{"id":"50a8b2c4-a2c5-4786-9613-c33ee7a61da0","arxiv_id":"2504.18731","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper derives LQG cost gradients with respect to sensor and actuator matrices and develops an ADMM alternating scheme that reaches stationary points for sparse, low-rank, and structure-constrained configurations.","lead":"The paper derives gradient formulas for the LQG control cost with respect to sensor and actuator matrices and embeds them in an ADMM-based alternating optimization for designing these matrices from scratch. It matters because most prior co-design work selects from a fixed candidate list, while this framework targets continuous design with sparsity, low-rank, or structural constraints.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's convergence guarantee rests on Proposition 1 and Remark 4, but Proposition 1's proof uses the false limit σ(A+B_iK_i^*)→0; in the scalar stabilizability-loss example the limit is -1. The convergence claim is therefore unproven as written.","rationale":"I read the paper as claiming Algorithm 2 converges to a stationary point of the original LQG configuration problem. For that to be true, the outer iterates must stay in S×D and have accumulation points, and the inner ADMM solves must give genuine descent or at least KKT in the limit. The first is what Proposition 1 is supposed to secure; the proof is wrong. The scalar example is not contrived—it is the canonical way stabilizability is lost. The second (Remark 4) is asserted. These are the weakest link in the central claim. I agree with the reader's assessment; a conditional verdict is appropriate. I did not find a defect in the gradient derivations or the three update formulas; they are standard and internally consistent. The simulations verify KKT residuals only, not global optimality, which is consistent with a stationarity claim. So no verdict change.","tokens_in":23391,"tokens_out":11643,"duration_ms":119104,"concrete_test":"Analytical test of Eq. (28): take the scalar system ẋ=x+εu+w with Q=R=1 and let ε→0. The CARE is 2P-ε²P²+1=0, whose positive solution is P(ε)=(1+√(1+ε²))/ε². Compute K*(ε)=-εP(ε) and σ(A+B(ε)K*(ε))=1-ε²P(ε)=-√(1+ε²). This tends to -1, not to 0, directly falsifying the claimed limit in Proposition 1. A correct coercivity proof would instead have to show J_LQG→∞ via P(ε)→∞; if no such proof is supplied, Theorem 4's convergence guarantee should be treated as unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the proof of Theorem 4, not the gradient formulas. Theorem 4's conclusion that Algorithm 2 yields a stationary point of (P0) is supported only by Proposition 1 and Remark 4. Proposition 1's proof is invalid: Eq. (28) asserts σ(A+B_iK_i^*)→σ(A+BK^*)=0 as B_i→B∈∂S, but this presupposes P_i→P. When stabilizability is being lost, P_i solving CARE diverges. For the scalar pair A=1, B_i=ε_i→0, Q=R=1, the CARE solution is P_i=(1+√(1+ε_i^2))/ε_i^2, hence K_i^*=-ε_iP_i and A+B_iK_i^*=1-ε_i^2P_i=-√(1+ε_i^2)→-1, not 0. Thus the spectral-abscissa argument does not establish coercivity, so the proof does not justify that the iterates remain in S or have an accumulation point. Remark 4 separately asserts the monotone decrease (36) from the inner ADMM, but ADMM for a nonconvex smooth term returns an approximate stationary point of the augmented Lagrangian, not a minimizer of J, so (36) is not automatic. If both premises fail, Theorem 4 lacks a valid proof; the gradient and KKT results in Theorems 1–3 appear unaffected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the joint design of the actuator matrix B and sensor matrix C for a continuous-time LQG system without assuming a pre-specified set of candidate components. The objective is J(B,C)=J_LQG+γ(Φ(B)+Ψ(C)) subject to convex configuration constraints and to stabilizability/detectability of (A,B) and (A,C). The main theoretical results are analytic gradient formulas for J_LQG with respect to B and C (Theorems 1 and 2), KKT-type necessary conditions (Theorem 3), and an ADMM-based alternating optimization algorithm (Algorithm 2) that is claimed to converge to a stationary point of the problem (Theorem 4). The paper also specializes the proximal updates to three scenarios: ℓ1 sparsity promotion, nuclear-norm low-rank promotion, and support-structure constraints. Numerical experiments on the REA1 benchmark compute returned matrices and verify