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The Chambolle-Pock method converges weakly with 0 < θ ≤ 1/2 and τσ|L|2 < 4θ(2−θ)/(1 − 2θ + 9θ2 − 4θ3)

  • Manu Upadhyaya
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Abstract

The Chambolle-Pock method, also known as the primal-dual hybrid gradient method, is a standard first-order algorithm for convex-concave saddle-point problems and composite convex optimization. We establish weak sequential convergence of its primal-dual iterates in real Hilbert spaces for every 0<θ≤ 1/2 whenever τσ|L|2<4θ(2−θ)/(1−2θ+9θ2−4θ3). This extends the weak-convergence theory to a previously unexplored range of extrapolation parameters.

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Cite

BibTeXRIS
BibTeX
@misc{upadhyaya2026chambollepock,
  title = {{The Chambolle-Pock method converges weakly with $0 < \theta \le 1/2$ and $\tau\sigma\|L\|^2 < 4\theta(2-\theta)/(1 - 2\theta + 9\theta^2 - 4\theta^3)$}},
  author = {Manu Upadhyaya},
  howpublished = {HAL},
  year = {2026},
  month = aug,
  eprint = {2604.06423},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05858429v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Chambolle-Pock method converges weakly with 0 < θ ≤ 1/2 and τσ|L|^2 < 4θ(2−θ)/(1 − 2θ + 9θ^2 − 4θ^3)
AU  - Manu Upadhyaya
PY  - 2026
DA  - 2026/08/20
PB  - HAL
UR  - https://hal.science/hal-05858429v1
AB  - The Chambolle-Pock method, also known as the primal-dual hybrid gradient method, is a standard first-order algorithm for convex-concave saddle-point problems and composite convex optimization. We establish weak sequential convergence of its primal-dual iterates in real Hilbert spaces for every 0<θ≤ 1/2 whenever τσ|L|^2<4θ(2−θ)/(1−2θ+9θ^2−4θ^3). This extends the weak-convergence theory to a previously unexplored range of extrapolation parameters.
ER  -