# A first-order primal-dual method with adaptivity to local smoothness

arXiv: https://arxiv.org/abs/2110.15148 · HTML: https://mlsanity.com/paper/2110.15148/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 28 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Adaptive Gradient Descent without Descent](https://mlsanity.com/paper/1910.09529/) (2019)
- [Golden Ratio Algorithms for Variational Inequalities](https://mlsanity.com/paper/1803.08832/) (2018)
- [Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems](https://mlsanity.com/paper/1406.5429/) (2014)
- [A first-order primal-dual algorithm with linesearch](https://mlsanity.com/paper/1608.08883/) (2016)
- [Adaptive Three Operator Splitting](https://mlsanity.com/paper/1804.02339/) (2018)

## Built on by (most cited first, 7 of 7)

- [Adaptive Proximal Gradient Method for Convex Optimization](https://mlsanity.com/paper/2308.02261/) (2023)
- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [Directional Smoothness and Gradient Methods: Convergence and Adaptivity](https://mlsanity.com/paper/2403.04081/) (2024)
- [A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems](https://mlsanity.com/paper/2407.03571/) (2024)
- [Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliers](https://mlsanity.com/paper/2410.01979/) (2024)
- [Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities](https://mlsanity.com/paper/2603.29128/) (2026)
- [Scheduling Recursive Reasoning in Looped Transformers](https://mlsanity.com/paper/2609.36653/) (2026)

## Abstract (arXiv)

We consider the problem of finding a saddle point for the convex-concave objective minx maxy f(x) + langle Ax, yrangle - g^*(y), where f is a convex function with locally Lipschitz gradient and g is convex and possibly non-smooth. We propose an adaptive version of the Condat-V~u algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving |A| and the norm of recently computed gradients of f. Under standard assumptions, we prove an O(k-1) ergodic convergence rate. Furthermore, when f is also locally strongly convex and A has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.