the corresponding stationarity residuals.","tokens_in":23666,"tokens_out":14731,"duration_ms":146721,"significance":"The analytic gradient formulas are the paper's clearest contribution: they are derived directly from the CARE/Lyapunov equations, contain no fitted parameters, and provide a practical stationarity check. The KKT conditions and the closed-form ADMM updates for the three scenarios are useful building blocks. The numerical verification is also a strength: the authors check the KKT residuals of the returned solutions rather than only plotting objective decrease. However, the central convergence claim of Algorithm 2 (Theorem 4) is not proven as written: its proof depends on an invalid step in Proposition 1 and on an unproven monotonicity assertion. The gradient and stationarity results of Theorems 1–3 appear sound, but the algorithmic guarantee needs substantial repair before the paper can be accepted.","major_comments":[{"comment":"The proof of Proposition 1 is invalid. Equation (28) asserts σ(A+B_iK_i^*)→σ(A+BK^*)=0 as B_i→B∈∂S, but K_i^*=−R^{-1}B_i^TP_i depends on the CARE solution P_i, and P_i may diverge as stabilizability is lost. For example, with scalar A=1, B_i=ε_i→0, and Q=R=1, the CARE solution is P_i=(1+√(1+ε_i^2))/ε_i^2, so A+B_iK_i^*=1−ε_i^2P_i=−√(1+ε_i^2)→−1, not 0. Consequently the Lyapunov lower bound (30) does not force tr(P_i)→∞. Because Proposition 1 is the basis for the coercivity used to keep iterates in S and to ensure accumulation points in Theorem 4, Theorem 4 is not established by the argument given.","section":"III.A, Proposition 1, Eq. (28)"},{"comment":"Remark 4 asserts the monotone decrease J(B_{h+1},C_h)≤J(B_h,C_h), but this does not follow from the ADMM inner loop as implemented. Algorithm 2 stops the inner ADMM at fixed positive tolerances ǫpri and ǫdual, so B_{h+1}=M_k is not an exact minimizer of (32); for a nonconvex smooth term, the ADMM stationarity relations (38)–(39) do not imply a decrease of the original objective J. The cited reference [24] concerns convergence of ADMM to a stationary point of the augmented or original problem under additional conditions; it does not supply the outer monotonicity (36). Without (36), the outer iterates need not converge, so the existence of the accumulation point (B^*,C^*) assumed in Theorem 4 is unjustified.","section":"III.A, Remark 4, Eq. (36)"},{"comment":"The proof of Theorem 4 has a double-limit gap. It derives stationarity for the inner iterate B_{k+1} while the outer variable is fixed at C_h, but then writes C^* and passes k→∞ and h→∞ simultaneously without justifying the interchange. It also replaces ∂J(B_{k+1},C^*)/∂B by ∂J(M_{k+1},C^*)/∂B on the strength of ‖B_{k+1}−M_{k+1}‖≤ǫpri; this requires a Lipschitz estimate for the gradient and a limit argument as ǫpri→0, and similarly for ρ(M_{k+1}−M_k)→0 as ǫdual→0. The tolerances are fixed positive constants in Algorithm 2, so the claimed limit is not immediate. As written, the conclusion that (B^*,C^*) satisfies (44)–(45) does not follow.","section":"III.A, Theorem 4 proof, Eqs. (38)–(44)"},{"comment":"The second, more specific part of Theorem 3 represents the normal cone N_{Ω_B}(B) as the set of all linear combinations of gradients of the equality and inequality descriptions with μ^j_B≥0. This representation requires a constraint qualification (for example, a Slater-type condition for the convex description); the statement and proof do not mention any such qualification. Without it, the displayed KKT conditions can fail at nonregular feasible points. The three concrete scenarios studied later have simple constraint sets where the issue is benign, but the general theorem as stated is missing an assumption.","section":"III.A, Theorem 3"}],"minor_comments":[{"comment":"In Theorem 2, 'Given a constant actuator matrix B∈D, for any sensor matrix C∈S' should be 'B∈S and C∈D'; the symbols appear swapped.","section":"II.A / Theorem 2"},{"comment":"In the increment of the CARE in the proof of Theorem 1, the term '∆P A^T' should read '∆P A'.","section":"III.A, Theorem 1 proof"},{"comment":"The proof of Lemma 5 begins 'Given that Φ(B)=0', but in this scenario Φ(M)=‖M‖_F^2; the zero-cost claim is a typo.","section":"III.D, Lemma 5 proof"},{"comment":"The displayed matrices B^* and C^* for the structure-constrained scenario appear to have misaligned rows and columns: B^* is printed with a single 0 in the