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Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that group-sparse and distributed LQ control can be solved directly, with no penalty parameter and no convex surrogate, by ADMM with Douglas-Rachford splitting, converging to stationary points and, for Hurwitz A, to the…

desk verdict The epi-composition reformulation and the Hurwitz/domain connection are worth a look, but the central convergence theorem rests on a nonexistent lemma and on an ℓ0 penalty term that is not lower semicontinuous, so the proof does not go through. read the letter →

arxiv 2507.19895 v4 pith:MB2MYDOS submitted 2025-07-26 math.OC cs.LG

Lechen Feng , Xun Li , Yuan-Hua Ni This is my paper · ORCID
classification math.OCcs.LG MSC 49M2790C2693B5290C22
keywords linear-quadraticproblemsparsefeedbackcontrolconstrainednonconvexoptimizationADMMDouglas-Rachfordsplittingepi-compositionfunctiongroupsparsityDCrelaxation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the group-sparse feedback linear-quadratic problem (SF-LQ) and the distributed LQ problem with fixed topology (DFT-LQ) can be solved directly, without a penalty parameter and without replacing the ℓ0-norm by a convex surrogate. The route is to rewrite the constrained problem as an unconstrained sum φ1(s)+φ2(s), where φ1 is an SDP value function arising by epi-composition and φ2 encodes the group-ℓ0 penalty, and then run Douglas-Rachford splitting, which the paper shows is equivalent to ADMM. The paper claims that when φ1 is L-smooth on its effective domain and the open-loop matrix A is Hurwitz, the iterates converge to a stationary point, and it characterizes that point as the global minimizer of a corresponding DFT-LQ problem. When smoothness fails, a projected subgradient descent method combined with a DC relaxation is proposed as an initialization mechanism. A sympathetic reader would care because the result promises sparsity-structured feedback design with convergence guarantees and no tuning of the penalty parameter, avoiding the spurious stationary points of penalty formulations.

What carries the argument

The load-bearing object is the epi-composition reformulation, φ1(s)=(Af)(s)=inf{f(W): AW=s} and φ2(s)=(Bg)(−s), which turns the affine constraint into an unconstrained sum. The iteration is the Douglas-Rachford splitting scheme un+1∈prox_{ηφ1}(sn), vn+1∈prox_{ηφ2}(2un−sn), sn+1=sn+ξ(vn+1−un+1), and Lemma 1 states that this is equivalent to the standard ADMM updates through a change of variables. Theorem 3 supplies the linchpin: A is Hurwitz if and only if dom(φ1) is the whole space, which is why the convergence analysis splits into the smooth, Hurwitz case and the nonsmooth case. The proximal subproblems are solved either through the ADMM form using the SDP-constrained methods of the companion paper [1], or through the explicit subdifferential formula derived in Theorem 1.

What would settle it

Compute the gradient of φ1 along a line that crosses a PSD-cone active-set boundary for a small Hurwitz LQ instance, such as the three-state Example 1; if the gradient jumps, Assumption 3 is violated and the smoothness premise of the theorem fails for that instance. Independently, inspect the cited Lemma 3: if it truly does not exist, the proof as written has a gap that must be repaired before the convergence claim is established.

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Extended reading notes

Core claim

The paper centers on the nonconvex, nonsmooth constrained problem min f(W)+g(P) subject to AW+BP=0, which underlies both SF-LQ and DFT-LQ, and it claims this problem can be attacked directly by ADMM through its equivalence with Douglas-Rachford splitting on min_s φ1(s)+φ2(s), where φ1(s)=(Af)(s) and φ2(s)=(Bg)(−s). It proves that φ1 is closed and convex, gives a subdifferential formula A⊤s∈∂f(W), and establishes the pivotal fact that dom(φ1)=$R^{{N*+mn}}$ if and only if A is Hurwitz. Assuming φ1 is L-smooth on its effective domain, Theorem 4 states that for Hurwitz A the DR iterates converge to a stationary point of φ=φ1+φ2, which the abstract identifies as the global minimizer of a corresponding DFT-LQ problem; Theorem 5 covers non-Hurwitz A with uk−vk converging to zero at rate o(1/√k) and every cluster point satisfying the first-order optimality condition. For the nonsmooth case, the paper proves stability of strict local minimizers of a DC-relaxed objective when A is Hurwitz and proposes subgradient descent to warm-start the DR iterations.

