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arxiv: 2605.15594 · v1 · pith:BKUTEOQ7new · submitted 2026-05-15 · 🧮 math.OC

Decomposition and Successive Decomposition Methods and Algorithms for Nonconvex Optimization

Yiqing Zhai , Ying Cui , Danny H. K. Tsang This is my paper

Pith reviewed 2026-05-20 17:58 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonconvex optimizationdecomposition methodsprimal decompositiondual decompositionsuccessive decompositionparallel algorithmsstationary pointsdistributed optimization
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The pith

Decomposition methods for nonconvex problems with general coupling structures produce stationary points and support parallel implementations.

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

The paper develops primal and dual decomposition methods plus their successive versions for nonconvex optimization problems whose coupling variables and constraints are more general than those treated before. These methods break the original problem into subproblems that can run in parallel or across distributed nodes while still reaching a stationary point of the full problem. A sympathetic reader would value this because it broadens the range of nonconvex problems that can be tackled with decomposition techniques, potentially improving speed and scalability in applications that already use such structures.

Core claim

The authors state that primal and dual decomposition and successive primal and successive dual decomposition methods and algorithms can be constructed for nonconvex problems possessing decomposition structures more general than those in existing literature. The methods exploit those structures, permit parallel and distributed implementations, generate stationary points of the original problems, and create opportunities for better convergence-computation tradeoffs. They generalize the standard decomposition methods for convex problems and extend prior successive methods for nonconvex problems, thereby enriching decomposition theory.

What carries the argument

Primal and dual decomposition that separates the problem via coupling variables and constraints, extended through successive iterations that refine subproblem solutions.

If this is right

  • The methods allow parallel and distributed implementations of the algorithms.
  • They produce stationary points of the original nonconvex problems.
  • They offer opportunities to achieve better tradeoffs between convergence performance and computation time.
  • They extend to indirect and two-level decomposition methods and algorithms.
  • They generalize convex decomposition methods and extend existing nonconvex ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The parallel capability could reduce wall-clock time for large nonconvex problems that arise in network design or resource allocation.
  • Numerical examples in the paper could be used to benchmark against other splitting techniques on the same test instances.
  • The framework might combine with existing convex solvers for the subproblems to handle mixed convex-nonconvex structures.

Load-bearing premise

The nonconvex problems must possess identifiable decomposition structures with coupling variables and constraints that let the algorithms converge to stationary points under the given conditions.

What would settle it

A concrete nonconvex test problem with coupling nonlinear equality constraints on which one of the proposed algorithms diverges or reaches a non-stationary point would disprove the convergence claim.

Figures

Figures reproduced from arXiv: 2605.15594 by Danny H. K. Tsang, Ying Cui, Yiqing Zhai.

Figure 1
Figure 1. Figure 1: Solution framework. (SDD-M), which converts the original nonconvex problem into a sequence of successively refined convex subproblems and master (dual) problems and does not rely on the original problem’s strong duality. Based on SDD-M, we present a Successive Dual Decomposition Algorithm (SDD-A), which slightly extends SCADD [20] to nonconvex problems with coupling linear equality constraints. 3) We compa… view at source ↗
Figure 2
Figure 2. Figure 2: Best objective values and minimum computation times for Examples 1-3 with coupling variables. [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Best objective values and minimum computation times for Examples 4-6 with coupling constraints. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

Existing results on decomposition methods and algorithms for nonconvex problems are minimal. Parallel decomposition algorithms do not exist for nonconvex problems with coupling nonlinear equality constraints. Besides, decomposition structures (i.e., coupling variables and constraints) are not fully exploited in designing decomposition methods and algorithms. In this paper, we consider nonconvex problems with decomposition structures that are more general than those handled in the existing literature. We propose primal and dual decomposition and successive primal and successive dual decomposition methods and algorithms for these nonconvex problems, which exploit decomposition structures, allow for parallel and distributed implementations, produce the original nonconvex problems' stationary points, and offer good opportunities to achieve superior tradeoff between convergence performance and computation time. Finally, we compare the proposed methods and algorithms, extend them to indirect and two-level decomposition methods and algorithms, and provide examples and numerical results to demonstrate their respective values. Notably, the proposed decomposition and successive decomposition methods and algorithms generalize the basic ones for convex problems, and the proposed successive decomposition methods and algorithms extend the existing ones for nonconvex problems, together enriching the decomposition theory.

