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PreprintCC BY 4.0arXiv

Stability of the Monge Map in Semi-Dual Optimal Transport

  • Anton Selitskiy
  • David Millard
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Abstract

This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.

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Cite

BibTeXRIS
BibTeX
@misc{selitskiy2026stability,
  title = {{Stability of the Monge Map in Semi-Dual Optimal Transport}},
  author = {Anton Selitskiy and David Millard},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.05569},
  eprint = {2605.05569},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.05569},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stability of the Monge Map in Semi-Dual Optimal Transport
AU  - Anton Selitskiy
AU  - David Millard
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2605.05569
UR  - https://arxiv.org/abs/2605.05569
AB  - This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.
ER  -