Load-bearing premise

The convergence theorems rest on Assumption 3, that φ1(s)=(Af)(s) is Lipschitz-smooth on its effective domain; the paper does not prove this for the LQ problem, and the proof of Theorem 4 cites a missing Lemma 3 to extend it to full-space smoothness when A is Hurwitz, so the argument from assumptions to the stated convergence is incomplete.

Editorial extensions

If this is right

  • Direct group-sparse feedback design becomes possible: solving the constrained problem by ADMM yields sparsity-structured stabilizing gains without tuning a penalty parameter ρ.
  • For Hurwitz A, the stationary point reached by DR splitting is the global minimizer of the corresponding DFT-LQ problem, meaning the nonconvex ℓ0 formulation is exact in that setting rather than merely a relaxation.
  • When A is not Hurwitz, the algorithm still provides subsequential convergence, with uk−vk converging to zero at rate o(1/√k) and every cluster point satisfying the first-order optimality condition.
  • In the nonsmooth case, subgradient descent on the DC relaxation reaches stable strict local minima for Hurwitz A and can serve as a warm start that moves the iterates toward the smooth regime.
  • Numerical comparisons indicate that solving the ℓ0 problem directly yields lower LQ cost than ℓ1-relaxation baselines when the feedback gain lies far from the origin, a regime where the ℓ1 surrogate dominates the objective.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the smoothness certificate for φ1 on its effective domain can be established, the theorem suggests a testable exactness property: ADMM on the nonconvex ℓ0 problem would recover the DFT-LQ global optimum, which could be checked against mixed-integer SDP reference solutions on random small systems.
  • The Hurwitz characterization points to a possible extension: pre-stabilizing a general system with a preliminary feedback that preserves block-sparsity structure might bring a wider class of stabilizable systems into the smooth regime.
  • The warm-start idea is directly amenable to benchmarking: one could measure how often DR splitting converges to a stationary point when initialized by the subgradient-DC method versus random initialization.
  • Because Assumption 3 is not verified in the paper for the LQ value function, a natural next step is to characterize exactly which LQ data make φ1 L-smooth on its domain, and whether SDP active-set changes create gradient discontinuities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper reformulates the distributed LQ (DFT-LQ) and sparse feedback LQ (SF-LQ) problems as constrained nonsmooth, nonconvex optimization problems with a group ℓ0 penalty, and then studies direct ADMM/DR splitting on the equivalent epi-composition form min φ1(s)+φ2(s). The main theoretical claims are: Theorem 4 gives convergence of DR iterates to a stationary point under Assumption 3 when A is Hurwitz; Theorem 5 gives cluster-point stationarity when A is not Hurwitz; and Section 4.3 proposes a subgradient/DC-relaxation method for the nonsmooth case. Numerical examples compare the method with MISDP, convex relaxations, LQRSP, S-PI, and the companion penalty-based PALM method.

Significance. If the convergence results were correct, the paper would usefully extend Douglas-Rachford splitting to problems in which both terms are nonsmooth and one is nonconvex, and it would provide a non-penalty method for group-sparse feedback design. The paper is also honest about limitations in the numerical section, noting the need to tune β and the loss of sparsity for large β. However, the central theorem is not sound as written: the closedness claim for φ2 is false, the proof of Theorem 4 cites a nonexistent lemma, and Assumption 3 is never verified. These are load-bearing gaps, so the advertised convergence guarantees are not established.

