Smoothness of the Augmented Lagrangian Dual in Convex Optimization

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.