# Asymptotic convergence of constrained primal-dual dynamics

arXiv: https://arxiv.org/abs/1510.02150 · HTML: https://mlsanity.com/paper/1510.02150/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2015.

## Built on by (most cited first, 8 of 8)

- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [Constrained Reinforcement Learning via Dissipative Saddle Flow Dynamics](https://mlsanity.com/paper/2212.01505/) (2022)
- [Fully Distributed GNE Algorithms for Multi-Robot Placement without Consensus on Multipliers](https://mlsanity.com/paper/2608.29388/) (2026)
- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)
- [Provable Accuracy Bounds for Hybrid Dynamical Optimization and Sampling](https://mlsanity.com/paper/2410.06397/) (2024)
- [Residual-Controlled Multiplier Learning for Stochastic Constrained Decision-Making](https://mlsanity.com/paper/2606.07088/) (2026)
- [Input-to-State Stability Framework for Fully Distributed Primal-Dual Dynamics for Quadratic GNEPs Without Multiplier Consensus](https://mlsanity.com/paper/2609.06983/) (2026)

## Abstract (arXiv)

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.