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 manuscript proposes primal and dual decomposition methods together with their successive variants for nonconvex optimization problems possessing general decomposition structures (coupling variables and nonlinear equality constraints). The central claims are that these algorithms exploit the structures for parallel and distributed implementations, generate sequences whose limit points are stationary points of the original undecomposed problem, and furnish favorable trade-offs between convergence rate and computational effort. The work also presents indirect and two-level extensions, comparisons among the variants, illustrative examples, and numerical results, while asserting that the new methods generalize the classical convex case and extend prior nonconvex decomposition approaches.

Significance. If the stationarity claims are placed on a rigorous footing, the contribution would meaningfully enlarge the set of decomposition techniques available for nonconvex problems, particularly those with nonlinear coupling equalities where parallel algorithms have been scarce. Successful validation would directly support distributed and parallel solution of large-scale nonconvex programs arising in engineering and operations research.

major comments (2)
  1. [Abstract and dual-decomposition convergence analysis] Abstract and the convergence statements for the dual and successive-dual algorithms: the assertion that these methods produce stationary points of the original nonconvex problem with nonlinear coupling equalities is load-bearing yet rests on unstated regularity conditions. The manuscript must supply an explicit constraint qualification (e.g., MFCQ or LICQ at limit points) and a precise argument showing how the multiplier updates recover the KKT conditions of the undecomposed problem; without this, the correspondence between decomposed stationary points and original stationary points remains unverified.
  2. [Successive dual decomposition section] Section presenting the successive dual decomposition algorithm: the proof that limit points satisfy first-order optimality for the original problem must address the nonsmoothness and nonconcavity of the dual function that arise once the coupling equalities are nonlinear; the current sketch appears to omit the necessary subdifferential or variational inequality argument that closes this gap.
minor comments (2)
  1. [Numerical results] The numerical examples would be strengthened by reporting iteration counts, wall-clock times, and feasibility residuals side-by-side with at least one established nonconvex decomposition baseline.
  2. [Notation and problem formulation] Notation for the coupling constraints and the associated multipliers should be unified across the primal, dual, and successive variants to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions. We address each major comment below and will make the necessary revisions to strengthen the convergence analysis.

read point-by-point responses

  1. Referee: [Abstract and dual-decomposition convergence analysis] Abstract and the convergence statements for the dual and successive-dual algorithms: the assertion that these methods produce stationary points of the original nonconvex problem with nonlinear coupling equalities is load-bearing yet rests on unstated regularity conditions. The manuscript must supply an explicit constraint qualification (e.g., MFCQ or LICQ at limit points) and a precise argument showing how the multiplier updates recover the KKT conditions of the undecomposed problem; without this, the correspondence between decomposed stationary points and original stationary points remains unverified.

    Authors: We acknowledge that the current presentation of the convergence results for the dual decomposition and successive dual decomposition algorithms would benefit from explicit regularity assumptions. In the revised version, we will state the Mangasarian-Fromovitz Constraint Qualification (MFCQ) at the limit points. Additionally, we will expand the proof to include a step-by-step derivation demonstrating how the multiplier updates in the algorithm correspond to the KKT conditions of the original problem, particularly the stationarity with respect to the coupling variables and constraints. This will be added to the relevant sections on dual decomposition. revision: yes

  2. Referee: [Successive dual decomposition section] Section presenting the successive dual decomposition algorithm: the proof that limit points satisfy first-order optimality for the original problem must address the nonsmoothness and nonconcavity of the dual function that arise once the coupling equalities are nonlinear; the current sketch appears to omit the necessary subdifferential or variational inequality argument that closes this gap.

    Authors: We agree that the proof sketch for the successive dual decomposition requires strengthening to handle the nonsmooth and nonconcave nature of the dual function induced by nonlinear equalities. We will revise this section to incorporate a subdifferential inclusion argument. Specifically, we will show that any limit point satisfies a variational inequality involving the subdifferential of the dual function, which in turn implies the first-order optimality conditions for the original nonconvex problem. This addition will close the identified gap. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic proposals and convergence claims are constructive, not self-referential

full rationale

The paper introduces primal/dual and successive decomposition algorithms for nonconvex problems with general coupling structures, claiming they produce stationary points of the original problem while generalizing convex cases. These are presented as new constructive methods with parallel/distributed implementations, not as derivations that reduce by construction to fitted inputs, self-definitions, or prior self-citations. No equations equate a 'prediction' to a fitted parameter, no uniqueness theorem is imported from the authors' own prior work to force the result, and the abstract's extension statements do not rename known results or smuggle ansatzes. The derivation chain consists of algorithm design steps whose validity rests on stated conditions rather than tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Without the full manuscript, specific free parameters, axioms, or invented entities cannot be identified; the abstract implies reliance on standard assumptions in nonconvex optimization such as differentiability and existence of stationary points.

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