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arxiv: 2606.20383 · v1 · pith:I6NMSF2Bnew · submitted 2026-06-18 · 🧮 math.OC

A Single-Loop Minorized Dual Decomposition Method for Nonsmooth Multi-Stage Stochastic Programming

Dan Luo , Hailin Sun , Lei Yang , Yang You This is my paper

Pith reviewed 2026-06-26 15:54 UTC · model grok-4.3

classification 🧮 math.OC
keywords multi-stage stochastic programmingdual decompositionalternating direction method of multipliersnonsmooth optimizationglobal convergencedecomposable structure
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The pith

A single-loop minorized dual decomposition method converges globally for nonsmooth multi-stage stochastic programs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a single-loop optimization algorithm for multi-stage stochastic programming problems with nonsmooth composite objectives. Each iteration builds a minorized version of the problem and its restricted Wolfe dual, then applies one iteration of symmetric Gauss-Seidel inexact alternating direction method of multipliers to the dual. This construction keeps the stage-wise and scenario-wise decomposability intact for parallel computation. The authors prove that the generated iterates converge globally for the three-stage case and extend the result to the general multi-stage setting.

Core claim

The method constructs a minorized subproblem and its restricted Wolfe dual at each step, then performs one iteration of the symmetric Gauss-Seidel based inexact ADMM on the dual to produce the next iterate. The updates preserve the intrinsic decomposable structure of the MSP problem. Global convergence of the generated iterates is established for the three-stage case, with the corresponding theorem extended to the general multi-stage setting.

What carries the argument

The single-loop minorized dual decomposition method that performs one iteration of symmetric Gauss-Seidel inexact ADMM on the restricted Wolfe dual of a minorized problem while preserving stage-wise and scenario-wise decomposability.

Load-bearing premise

The minorized subproblem and its restricted Wolfe dual, when updated with one iteration of symmetric Gauss-Seidel inexact ADMM, produce iterates whose convergence can be established under the stage-wise and scenario-wise decomposable structure of the MSP problem.

What would settle it

A concrete three-stage stochastic program instance with nonsmooth objectives on which the generated iterates fail to converge to a solution would falsify the global convergence claim.

Figures

Figures reproduced from arXiv: 2606.20383 by Dan Luo, Hailin Sun, Lei Yang, Yang You.

Figure 1
Figure 1. Figure 1: Flowchart for Solving Problem (2.2). Remark 2.1. In Algorithm 1, the symbol “≈” indicates that the corresponding block sub￾problem is solved inexactly. Specifically, for example, the update in (2.8) means that w˜ k+1 1 is computed such that there exists an error vector d k w˜1 satisfying d k w˜1 ∈ ∂w˜1Lb x k 1:3 σ  v k 1:3,( ˜w k+1 1 , wk 2 , wk 3 ), yk 1:3, zk 1:3; x k 1:3 with ∥d k w˜1 ∥ ≤ εk, where εk… view at source ↗
Figure 2
Figure 2. Figure 2: Instance 1 (Experiment 1): ˜d = 5, T = 3 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Instance 2 (Experiment 1): ˜d = 200, T = 3 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Instance 3 (Experiment 1): ˜d = 400, T = 3 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Instance 4 (Experiment 1): ˜d = 5, T = 4 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Instance 5 (Experiment 1): ˜d = 200, T = 4 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Instance 6 (Experiment 1): ˜d = 400, T = 4 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Instance 7 (Experiment 1): ˜d = 5, T = 5 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Instance 1 (Experiment 2): d = 5, T = 3 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Instance 2 (Experiment 2): d = 200, T = 3 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Instance 3 (Experiment 2): d = 400, T = 3 For the case of exponential–ℓ1 composite objective functions, the results reported in Ta￾bles 4–6 and Figures 9–15 further illustrate the computational advantage of the SL-MDD method. In particular, Figures 9–15 show that it attains an optimal solution satisfying the KKT conditions in substantially fewer iterations than P-sGS-iADMM. Moreover, as the prob￾lem dimen… view at source ↗
Figure 12
Figure 12. Figure 12: Instance 4 (Experiment 2): d = 5, T = 4 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Instance 5 (Experiment 2): d = 200, T = 4 (a) Function value vs iteration (b) KKT residuals vs iteration (c) Parallel time vs iteration [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Instance 6 (Experiment 2): d = 400, T = 4 28 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Instance 7 (Experiment 2): d = 5, T = 5 [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
read the original abstract

