Finite termination when $L_\beta$ reaches its limit value

Proved

NonconvexSplitting.ADMMKL.finite_termination

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}).

Let $(x^*,y^*,z^*)$ be a cluster point and $l^*:=L_\beta(x^*,y^*,z^*)$ . If $L_\beta(x^t,y^t,z^t)=l^*$ for some $t\ge1$ , then

x^{t+k}=x^t,\quad z^{t+k}=z^t\quad(k\ge0),\qquad y^{t+k}=y^{t+1}\quad(k\ge1),

so the algorithm terminates finitely.

This disposes of the degenerate case, so that the KL argument may assume $L_\beta(x^t,y^t,z^t)>l^*$ for all $t\ge1$ .

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.

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

/-- Finite termination (Li–Pong, p. 14, after (39)): along the ADMM under Assumption 1 with
`T₁ = 0` with a cluster point `(x*, y*, z*)`, if `L_β(x^t, y^t, z^t) = L_β(x*, y*, z*)` for some
`t ≥ 1`, then `x^{t+k} = x^t` and `z^{t+k} = z^t` for all `k ≥ 0` and `y^{t+k} = y^{t+1}` for all
`k ≥ 1`. -/
theorem finite_termination {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)))
    (t : ℕ) (ht : 1 ≤ t) (hl : augLag h P M β (x t) (y t) (z t) = augLag h P M β xs ys zs) :
    (∀ k : ℕ, x (t + k) = x t ∧ z (t + k) = z t) ∧ ∀ k : ℕ, 1 ≤ k → y (t + k) = y (t + 1) := by sorry

end NonconvexSplitting.ADMMKL

Source

Li & Pong, Global Convergence of Splitting Methods for Nonconvex Composite Optimization, arXiv:1407.0753v6, p. 14, the paragraph after (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

Finite termination when LβL_\betaLβ​ reaches its limit value

Finite termination when $L_\beta$ reaches its limit value