Eq. (39) — $L_\beta(x^t,y^t,z^t)\to L_\beta(x^,y^,z^*)$ at a cluster point

Proved

NonconvexSplitting.ADMMKL.eq39_lagrangian_limit

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}).

If $(x^*,y^*,z^*)$ is a cluster point of $(x^t,y^t,z^t)$ , then

\lim_{t\to\infty}L_\beta(x^t,y^t,z^t)=l^*:=L_\beta(x^*,y^*,z^*).

The limit value $l^*$ is the level at which the KL inequality is applied.

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 limit is taken in EReal. "Cluster point" means the limit of a subsequence $(x^{s(i)},y^{s(i)},z^{s(i)})$ with $s$ strictly increasing.

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. (39) of Li–Pong (p. 14): along the ADMM under Assumption 1 with `T₁ = 0`, if
`(x*, y*, z*)` is a cluster point then `L_β(x^t, y^t, z^t) → L_β(x*, y*, z*)` (in `EReal`). -/
theorem eq39_lagrangian_limit {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)
    (xs : EuclideanSpace ℝ (Fin n)) (ys zs : EuclideanSpace ℝ (Fin m))
    (hclu : ∃ s : ℕ → ℕ, StrictMono s ∧
      Tendsto (fun i => (x (s i), y (s i), z (s i))) atTop (𝓝 (xs, ys, zs))) :
    Tendsto (fun t => augLag h P M β (x t) (y t) (z t)) atTop
      (𝓝 (augLag h P M β xs ys zs)) := by sorry

end NonconvexSplitting.ADMMKL

Source

Li & Pong, Global Convergence of Splitting Methods for Nonconvex Composite Optimization, arXiv:1407.0753v6, p. 14, Eq. (39)

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. (39) — Lβ(xt,yt,zt)→Lβ(x∗,y∗,z∗)L_\beta(x^t,y^t,z^t)\to L_\beta(x^*,y^*,z^*)Lβ​(xt,yt,zt)→Lβ​(x∗,y∗,z∗) at a cluster point

Eq. (39) — $L_\beta(x^t,y^t,z^t)\to L_\beta(x^,y^,z^*)$ at a cluster point