{"id":"067d4834-16ee-4cad-9d74-c6950d52831d","arxiv_id":"2501.14191","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For quadratic cone programs with strongly convex objectives, choosing the objective scaling factor as sqrt(sigma_min / 2) provably minimizes the KKT condition number, and packaging this with hypersphere and row-normalization steps yields a fast, factorization-free preconditioner.","lead":"This paper proposes a three-step preconditioning recipe for convex quadratic cone programs that makes first-order solvers faster on ill-conditioned problems. The key new piece is a closed-form choice of the objective scaling factor that minimizes the condition number of the KKT matrix.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Block row-normalization for SOC/PSD blocks is a single-scalar heuristic, so an ill-scaled anisotropic cone block can leave HH^T ill-conditioned; the paper's general-cone benefit is untested in this regime.","rationale":"The reader's weakest assumption and my concern coincide: Theorem 1 is mathematically sound, and I verified the monotonicity argument in its proof; Lemma 1 and Corollary 1 are internally consistent. The open issue is step 2. The paper is honest about the heuristic nature of row normalization, and the numerical results are plausible, but the two experiments are not designed to detect the failure mode just described. This does not undermine the theorem or the demonstrated improvement on the reported problems; it only means the general claim for QCPs with nonlinear cone blocks is conditional on further evidence. The code is available, and the proposed test is a small modification of the existing benchmark, so it should settle the question without changing the reader's verdict.","tokens_in":12705,"tokens_out":20965,"duration_ms":207354,"concrete_test":"Run the Section III convex example with the 2-norm ball constraints restored (SOC blocks) and with A=diag(1,10^6) in a few blocks, while keeping the rest of the data unchanged. Measure kappa(HH^T) before and after block row-normalization, and record PIPG iterations and combined solve time for hypersphere versus modified Ruiz equilibration. If kappa(HH^T) does not decrease, or PIPG loses to Ruiz in this setting, the row-normalization heuristic fails on anisotropic SOC data and the paper's general-cone claim must be qualified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem in Sec. II-C is correct for the H it is handed, but the overall three-step method's practical claim rests on Sec. II-B's block row-normalization. For zero/nonnegative orthant blocks, each row can be independently normalized; for an SOC or PSD block, Eq. (2b) forces E to be a single positive scalar per block. Thus anisotropic data inside such a block cannot be equilibrated. Example: an SOC constraint ||A x||_2 <= t with A=diag(1,1e6), t=1 gives rows of norms 1,1e6,1; the max-element rule with c=1/max{1,1e6,1}=1e-6 yields rows of norms 1e-6,1,1e-6, making sigma_min(HH^T) ~ 1e-12, so kappa(K) at lambda* remains enormous. The paper explicitly labels the step heuristic (Sec. I-A, 'although not guaranteed to reduce the condition number') and offers no analysis of when it helps. The convex experiment replaces 2-norm balls with infinity-norm balls (linear rows), and the SOC thrust constraint in the rocket-landing example is isotropic (A=I), so neither experiment stresses this failure mode. Consequently, the central claim that the three-step preconditioner improves ill-conditioned general QCPs is not yet supported for anisotropic SOC/PSD data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a three-step preconditioning procedure, called the hypersphere preconditioner, for strongly convex quadratic cone programs (QCPs). Step 1 transforms the Hessian via its Cholesky factor so that the objective becomes λ/2 ||z||^2. Step 2 applies block row-normalization to the constraint matrix, respecting cone block structure through a positive diagonal scaling. Step 3 chooses the objective scaling λ to minimize the condition number of the KKT matrix K = [λI H^T; H 0]; the main theoretical result (Theorem 1) gives the closed-form λ* = sqrt(σ_min(HH^T)/2). The paper also proves a relationship between this scaling and the step-size ratio ω in the PIPG first-order solver (Theorem 2), and presents numerical comparisons against modified Ruiz equilibration and a QR preconditioner on a convex optimal control problem and a nonconvex rocket-landing problem solved via sequential conic optimization.","tokens_in":13001,"tokens_out":9775,"duration_ms":89483,"significance":"The closed-form expression for λ* and the connection between objective scaling and PIPG step-size tuning are clean, useful, and correctly derived. The proofs of Lemma 1, Corollary 1, and Theorem 1 are self-contained and correct under the stated assumptions n > m and rank H = m. The numerical section is reproducible (public code), uses ground-truth interior-point solutions, and shows substantial iteration-count and wall-clock improvements on the tested problems. The main caveat is that the practical benefit of