first row, and C^* appears to have five rows. Please correct the typesetting.","section":"IV.C"},{"comment":"Algorithm 1 does not specify a stopping criterion for the inner gradient descent, although Theorem 4's proof treats B_{k+1} as an exact stationary point of (37); the dependence on the inner tolerance should be stated.","section":"III.A, Algorithm 1"},{"comment":"The choice of ρ is only described as 'sufficiently large'; the simulations fix ρ=1 with no sensitivity study, so the practical reliability of the convexification argument is not demonstrated.","section":"III.A, Remark 4"}],"recommendation":"major_revision","confidential_remarks":"I see no evidence of circularity or self-promotion in the derivation. The paper fits the scope of the journal. The main issue is the unsupported convergence theorem; if the authors can prove coercivity and outer convergence under stated assumptions, the paper would be acceptable. The current version is not ready."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe headline is this: the paper has a genuinely useful formulation and some correct-looking gradient mathematics, but the convergence proof for its main algorithm has a real gap. The idea of designing B and C from scratch rather than selecting from a fixed candidate set is a nice reframing, and the CARE-based gradient identities in Theorems 1 and 2 appear to be derived carefully. The KKT analysis and the three tailored scenarios (sparsity, low-rank, structure) are well organized and give explicit update rules. The simulations check stationarity residuals on a benchmark, which is a step beyond just showing cost decrease.\n\nWhere it gets shaky is Theorem 4. The proof leans on Proposition 1's coercivity claim, but that proof uses the limit σ(A+B_i K_i^*)→0 as B_i approaches the boundary of the stabilizable set. That is not true in general. For a scalar system A=1, B_i=ε_i with ε_i→0, the CARE solution P_i ~ 2/ε_i^2, so K_i^*=-ε_i P_i ~ -2/ε_i, and A+B_i K_i^* ~ -1. The spectral abscissa goes to -1, not 0. So the lower bound on tr(P_i) from the Lyapunov argument does not diverge, and the proof does not establish coercivity. The proposition itself may be true, but the proof as written is invalid. Remark 4 also asserts monotone decrease of J in the alternating scheme without proof; that is not automatic for nonconvex ADMM. So the claim that Algorithm 2 converges to a stationary point is currently unproven.\n\nThe rest of the paper seems fine. The gradient formulas and KKT conditions do not depend on that proposition. The citation pattern is standard and not self-serving. The experiments are honest but minimal: one benchmark, no comparisons to existing selection/placement methods, and no exploration of different random starts.\n\nOverall, this is a paper worth engaging with. A serious referee would likely require a corrected coercivity argument and a more careful convergence analysis, but those are fixable. I would accept it for peer review and see what the authors come back with. I might cite the gradient formulas in future work, but I would not rely on the algorithm's convergence guarantee yet.\n\nBest,\n[You]","headline":"Useful continuous co-design formulation and solid gradient derivations, but the Algorithm 2 convergence proof rests on an invalid spectral abscissa claim.","tokens_in":24214,"tokens_out":3027,"would_cite":true,"duration_ms":27978,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E20","90C26","93B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"LQG sensor and actuator design becomes a differentiable optimization problem","keywords":["LQG control","sensor configuration","actuator configuration","co-design","ADMM","algebraic Riccati equation","sparsity promotion","low-rank promotion"],"falsifier":"A numerical scan that approaches the boundary of the stabilizable region and finds bounded LQG cost while the optimal feedback gain grows without bound would falsify the coercivity premise behind Theorem 4.","tokens_in":1852,"feed_emoji":"🎛️","tokens_out":2562,"duration_ms":85699,"temperature":0.7,"pith_summary":"The paper claims that the LQG performance cost is differentiable with respect to the actuator matrix $B$ and the sensor matrix $C$, with gradients given by closed-form expressions built from the algebraic Riccati solutions and a pair of Lyapunov equations. This matters because existing methods typically select