-0187


Submitted: 9 March 2022
Accepted: 24 August 2022
Published online: 20 December 2022


  1. primal-dual method
  2. optimal transport
  3. Wasserstein barycenter
  4. saddle-point

MSC codes

  1. Primary: 65Y20
  2. 49Q22; Secondary: 90C05
  3. 90C06
  4. 90C08
  5. 90C47



Antonin Chambolle
CEREMADE, CNRS & Université Paris-Dauphine, PSL University, Paris, 91128 France.
Universidad Adolfo Ibáñez, Facultad de Ingeniería y Ciencias, 7941169 Santiago, Chile, and Departamento de Ingeniería Industrial, Universidad Católica del Norte, 1780000 Antofagasta, Chile.

Funding Information

ANID-PFCHA/Doctorado Nacional: 2019-21190161
Funding. The work of the second author was supported by a doctoral scholarship from ANID-PFCHA/Doctorado Nacional/2019-21190161.

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