the full three-step method rests on the block row-normalization heuristic, which is not analyzed and is not exercised in the numerically demanding anisotropic SOC/PSD regime; the paper's general-cone claims are therefore broader than the evidence.","major_comments":[{"comment":"The paper's advertised scope is general QCPs, but the block row-normalization step is the only component that attempts to condition the constraint matrix, and for SOC/PSD blocks it is a per-block scalar scaling. For the SOC example in Eq. (4) with A = diag(1, 1e6) and t = 1, the max-element rule gives c = 1e-6, so the normalized rows have norms 1e-6, 1, 1e-6 and σ_min(HH^T) is approximately 1e-12; κ(λ*) is then of order 1e6, so the preconditioner does not resolve the ill-conditioning. The numerical experiments deliberately avoid this regime: the convex example replaces 2-norm balls with infinity-norm balls (linear rows), and the rocket-landing SOC thrust constraint is isotropic. Since the paper itself labels block row-normalization as 'not guaranteed to reduce the condition number' (Section I-A, item 2), the conclusion that the hypersphere preconditioner improves ill-conditioned general QCPs is not supported for anisotropic SOC/PSD data. Please either add experiments with anisotropic SOC/PSD constraints or explicitly state in the abstract, introduction, and conclusion that the demonstrated benefits are for problems where block row-normalization is effective, such as problems with linear cone blocks.","section":"Section II-C / Lemma 1"},{"comment":"The optimal scaling result requires n > m and rank H = m, but this assumption is stated only inside Lemma 1 and not in the problem formulation or the abstract. If n ≤ m or H is rank-deficient, the spectrum of K contains additional eigenvalues (zero eigenvalues when H^T has a nontrivial nullspace), and the formula for λ* in Theorem 1 does not apply. Since the paper presents the method for a general QCP template, the scope of the theoretical claim should be stated prominently and the behavior in the excluded regimes should be discussed at least briefly.","section":"Section II-C / Lemma 1"}],"minor_comments":[{"comment":"The notation 'L2_SOC' is confusing; it should be L_SOC or L_SOC^2. Also, the scaling rule c = 1/max{|A11|, |A22|, 1} is stated without explaining that it is forced by the requirement that the entire block be scaled by a single scalar; a sentence making this explicit would help.","section":"Section II-B, Eq. (4)"},{"comment":"The caption and axis labels of Fig. 3 appear garbled: the y-axis is labeled 'Combined Solve Time [ms]' but the caption mentions 'Performance Robustness'. Please clarify what is plotted and whether the vertical bars or lines correspond to the named preconditioners.","section":"Section III, Fig. 3"},{"comment":"The phrases 'optimal preconditioning' and 'optimal objective function scaling factor' should be qualified as optimal with respect to the KKT-matrix condition number for a fixed H. The overall three-step procedure is not optimal because the block row-normalization step is heuristic; the current wording in the conclusion can be read as claiming a stronger optimality.","section":"Abstract and Conclusion"},{"comment":"The assumptions n > m and rank H = m are used critically in the proof (to ensure that the eigenvalue λ has multiplicity n−m and that HH^T is nonsingular). I recommend adding a short remark after Lemma 1 explaining what changes when these assumptions fail, for example the appearance of zero eigenvalues when rank H < m.","section":"Section II-C, after Lemma 1"},{"comment":"The presolve time is reported only as a percentage of combined solve time and is said to be dominated by shifted power iteration, but the regular power iteration for σ_max also contributes. Reporting the two contributions separately would make the online-cost discussion more informative.","section":"Table II and Algorithm 1"},{"comment":"The shifted power iteration requires a spectral gap assumption for the shifted matrix M − σ_max I; the text mentions its slow convergence for clustered small eigenvalues only in Section III. A sentence in Appendix B stating this assumption and its practical implication would make the algorithm's limitations clearer.","section":"Appendix B, Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core is sound and the numerical results are convincing for the tested problem classes, but the general-cone claim is broader than the evidence. The manuscript would be suitable for publication after either adding experiments with anisotropic SOC/PSD constraints or explicitly narrowing the claimed applicability. The contribution is incremental over the authors' prior work, but the closed-form λ* and the step-size equivalence are sufficiently useful for a journal paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — worth a read. Theorem 1 is the real thing: for a saddle-point KKT matrix with n>m and full-row-rank H, λ*=sqrt(σ_min/2) minimizes the condition number. I checked the spectrum argument; it's correct, and it's new in closed form as far as I can tell. Prior work in this group tuned this scalar by hand; now it's derived. The numerical work is honest: ground-truth IPM solutions, iteration counts, code on GitHub, no fitted parameters. The three-step recipe (hypersphere transform, block row-normalization, optimal λ) does improve PIPG on the tested ill-conditioned problems, and the rocket-landing SeCO example shows genuine speedups.