Eq. (36) — sufficient decrease of $L_\beta$ along the ADMM

Proved

NonconvexSplitting.ADMMKL.eq36_sufficient_decrease

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 closed, $\mathcal M:\mathbb R^n\to\mathbb R^m$ linear and $\beta>0$ , with augmented Lagrangian $L_\beta(x,y,z)=h(x)+P(y)-\langle z,\mathcal Mx-y\rangle+\frac\beta2\|\mathcal Mx-y\|^2$ . Suppose Assumption 1 holds with $\mathcal T_1=0$ (hence $\phi=0$ ), with witnesses $\sigma,\mathcal Q_1,\mathcal Q_2,\delta,\mathcal Q_3,\gamma$ , and let $(x^t,y^t,z^t)_{t\ge0}$ be generated by the ADMM

y^{t+1}\in\operatorname*{Arg\,min}_y L_\beta(x^t,y,z^t),\qquad x^{t+1}\in\operatorname*{Arg\,min}_x L_\beta(x,y^{t+1},z^t),\qquad z^{t+1}=z^t-\beta(\mathcal Mx^{t+1}-y^{t+1}).

Then there is a constant $D>0$ such that for every $t\ge1$

L_\beta(x^t,y^t,z^t)-L_\beta(x^{t+1},y^{t+1},z^{t+1})\ge D\,\|x^{t+1}-x^t\|^2 .

In particular $\bigl(L_\beta(x^t,y^t,z^t)\bigr)_{t\ge1}$ is non-increasing.

This is the "sufficient decrease" half of the KL argument.

Formalization Note The ADMM is the relation IsProxADMMSeq with $\phi = 0$ (the Bregman term vanishes); Assumption 1 is instantiated with $\phi = 0$ and $\mathcal T_1 = 0$ , and $\mathcal T_2$ is left as a parameter (Assumption 1 forces $0 \preceq \mathcal T_2 \preceq \mathcal T_1 = 0$ ). Values of $L_\beta$ and $P$ live in EReal; iterates are indexed from $0$ , and $y^0$ is not constrained. The inequality is written additively, $L_\beta(x^{t+1},y^{t+1},z^{t+1})+D\|x^{t+1}-x^t\|^2\le L_\beta(x^t,y^t,z^t)$ in EReal. The paper's middle member $-\frac12\|x^{t+1}-x^t\|^2_{\frac{2}{\sigma\beta\gamma}\mathcal Q_3-\delta\mathcal I}$ is omitted.

Preamble

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

open Filter Topology
open scoped InnerProductSpace

Formal statement

namespace NonconvexSplitting.ADMMKL

/-- Eq. (36) of Li–Pong (p. 14): along the ADMM under Assumption 1 with `T₁ = 0`, the augmented
Lagrangian decreases by at least `D ‖x^{t+1} - x^t‖²` for a constant `D > 0` and all `t ≥ 1`
(written additively in `EReal`). -/
theorem eq36_sufficient_decrease {n m : ℕ} (h : EuclideanSpace ℝ (Fin n) → ℝ) (P : EuclideanSpace ℝ (Fin m) → EReal)
    (M : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin m)) (β : ℝ)
    (hstd : StandingAssumptions h P (fun _ => 0) β)
    (σ δ γ : ℝ) (Q1 Q2 T2 Q3 : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin n))
    (hA : Assumption1 h (fun _ => 0) M β σ Q1 Q2 0 T2 δ Q3 γ)
    (x : ℕ → EuclideanSpace ℝ (Fin n)) (y z : ℕ → EuclideanSpace ℝ (Fin m))
    (hseq : IsProxADMMSeq h P (fun _ => 0) M β x y z) :
    ∃ D : ℝ, 0 < D ∧ ∀ t : ℕ, 1 ≤ t →
      augLag h P M β (x (t + 1)) (y (t + 1)) (z (t + 1)) + ((D * ‖x (t + 1) - x t‖ ^ 2 : ℝ) : EReal) ≤
        augLag h P M β (x t) (y t) (z t) := by sorry

end NonconvexSplitting.ADMMKL

Source

Li & Pong, Global Convergence of Splitting Methods for Nonconvex Composite Optimization, arXiv:1407.0753v6, p. 14, Eq. (36)

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).

View graph

Eq. (36) — sufficient decrease of LβL_\betaLβ​ along the ADMM

Eq. (36) — sufficient decrease of $L_\beta$ along the ADMM