sensors and actuators from a predetermined candidate set, whereas this formulation allows designing them from scratch under convex configuration costs and constraints. The paper derives KKT stationarity conditions and proposes an ADMM-based alternating algorithm whose accumulation point is claimed to be a stationary point. It then specializes the framework to sparsity, low-rank, and structure-constrained configuration costs, with explicit proximal updates for each.","feed_headline":"Closed-form gradients enable from-scratch LQG sensor and actuator design","feed_subtitle":"Analytic gradients plus ADMM alternating optimization reach provable stationary points under convex costs.","key_machinery":"The central object is the pair of continuous-time algebraic Riccati equations (CAREs) for the optimal observer and controller, whose unique positive-definite solutions $X$ and $P$ exist under the stabilizability and detectability assumption. The paper perturbs $B$ and $C$ through these CAREs, rearranges trace identities, and obtains gradient formulas expressed through Lyapunov equations; this is what turns the implicit LQG cost into an explicitly differentiable objective. The optimization machinery is an ADMM splitting that separates the smooth nonconvex LQG term from the convex nonsmooth configuration cost, with auxiliary-variable updates that have closed forms for the three scenarios considered.","core_discovery":"The central claim is that the LQG cost $J_{\\mathrm{LQG}}$, despite depending on $B$ and $C$ only implicitly through two continuous-time algebraic Riccati equations, has closed-form gradients: $\\partial J_{\\mathrm{LQG}}/\\partial B = -2P(G_1+G_2)PBR^{-1}$ and $\\partial J_{\\mathrm{LQG}}/\\partial C = -2\\Pi_v^{-1}CX(H_1+H_2)X$, where $P,X$ are the Riccati solutions and $G_1,G_2,H_1,H_2$ solve Lyapunov equations built from the optimal closed-loop and observer matrices. Theorem 3 converts these gradients into necessary first-order optimality conditions for the constrained joint configuration problem, and Theorem 4 states that the accumulation point of Algorithm 2, an ADMM-based alternating optimization, satisfies these conditions and is therefore at least a stationary point. The paper also gives closed-form proximal updates for three representative scenarios: soft thresholding for $\\ell^1$ sparsity, singular value thresholding for the nuclear norm, and a mask projection for structure constraints.","pith_inferences":["The same Riccati-perturbation derivation is likely to produce analogous gradient formulas for discrete-time LQG or $H_2$ performance, although the paper does not state these extensions.","The stationarity guarantee is local, so global optimality should not be assumed; restarting Algorithm 2 from several initial pairs is a natural safeguard that the paper does not discuss.","If the true design problem is binary (a sensor or actuator is either installed or not), the continuous matrices returned by this framework would need a rounding or mixed-integer layer, which the paper leaves implicit.","The coercivity premise behind the boundary argument is directly testable: a numerical scan of near-unstabilizable pairs would reveal whether the LQG cost actually diverges when the optimal feedback gain itself blows up."],"forward_implications":["Designers can optimize sensor and actuator matrices from scratch, with no predefined candidate set, while the stabilizability and detectability constraints are enforced by the optimization itself.","A returned configuration $(B^*,C^*)$ can be checked for stationarity by directly evaluating the two gradient formulas and the corresponding normal-cone conditions.","Varying the relative weight $\\gamma$ traces a performance-versus-configuration-cost frontier, as demonstrated numerically for sparsity, low-rank, and structure-constrained designs.","Low-rank solutions come with explicit factorizations $B=B_1B_2$ and $C=C_1C_2$, so the designed matrices translate directly into a reduced set of actuators and sensors.","Because the outer loop alternates and each inner ADMM update is either a smooth gradient step or a convex proximal step, the framework can be embedded in numerical solvers with modest per-iteration cost."],"supporting_citations":[{"why":"Supplies the CARE-based characterization of the optimal LQG controller and the existence and uniqueness of the Riccati solutions used throughout