\n\nThe soft spot is the one your report flags: the general-cone benefit is not established for anisotropic SOC/PSD blocks. Block row-normalization scales each separable cone block by a single scalar. If a SOC block is anisotropic — say ||diag(1,1e6)x||_2 <= t — the max-element rule leaves row norms 1e-6, 1, 1e-6, so χ(HH^T) is still ~1e12 and the optimal λ does not save you. The paper explicitly calls row normalization a heuristic, which I respect, but the title and abstract sell a general QCP preconditioner. The experiments replace 2-norm balls by ∞-norm balls precisely to avoid this, and the rocket SOC constraints are isotropic, so the failure mode is never exercised. That is a real gap between claim and evidence.\n\nNothing else worries me. The shifted power iteration caveat is stated. The full-rank assumption is standard. The citation pattern is fine; the self-citations are to the work this builds on.\n\nVerdict: accept for peer review, not desk reject. A serious referee should ask the authors to state the SOC/PSD limitation up front and include at least one anisotropic SOC example, even if it shows the preconditioner not helping — that would be informative. The theorem and central recipe are worth publishing; the gap is in the scope claim, not the math.","headline":"A correct and useful closed-form λ* for KKT conditioning, with an honest but unexplored limitation: the general-cone promise outruns the evidence for anisotropic SOC/PSD blocks.","tokens_in":84,"tokens_out":5293,"would_cite":true,"duration_ms":67707,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C46","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives the closed-form objective scaling that minimizes the condition number of the KKT matrix for full-rank quadratic cone programs, and wraps it into a three-step factorization-free preconditioner that speeds up first-order…","keywords":["optimal preconditioning","quadratic cone programming","first-order methods","KKT matrix","condition number","hypersphere preconditioner","sequential conic optimization","shifted power iteration"],"falsifier":"To test Theorem 1 directly, take a small full-rank $H$ (for example $m=2,n=3$), evaluate $\\kappa(\\lambda)$ from Corollary 1 across a fine grid of $\\lambda$, and check whether the minimum occurs at $\\lambda^*=\\sqrt{\\sigma_{\\min}/2}$; a counterexample would disprove the theorem. To test the practical claim, construct a second-order cone program whose block row-normalization step provably leaves $H$ ill-conditioned and compare PIPG iterations with and without the full preconditioner; if iteration counts do not improve, the heuristic step fails in that setting.","tokens_in":12543,"feed_emoji":"⚡","tokens_out":11207,"duration_ms":91347,"temperature":0.7,"pith_summary":"This paper argues that a three-step 'hypersphere preconditioner' can make first-order solvers practical on ill-conditioned quadratic cone programs with strongly convex objectives. First, a Cholesky-based change of variables turns the objective Hessian into a scaled identity, so its condition number is one. Second, the constraint matrix rows are normalized block-wise. Third, the objective is scaled by the closed-form value $\\lambda^* = \\sqrt{\\sigma_{\\min}/2}$, where $\\sigma_{\\min}$ is the smallest eigenvalue of $H H^{\\top}$; the paper proves this value minimizes the condition number of the KKT matrix. The result matters for online applications such as model predictive control and sequential convex programming, where the same problem structure is solved repeatedly with changing data and matrix factorizations are too expensive.","feed_headline":"Closed-form scaling minimizes the KKT condition number","feed_subtitle":"It proves the optimal objective scaling and shows faster PIPG solves on ill-conditioned trajectories.","key_machinery":"The object that carries the argument is the KKT matrix $K(\\lambda) = \\begin{bmatrix} \\lambda I & H^{\\top} \\\\ H & 0 \\end{bmatrix}$ of the preconditioned problem, together with the spectrum formula that reduces its condition number to a one-dimensional function of $\\lambda$. The proof of optimality compares the increasing function $f_1(\\lambda)=\\lambda$ and the decreasing function $f_2(\\lambda)=(\\sqrt{\\lambda^2+4\\sigma_{\\min}}-\\lambda)/2$; the minimizer sits at their intersection, which is $\\lambda^*=\\sqrt{\\sigma_{\\min}/2}$. To keep the procedure factorization-free, $\\sigma_{\\min}$ is estimated by shifted power iteration on $H H^{\\top}$, so no inverse of $H$ or $H H^{\\top}$ is ever needed. The second step, block row-normalization, is the only heuristic in the loop: it scales each cone block's rows by a scalar so that the cone structure in the problem's