Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints

@misc{he2026accelerated,
  title = {{Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints}},
  author = {Xin He},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01415},
  eprint = {2609.01415},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01415},
}

TY  - UNPB
TI  - Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints
AU  - Xin He
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.01415
UR  - https://arxiv.org/abs/2609.01415
AB  - We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping α/t, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and α≥3, we establish 𝒪(t^−2) convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime α>3, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to o(t^−2). We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields 𝒪(k^−2) rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to o(k^−2). Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.
ER  -