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arxiv: 2604.10422 · v1 · submitted 2026-04-12 · 🧮 math.OC · cs.SY· eess.SY

Distributed Optimization with Coupled Constraints over Time-Varying Digraph

Yeong-Ung Kim , Hyo-Sung Ahn This is my paper

Pith reviewed 2026-05-10 16:38 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords distributed optimizationtime-varying digraphcoupled constraintsprimal-dual algorithmnonsmooth convex functionsO(1/k) convergenceduality analysis
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The pith

A distributed algorithm achieves O(1/k) convergence for convex optimization with coupled constraints over time-varying digraphs.

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

The paper develops a fully distributed algorithm for convex optimization problems that have nonsmooth local objective functions and network-wide coupled equality and inequality constraints. It combines right-hand-side allocation decomposition with primal-dual methods to operate over time-varying directed graphs without requiring the exchange of primal variables. The algorithm is shown to converge at an O(1/k) rate in optimality through duality analysis, provided the local objectives are strongly convex and the subdifferentials of the local inequalities are bounded.

Core claim

The proposed algorithm integrates decomposition by right hand side allocation and primal-dual methods to solve distributed convex optimization problems with general nonsmooth local objectives and coupled equalities and inequalities over time-varying directed graphs in a fully distributed manner, achieving an O(1/k) rate of convergence in terms of optimality under strong convexity of local objectives and bounded subdifferential of local inequalities.

What carries the argument

Integration of right-hand-side allocation decomposition and primal-dual methods for handling coupled constraints over time-varying digraphs.

If this is right

  • The algorithm handles problems such as economic dispatch and network utility maximization in a distributed way.
  • Privacy is preserved since primal variables are not communicated.
  • The convergence holds for time-varying directed graphs satisfying the necessary connectivity conditions.
  • The method applies to nonsmooth convex functions without requiring differentiability.

Where Pith is reading between the lines

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

  • The duality analysis enables the method to work when the digraph is time-varying but remains connected in a joint sense over intervals.
  • Similar decomposition techniques could extend to related problems with different coupling structures if the strong convexity holds.

Load-bearing premise

Local objective functions are strongly convex, the subdifferential of local inequalities is bounded, and the time-varying digraph satisfies connectivity conditions for the duality analysis.

What would settle it

A numerical example satisfying strong convexity and bounded subdifferential where the optimality gap fails to decrease at O(1/k) rate would falsify the convergence guarantee.

Figures

Figures reproduced from arXiv: 2604.10422 by Hyo-Sung Ahn, Yeong-Ung Kim.

Figure 1
Figure 1. Figure 1: Convergence performance with 20 agents (a) Primal variables. (b) Objective value. (c) Feasibility gap. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Proof Diagram of Theorem 1 C. Convergence Analysis Methodology We summarize a methodology for convergence analysis called aggregate lower-bounding (ALB) proposed in [25]. In the convergence analysis based on Lyapunov method, one seeks a Lyapunov function satisfying L(x k+1) − L(x k ) ≤ −V (x k ) where V is a positive definite function. As the assumptions become harsher and the algorithm becomes more comple… view at source ↗
read the original abstract

In this paper, we develop a distributed algorithm for solving a class of distributed convex optimization problems where the local objective functions can be a general nonsmooth function, and all equalities and inequalities are network-wide coupled. This type of problem arises from many areas, such as economic dispatch, network utility maximization, and demand response. Integrating the decomposition by right hand side allocation and primal-dual methods, the proposed algorithm is able to handle the distributed optimization over networks with time-varying directed graph in fully distributed fashion. This algorithm does not require the communication of sensitive information, such as primal variables, for privacy issues. Further, we show that the proposed algorithm is guaranteed to achieve an $O(1/k)$ rate of convergence in terms of optimality based on duality analysis under the condition that local objective functions are strongly convex but not necessarily differentiable, and the subdifferential of local inequalities is bounded. We simulate the proposed algorithm to demonstrate its remarkable performance.

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

0 major / 2 minor

Summary. The manuscript develops a distributed primal-dual algorithm for convex optimization problems with network-wide coupled equality and inequality constraints over time-varying directed graphs. It combines right-hand-side allocation with primal-dual updates to accommodate nonsmooth strongly convex local objectives while avoiding communication of primal variables. The central result is an O(1/k) optimality convergence rate derived via duality analysis, under the assumptions that local objectives are strongly convex, the subdifferential of local inequality functions is bounded, and the time-varying digraph satisfies suitable connectivity conditions.

Significance. If the stated O(1/k) rate holds, the result is a useful extension of primal-dual distributed optimization methods to time-varying digraphs with coupled constraints and nonsmooth objectives. The fully distributed implementation and privacy-preserving property (no primal-variable exchange) are practical strengths for applications such as economic dispatch and network utility maximization. The approach relies on standard strong-convexity and bounded-subdifferential assumptions rather than ad-hoc parameters.

minor comments (2)
  1. The abstract states the O(1/k) rate but does not list the precise graph-connectivity assumption (e.g., uniform joint strong connectivity or similar) required for the duality argument; this should be stated explicitly in the main text near the theorem statement.
  2. The bounded-subdifferential assumption on local inequalities is used for step-size selection and error bounds; an explicit statement of how this bound enters the O(1/k) derivation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as the recommendation for minor revision. The referee's description correctly identifies the key elements of the work: the fully distributed primal-dual algorithm for nonsmooth strongly convex problems with coupled constraints over time-varying digraphs, the O(1/k) convergence rate, and the privacy-preserving property of not communicating primal variables.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its O(1/k) optimality convergence claim via standard duality analysis applied to a primal-dual algorithm that combines right-hand-side allocation with subgradient updates. This rests on explicit external assumptions (strong convexity of local objectives, bounded subdifferentials of local inequalities, and connectivity of the time-varying digraph sequence) that are not defined in terms of the target rate or any fitted parameter. No step reduces by construction to a self-definition, a renamed empirical pattern, or a load-bearing self-citation whose validity is presupposed inside the paper. The argument structure is self-contained against independent optimization benchmarks and does not invoke uniqueness theorems or ansatzes imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on domain assumptions of strong convexity and bounded subdifferentials; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Local objective functions are strongly convex
    Invoked to obtain the O(1/k) rate via duality analysis.
  • domain assumption Subdifferential of local inequalities is bounded
    Required for the convergence proof to hold.

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