major comments (5)
  1. [Theorem 2, Section 4.1] Theorem 2 asserts that φ2 is a proper closed function, but the displayed formula φ2(s) = γ∥π(−[0_{mn×N*} I_{mn}]s)∥0 + δ_{A(s)}(s) is not lower semicontinuous. Take s_k = (0, e_1/k) with e_1 a unit vector in R^{mn}; then φ2(s_k) = γ for every k while φ2(0) = 0, so liminf_k φ2(s_k) = γ > φ2(0). Thus φ2 is not lsc at 0. Since Theorem 4 invokes Theorem 4.4 of [14], which requires the second function to be proper and lower semicontinuous, the appeal to [14] is invalid. Theorem 5 is affected as well, because its proof uses subdifferentials of φ2, which are only defined for lower semicontinuous functions in the framework of Section 2. This obstruction is independent of Assumption 3.
  2. [Assumption 3 and Theorems 4-5, Section 4.2] Assumption 3, that φ1 is L-smooth on dom(φ1), is never verified for the LQ problem. Theorem 1 establishes only convexity and closedness of φ1, and Theorem 3 relates the Hurwitz property to dom(φ1) = R^{N*+mn}; neither result implies L-smoothness on the domain. Since φ1 is the value function of an SDP with PSD cone constraints, nonsmoothness at active-set changes is a standard phenomenon, so this is not a harmless technical condition. The paper must either prove Assumption 3 for the SF-LQ/DFT-LQ instance or replace it with a verifiable sufficient condition; as written, both Theorem 4 and Theorem 5 rest on an unproved hypothesis.
  3. [Proof of Theorem 4, Section 4.2] The proof states: 'By Lemma 3 and Assumption 3, we know that φ1 is Lφ1-smooth on R^{N*+mn}.' No Lemma 3 exists in the manuscript; the numbered lemmas are Lemma 1 and, in the appendix, Lemma 2. Moreover, even the Hurwitz property would only give dom(φ1) = R^{N*+mn} by Theorem 3, not L-smoothness on the whole space. The chain from assumptions to the conclusion of Theorem 4 is therefore incomplete.
  4. [Abstract and Theorem 4] The abstract claims that the DR splitting converges to a stationary point and that 'we characterize this stationary point as the global minimizer of a corresponding DFT-LQ problem.' No theorem in the manuscript proves global optimality: Theorem 4 gives only stationarity of φ, and Theorem 5 gives cluster-point stationarity. If the global-minimizer statement is intended, it must be proved or removed; otherwise the abstract overstates the results.
  5. [Section 5, Example 1] The numerical experiments do not support the advertised advantage of avoiding parameter tuning. In Example 1, with β = 10 and β = 100 the ADMM iterates only show cluster-point or subsequence behavior, and full convergence is obtained only by increasing β to 300. The paper acknowledges this and states that β must be chosen case by case as a balance between oscillation and sparsity. This is effectively a parameter-tuning requirement similar to the penalty parameter ρ that the paper claims to avoid, and it should be stated as a limitation in the contributions section.
minor comments (5)
  1. [Lemma 1, Section 4.2] The proof of Lemma 1 is omitted with the remark that it is 'commonly found in the literature.' Since the equivalence between ADMM (15) and DR splitting (19) is central to the paper, a proof or a precise reference to the exact statement should be provided.
  2. [Notation, Section 2] The quantity N* is defined only as 'the total number of rows in bA'; since it appears throughout the theoretical results, it should be defined explicitly in terms of the problem dimensions and the constraints defining bA.
  3. [Section 4.3, Eq. (21)] The proximal subproblem for prox_{ηφ1} is not solved in closed form; the paper proposes a subgradient method to solve (21), but the inner-loop stopping criterion and its effect on the outer DR convergence analysis are not discussed. The convergence results in Section 4.2 assume exact evaluations of prox_{ηφ1}.
  4. [Theorem 3, Section 4.1] The proof of Theorem 3 is only sketched, and phrases such as 'as implied by the properties of diagonally dominant matrices' are not a complete argument. In particular, the converse direction assumes wi ∈ ker(bn) for all i without justifying that this is without loss of generality for arbitrary s2.
  5. [Abstract, general] There are two versions of the abstract in the submitted material, and they differ on whether the stationary point is claimed to be a global minimizer of the corresponding DFT-LQ problem. The final version should be consistent and should not claim global optimality unless a theorem supports it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the DR/ADMM convergence claim is an external-theorem application, not a self-referential derivation; the main gaps (nonexistent Lemma 3, false closedness of the ℓ0 epi-composition) are correctness failures, not circular reductions.

