# Smoothness of the Augmented Lagrangian Dual in Convex Optimization

arXiv: https://arxiv.org/abs/2505.01824 · HTML: https://mlsanity.com/paper/2505.01824/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2025.

## Built on by (most cited first, 1 of 1)

- [Augmented Lagrangian Method for Last-Iterate Convergence for Constrained MDPs](https://mlsanity.com/paper/2605.11694/) (2026)

## Abstract (arXiv)

This paper focuses on the general linearly constrained optimization problem: minx ∈ ℝd f(x)  s.t.  Ax = b, where f: ℝd rightarrow ℝ cup +∞ is a closed proper convex function, A ∈ ℝp × d, and b ∈ ℝp. We define the standard dual function φ(λ) = infx f(x) + langle λ, A x - b rangle, the augmented Lagrangian Lρ(x, λ) = f(x) + langle λ, Ax - b rangle + fracρ2|Ax - b|2 (ρ > 0), and the augmented Lagrangian dual function φρ(λ) = infx Lρ(x, λ). Under the fundamental condition that dom  φ neq emptyset, we establish that: (1) φρ is frac1ρ-smooth everywhere; and (2) the solution to minx ∈ ℝd Lρ(x, λ) exists for any λ ∈ ℝp. These theoretical findings substantially weaken the stringent assumptions typically imposed in the literature to ensure such properties.