# A Universal Primal-Dual Convex Optimization Framework

arXiv: https://arxiv.org/abs/1502.03123 · HTML: https://mlsanity.com/paper/1502.03123/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2015.

## Built on by (most cited first, 5 of 5)

- [Last-Iterate Convergent Policy Gradient Primal-Dual Methods for Constrained MDPs](https://mlsanity.com/paper/2306.11700/) (2023)
- [First-Order Methods for Convex Optimization](https://mlsanity.com/paper/2101.00935/) (2021)
- [Fast, Scalable, Warm-Start Semidefinite Programming with Spectral Bundling and Sketching](https://mlsanity.com/paper/2312.11801/) (2023)
- [Adaptive proximal gradient methods are universal without approximation](https://mlsanity.com/paper/2402.06271/) (2024)
- [Nonsmooth Projection-Free Optimization with Functional Constraints](https://mlsanity.com/paper/2311.11180/) (2023)

## Abstract (arXiv)

We propose a new primal-dual algorithmic framework for a prototypical constrained convex optimization template. The algorithmic instances of our framework are universal since they can automatically adapt to the unknown Holder continuity degree and constant within the dual formulation. They are also guaran- teed to have optimal convergence rates in the objective residual and the feasibility gap for each Holder smoothness degree. In contrast to existing primal-dual algorithms, our framework avoids the proximity operator of the objective function. We instead leverage computationally cheaper, Fenchel-type operators, which are the main workhorses of the generalized conditional gradient (GCG)-type methods. In contrast to the GCG-type methods, our framework does not require the objective function to be differentiable, and can also process additional general linear inclusion constraints, while guarantees the convergence rate on the primal problem