Eq. (20): one-step estimate of the augmented Lagrangian along the proximal ADMM

Proved

NonconvexSplitting.ADMMBounded.eq20_one_step

by mikedeng1 · Sep 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

admmnonconvex-optimizationp2o-batch-p200bp2o-gran-per-chapterp2o-plan-paperp2o-v1

Let $h : \mathbb{R}^n \to \mathbb{R}$ be twice continuously differentiable with bounded Hessian, $P : \mathbb{R}^m \to (-\infty, +\infty]$ proper and lower semicontinuous, $\phi : \mathbb{R}^n \to \mathbb{R}$ twice continuously differentiable and convex, $\mathcal M : \mathbb{R}^n \to \mathbb{R}^m$ linear with adjoint $\mathcal M^*$ , and $\beta > 0$ . Let $(x^t, y^t, z^t)_{t \ge 0}$ be a sequence generated by the proximal ADMM with penalty $\beta$ and proximal function $\phi$ (initial points $x^0, z^0$ arbitrary, $y^0$ unconstrained). Suppose Assumption 1 holds with witnesses $\sigma, \delta, \gamma$ and $\mathcal Q_1, \mathcal Q_2, \mathcal T_1, \mathcal T_2, \mathcal Q_3$ . For a self-map $\mathcal T$ write $\|x\|^2_{\mathcal T} := \langle x, \mathcal T x\rangle$ .

Then for every $t \ge 1$ ,

L_\beta(x^{t+1}, y^{t+1}, z^{t+1}) \le L_\beta(x^t, y^t, z^t) + \frac12\|x^{t+1} - x^t\|^2_{\frac{2}{\sigma\beta\gamma}\mathcal Q_3 - \delta\mathcal I - \mathcal T_2} + \frac12\|x^t - x^{t-1}\|^2_{\frac{2}{\sigma\beta(1-\gamma)}\mathcal T_1^2}.

This is the basic descent estimate of the proximal ADMM; summed along the iterations it gives the monotonicity of the merit sequence in Eq. (30).

Formalization Note The augmented Lagrangian $L_\beta$ takes values in $(-\infty, +\infty]$ , so the inequality is stated in EReal in additive form (the real terms moved to the side where they are added), which avoids extended-real subtraction. The weighted "norm" $\|x\|^2_{\mathcal T} = \langle x, \mathcal T x\rangle$ is defined for every self-map and can be negative when $\mathcal T$ is indefinite, as $\frac{2}{\sigma\beta\gamma}\mathcal Q_3 - \delta\mathcal I - \mathcal T_2$ may be. Iterates are indexed from $0$ ; $t \ge 1$ makes $x^{t-1}$ meaningful.

Preamble

import Mathlib
import Definitions.Def_NonconvexSplitting_Shared_StandingAssumptions
import Definitions.Def_NonconvexSplitting_Shared_IsProxADMMSeq
import Definitions.Def_NonconvexSplitting_Shared_Assumption1
open NonconvexSplitting.Shared

open Filter Topology
open scoped InnerProductSpace

Formal statement

namespace NonconvexSplitting.ADMMBounded

/-- Eq. (20) of Li–Pong (p. 8): one-step estimate for the augmented Lagrangian along the
proximal ADMM, for `t ≥ 1`, written additively in `EReal`. -/
theorem eq20_one_step {n m : ℕ} (h : EuclideanSpace ℝ (Fin n) → ℝ) (P : EuclideanSpace ℝ (Fin m) → EReal)
    (phi : EuclideanSpace ℝ (Fin n) → ℝ) (M : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin m)) (β : ℝ)
    (hstd : StandingAssumptions h P phi β)
    (σ δ γ : ℝ) (Q1 Q2 T1 T2 Q3 : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin n))
    (hA : Assumption1 h phi M β σ Q1 Q2 T1 T2 δ Q3 γ)
    (x : ℕ → EuclideanSpace ℝ (Fin n)) (y z : ℕ → EuclideanSpace ℝ (Fin m))
    (hseq : IsProxADMMSeq h P phi M β x y z)
    (t : ℕ) (ht : 1 ≤ t) :
    augLag h P M β (x (t + 1)) (y (t + 1)) (z (t + 1)) ≤
      augLag h P M β (x t) (y t) (z t) +
        ((1 / 2 * wnormSq ((2 / (σ * β * γ)) • Q3 - δ • (1 : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin n)) - T2) (x (t + 1) - x t) +
          1 / 2 * wnormSq ((2 / (σ * β * (1 - γ))) • (T1 * T1)) (x t - x (t - 1)) : ℝ) : EReal) := by sorry

end NonconvexSplitting.ADMMBounded

Source

Li & Pong, Global Convergence of Splitting Methods for Nonconvex Composite Optimization, arXiv:1407.0753v6, p. 8, Eq. (20)

Human review

Endorsed by Shuze Chen · Oct 1, 2026
Confirmed by the moderator at approval.
Endorsed by mikedeng1 · Oct 1, 2026
Confirmed by the mission captain (proposal self-audit).

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