full rationale

No significant circularity found. The chain from the constrained problem (12) to the unconstrained form (16)-(18) is an exact epi-composition equivalence obtained by taking infima over auxiliary variables, not a definition of the conclusion in terms of the conclusion. The convergence statement in Theorem 4 is imported from Themelis-Patrinos [14], a genuinely external theorem, with Assumption 3 and Theorem 3 used only to argue that in the Hurwitz case the problem falls into the smooth full-domain regime; no parameter is fitted and later renamed as a prediction. The paper does lean on the same-author companion work [1] for the original SF-LQ/DFT-LQ reformulation, for the semi-algebraicity of φ2, and for subproblem solvers; these are citations rather than circular definitions, and they do not encode the target convergence result. The Hurwitz case is explicitly an application of [14], so the contribution is reduced novelty rather than a circular derivation. Two in-scope proof gaps should be weighed as correctness risks rather than circularity: Theorem 4's proof cites a nonexistent 'Lemma 3' to pass from Assumption 3 to global L-smoothness of φ1, and Theorem 2's assertion that φ2 is closed/lower semicontinuous is false for the group-ℓ0 term (for instance, s_k = e_1/k gives φ2(s_k) = γ while φ2(0) = 0, so liminf φ2(s_k) > φ2(0)); consequently the lower-semicontinuity hypothesis of [14, Thm 4.4] is not met. These are gaps or false assertions, not self-referential reductions, so the circularity score remains 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities or mediators are introduced; the penalty parameter β and sparsity weight γ are algorithmic, not entities. The central claim rests on Assumption 3 and imported theorems rather than on fitted constants.

free parameters (1)
  • β (augmented Lagrangian/ADMM parameter) = chosen per example (10, 30, 50, 100, 300)
    The theoretical convergence only requires an inequality on η=1/β, but the demonstrated convergence in Example 1 is obtained by increasing β until oscillations vanish; the paper calls this a case-by-case trade-off with sparsity.
assumptions (4)
  • domain assumption Assumption 1: C^T D=0, D^T D≻0, C^T C≻0, B1 B1^T≻0, (A,B2) stabilizable, (A,C) has no unobservable modes on the imaginary axis.
    Standard assumptions for the LQ problem; not restrictive for the target setting.
  • ad hoc to paper Assumption 3: φ1 is L-smooth on dom(φ1).
    This is the load-bearing smoothness condition. The paper does not verify it, and for an SDP value function it is likely false at active-set changes. It appears in Assumption 3, Section 4.2.
  • standard math Theorem 4.4 of Themelis and Patrinos [14] applies.
    The paper imports the convergence conclusion for the smooth case from [14]; no self-contained proof is provided.
  • domain assumption A is Hurwitz implies dom(φ1)=R^{N*+mn} (Theorem 3).
    Used to reduce Assumption 3 to global smoothness in Theorem 4; the proof of Theorem 3 is itself abbreviated and partly asserted.

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Cite this review

Pith. "Pith review of Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control." pith.science (2026). https://pith.science/paper/MB2MYDOS

@misc{pith2026250719895,
  author       = {Pith},
  title        = {Pith review of: Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MB2MYDOS}},
  note         = {Machine review of arXiv:2507.19895}
}
abstract

In this paper, we study the distributed linear quadratic problem with fixed communication topology (DFT-LQ) and the sparse feedback linear quadratic (SF-LQ) problem through a unified optimization framework. Specifically, both problems are formulated as a nonconvex, nonsmooth optimization problem equipped with an $\ell_0$-penalty under affine constraints. To solve this problem, we first investigate the application of the Douglas-Rachford (DR) splitting algorithm. Under the local condition that the generated iterates remain on a fixed smooth manifold, we establish the convergence of the DR splitting to a stationary point. Furthermore, we characterize this stationary point as the global minimizer of a corresponding DFT-LQ problem. To bypass the restriction of the smooth manifold assumption, we introduce a projected subgradient descent algorithm that achieves global convergence without relying on smooth-manifold structures. This algorithm may serve as a warm-start mechanism that effectively drives the iterates toward the desired smooth manifolds, thereby establishing a favorable initialization where the convergence theory of the DR splitting algorithm becomes fully applicable. Numerical experiments shed light on the effectiveness of the proposed methods in distributed group-sparse controller design.

Figures

Figures reproduced from arXiv: 2507.19895 by the authors.

Figure 1
Figure 1. Figures of Section 5 18 [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. ADMM Convergence Pattern: along the direction indicated by the arrows, the parameter [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Nonconvex Optimization Framework for Group-Sparse Feedback Linear-Quadratic Optimal Control: Penalty Approach

    math.OC 2025-07 reject novelty 5.0 of 10

    A penalty-based PALM algorithm solves a group-ℓ0 regularized LQ problem and converges to a critical point under explicit parameter conditions.

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