Eq. (16) — primal steps vanish when a cluster point exists

Proved

NonconvexSplitting.ProxADMM.eq16_primal_steps_vanish

by mikedeng1 · Sep 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

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

Assume the standing assumptions and Assumption 1, and let $(x^t, y^t, z^t)$ be generated by the proximal ADMM. If the sequence $\{(x^t, y^t, z^t)\}$ has a cluster point, then

\lim_{t\to\infty}\|x^{t+1} - x^t\| = 0 .

Combined with (14) and (15), this gives (9), the vanishing of all successive differences.

Formalization Note "Has a cluster point" is encoded as the existence of a strictly increasing $t_i$ along which $(x^{t_i}, y^{t_i}, z^{t_i})$ converges.

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.ProxADMM

/-- Eq. (16) of Li–Pong (pp. 8–9): if the proximal ADMM sequence has a cluster point, the
successive differences of the primal iterates tend to zero. -/
theorem eq16_primal_steps_vanish {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)
    (hcl : ∃ (s : ℕ → ℕ) (xs : EuclideanSpace ℝ (Fin n)) (ys zs : EuclideanSpace ℝ (Fin m)), StrictMono s ∧
      Tendsto (fun i => (x (s i), y (s i), z (s i))) atTop (𝓝 (xs, ys, zs))) :
    Tendsto (fun t => ‖x (t + 1) - x t‖) atTop (𝓝 0) := by sorry

end NonconvexSplitting.ProxADMM

Source

Li & Pong, Global Convergence of Splitting Methods for Nonconvex Composite Optimization, arXiv:1407.0753v6, p. 8, Eq. (16) (proved on p. 9)

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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