the gradient derivations.","marker":"[18]"},{"why":"Provides the variational-inequality and normal-cone optimality conditions that underlie Theorem 3 and the ADMM update structure in Algorithm 2.","marker":"[20]"},{"why":"Gives the Lyapunov lower bound used in Proposition 1 to argue that the LQG cost diverges near the stabilizability boundary.","marker":"[21]"},{"why":"Supports the gradient-descent-with-backtracking inner solver and the claim that this smooth subproblem reaches a stationary point.","marker":"[22]"},{"why":"Cited for the convergence of ADMM in nonconvex nonsmooth optimization, the basis for Remark 4 and Theorem 4.","marker":"[24]"},{"why":"Supplies the convex-analysis result that gives the soft-thresholding closed-form solution for the sparsity-promoting auxiliary update.","marker":"[26]"},{"why":"Gives the subdifferential of the nuclear norm used in the low-rank scenario's optimality conditions and in verifying the singular value thresholding update.","marker":"[28]"},{"why":"Provides the chemical-reactor benchmark system used in the numerical demonstrations of all three configuration scenarios.","marker":"[29]"}],"fun_headline_variants":["From-scratch LQG co-design: closed-form gradients meet ADMM","Riccati gradients power joint LQG sensor/actuator design","Unified ADMM for LQG configuration with no preset components","Provable stationary points for LQG co-design via ADMM","Closed-form Riccati gradients enable from-scratch LQG design"],"cache_read_input_tokens":26240,"weakest_assumption_plain":"The proof that Algorithm 2 reaches a stationary point rests on two unsupported premises: that the LQG cost diverges at the boundary of the stabilizable/detectable region even when the optimal gains diverge, and that each alternating update never increases the total cost.","fun_headline_variants_meta":{"raw":{"variants":["From-scratch LQG co-design: closed-form gradients meet ADMM","Riccati gradients power joint LQG sensor/actuator design","Unified ADMM for LQG configuration with no preset components","Provable stationary points for LQG co-design via ADMM","Closed-form Riccati gradients enable from-scratch LQG design"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001114,"raw_usage":{"total_tokens":4657,"prompt_tokens":980,"completion_tokens":3677,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":3584}},"tokens_in":596,"tokens_out":3677,"duration_ms":26807,"temperature":1.0,"reasoning_tokens":3584,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:11:51.551344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical scan that approaches the boundary of the stabilizable region and finds bounded LQG cost while the optimal feedback gain grows without bound would falsify the coercivity premise behind Theorem 4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CARE-based characterization of the optimal LQG controller and the existence and uniqueness of the Riccati solutions used throughout the gradient derivations."},{"cited_title":"Upper and lower bounds on the sol ution of the algebraic Riccati equation,","cited_arxiv_id":null,"evidence_quote":"Gives the Lyapunov lower bound used in Proposition 1 to argue that the LQG cost diverges near the stabilizability boundary."},{"cited_title":"Nocedal and S","cited_arxiv_id":null,"evidence_quote":"Supports the gradient-descent-with-backtracking inner solver and the claim that this smooth subproblem reaches a stationary point."},{"cited_title":"Global convergence of ADMM in nonconvex nonsmooth optimization,","cited_arxiv_id":null,"evidence_quote":"Cited for the convergence of ADMM in nonconvex nonsmooth optimization, the basis for Remark 4 and Theorem 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the convex-analysis result that gives the soft-thresholding closed-form solution for the sparsity-promoting auxiliary update."},{"cited_title":"Characterization of the subdifferentia l of some matrix norms,","cited_arxiv_id":null,"evidence_quote":"Gives the subdifferential of the nuclear norm used in the low-rank scenario's optimality conditions and in verifying the singular value thresholding update."},{"cited_title":"Description of the bench mark examples in COMPleib 1.0 (Tech. Report),","cited_arxiv_id":null,"evidence_quote":"Provides the chemical-reactor benchmark system used in the numerical demonstrations of all three configuration scenarios."}],"review_version":1}