In this paper, we study multi-stage stochastic programming (MSP) problems with nonsmooth composite objectives. Tailored to their intrinsic stage-wise and scenario-wise structure, we develop a single-loop minorized dual decomposition method, in which each iteration constructs a minorized problem and its restricted Wolfe dual, and then performs \textit{one iteration} of the symmetric Gauss--Seidel based inexact alternating direction method of multipliers on the resulting dual problem to generate the next iterate. A key feature of the proposed optimization framework is that the resulting updates preserve the stage-wise and scenario-wise decomposable structure of the MSP problem and are suitable for parallel implementation. We establish global convergence of the generated iterates for the three-stage case and further establish the corresponding global convergence theorem for the general multi-stage setting. Numerical experiments illustrate the computational viability of the proposed framework and its favorable scaling behavior with respect to the stage-wise and scenario-wise structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a single-loop minorized dual decomposition algorithm for nonsmooth multi-stage stochastic programs. At each iteration a minorized subproblem and its restricted Wolfe dual are formed; a single symmetric Gauss-Seidel iteration of inexact ADMM is then applied to the dual, producing an update that preserves stage-wise and scenario-wise decomposability. Global convergence of the iterates is claimed for the three-stage case, with an inductive extension asserted for the general multi-stage setting. Numerical results illustrate practical performance and scaling.

Significance. If the stated convergence theorems hold, the framework would supply a structure-preserving, parallelizable single-loop method for a practically important class of nonsmooth MSP problems. The explicit use of one inexact ADMM step while retaining global convergence guarantees is a noteworthy technical feature.

major comments (2)
  1. [proof of the general multi-stage convergence theorem] The global convergence theorem for the general multi-stage case (abstract and the corresponding theorem statement) rests on an inductive argument that transfers the sufficient-descent or Lyapunov inequality established for three stages to an arbitrary number of stages. It is not shown that the one-iteration symmetric Gauss-Seidel inexactness error remains controlled by constants independent of the number of stages and of the Lipschitz moduli of the nonsmooth terms; this step is load-bearing for the general claim.
  2. [three-stage convergence analysis] The three-stage convergence analysis (presumably the section establishing the base case) must verify that the minorized subproblem plus one restricted-Wolfe-dual ADMM step produces a uniform error bound compatible with the outer minorization sequence. Without an explicit statement of the admissible inexactness tolerance and its dependence on the stage-wise data, it is unclear whether the induction hypothesis can be closed.
minor comments (2)
  1. Notation for the restricted Wolfe dual and the symmetric Gauss-Seidel ordering should be introduced with an explicit algorithmic listing (e.g., Algorithm 1) rather than only in the text.
  2. The numerical section would benefit from a table reporting iteration counts and wall-clock times versus number of stages and scenarios to quantify the claimed scaling behavior.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address the major comments point by point below and will make revisions to clarify the convergence analysis as needed.

read point-by-point responses

  1. Referee: [proof of the general multi-stage convergence theorem] The global convergence theorem for the general multi-stage case (abstract and the corresponding theorem statement) rests on an inductive argument that transfers the sufficient-descent or Lyapunov inequality established for three stages to an arbitrary number of stages. It is not shown that the one-iteration symmetric Gauss-Seidel inexactness error remains controlled by constants independent of the number of stages and of the Lipschitz moduli of the nonsmooth terms; this step is load-bearing for the general claim.

    Authors: We agree that the control of the inexactness error in the inductive step is crucial. In the current manuscript, the error bound is established using the uniform boundedness of the dual variables and the properties of the minorized subproblems, which ensure the constants are independent of the number of stages. However, to address this concern explicitly, we will add a dedicated lemma in the revision that proves the inexactness tolerance can be chosen independently of the stage count and depends only on local Lipschitz constants in a way that closes the induction. revision: yes

  2. Referee: [three-stage convergence analysis] The three-stage convergence analysis (presumably the section establishing the base case) must verify that the minorized subproblem plus one restricted-Wolfe-dual ADMM step produces a uniform error bound compatible with the outer minorization sequence. Without an explicit statement of the admissible inexactness tolerance and its dependence on the stage-wise data, it is unclear whether the induction hypothesis can be closed.

    Authors: The three-stage analysis derives the error bound from the one-step ADMM update on the restricted Wolfe dual, using the strong convexity of the augmented Lagrangian and the minorization error. The admissible inexactness is tied to the minorization parameter sequence, which decreases appropriately. We will revise the manuscript to include an explicit statement of the tolerance and its dependence on the stage-wise Lipschitz moduli to make the base case clearer and facilitate the induction. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence proofs rely on independent descent analysis and induction

full rationale

The paper constructs the minorized dual decomposition method from standard ADMM and Wolfe dual techniques applied to the MSP structure. Global convergence for the three-stage case is established via a Lyapunov inequality or sufficient-descent property under the one-iteration symmetric Gauss-Seidel inexact ADMM update. The general multi-stage case extends this by induction on the number of stages, preserving the same error bounds and decomposability. No step reduces by definition to its inputs, no parameters are fitted and relabeled as predictions, and any self-citations are not load-bearing for the central claims. The derivation is self-contained against external benchmarks such as standard inexact ADMM convergence theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical assumptions for convergence of inexact ADMM and minorization techniques in convex optimization; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Standard assumptions on the MSP problem (convexity, properness, and decomposability) that enable the minorized dual to preserve structure and allow convergence analysis.
    Invoked to justify the global convergence claims for three-stage and general cases.

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