equivalence requirement is preserved.","core_discovery":"The central claim is Theorem 1: for any $m \\times n$ constraint matrix $H$ with $n > m$ and rank $H = m$, the condition number of $K(\\lambda) = \\begin{bmatrix} \\lambda I & H^{\\top} \\\\ H & 0 \\end{bmatrix}$ is minimized at $\\lambda^* = \\sqrt{\\sigma_{\\min}/2}$, where $\\sigma_{\\min}$ is the smallest eigenvalue of $H H^{\\top}$. The proof uses an explicit spectrum of $K$: $n-m$ copies of $\\lambda$ and $m$ pairs $\\theta_{\\pm}(\\lambda,\\sigma_k) = (\\lambda \\pm \\sqrt{\\lambda^2+4\\sigma_k})/2$, so the largest eigenvalue is increasing in $\\lambda$ while the smallest in magnitude is the minimum of an increasing and a decreasing function; balancing those functions yields $\\lambda^*$. Corollary 2 shows the resulting condition number is at least 2, with equality when $H H^{\\top}$ is perfectly conditioned. Theorem 2 then shows that scaling the objective by $\\lambda$ is equivalent to scaling the PIPG step-size ratio by $1/\\sqrt{\\lambda}$, so $\\lambda^*$ also prescribes the optimal ratio. On top of these analytical results, the paper reports substantial iteration-count and wall-clock-time improvements for PIPG on convex optimal control and nonconvex rocket landing guidance problems, especially in the ill-conditioned regime.","pith_inferences":["A natural extension the paper does not pursue is applying the same optimal-scaling logic to the strongly convex part of a non-strongly-convex objective, which could broaden the method's reach to more MPC and SCP formulations.","Because Theorem 2 makes $\\lambda$ and $\\omega$ equivalent, an implementer could use $\\lambda^*$ as an automatic step-size rule for PIPG, re-estimating $\\sigma_{\\min}$ only when the constraint matrix changes; this is not explicitly proposed in the paper.","The block row-normalization heuristic could be replaced or augmented by a cone-block-aware scaling that also balances dual variables, and the optimality proof for $\\lambda^*$ would still apply to whatever $H$ results; this is an untested variant.","For dense ill-conditioned QCPs, the QR preconditioner's densification might be tolerable; applying $\\lambda^*$ on top of it would provide a principled scaling that the paper does not test."],"forward_implications":["If the central claim is correct, first-order conic solvers can be preconditioned without any matrix factorizations, preserving their sparsity and per-iteration cost.","The closed-form $\\lambda^*$ replaces manual tuning of the objective scaling (or equivalently the PIPG step-size ratio $\\omega$) with a computation from $\\sigma_{\\min}$.","The lower bound $\\kappa(\\lambda^*) \\geq 2$ states a hard limit: once the objective is identity-scaled and rows are normalized, no choice of $\\lambda$ can make the KKT matrix better conditioned than 2.","For online solves with changing data, $\\sigma_{\\min}$ can be re-estimated by shifted power iteration and the solver parameters updated, keeping the preconditioner aligned with the current problem instance.","In the numerical tests, the preconditioner is reported to reduce PIPG iterations from the solver's maximum to a few hundred on the hardest convex cases, and to cut both KKT condition number and iterations in the rocket landing guidance problem."],"supporting_citations":[{"why":"Introduces the hypersphere preconditioning idea for problems where the cone contains only the zero cone; this paper extends it to general cones.","marker":"[4]"},{"why":"Defines the PIPG first-order primal-dual method and the convex optimal control problem used as a test case.","marker":"[5]"},{"why":"Presents the multi-phase rocket landing guidance problem solved by sequential conic optimization, used here for the nonconvex demonstration.","marker":"[6]"},{"why":"Specifies the block row-normalization requirement for cone programs that the second step of the preconditioner follows.","marker":"[19]"},{"why":"Provides the modified Ruiz equilibration baseline against which the hypersphere preconditioner is compared.","marker":"[20]"},{"why":"Supplies the QR factorization-based constraint preconditioner used as the second baseline.","marker":"[32]"},{"why":"The shifted power iteration algorithm used to estimate the smallest singular value $\\sigma_{\\min}$ without factorizations.","marker":"[33]"},{"why":"Defines the extrapolated PIPG variant and the step-size ratio $\\omega$ that Theorem 2 equates with objective scaling.","marker":"[22]"}],"fun_headline_variants":["Optimal scaling proven for ill-conditioned quadratic cone programs","Theorem reveals optimal step-size ratio for first-order conic solves","Hypersphere preconditioner: closed-form optimum for QCPs","Closed-form scaling minimizes KKT condition number"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method's practical benefit depends on the unproven guess that normalizing each block of constraint rows makes the constraint matrix easier to solve; if that guess is wrong, the optimal scaling is computed for a constraint matrix that is still hard to solve.","fun_headline_variants_meta":{"raw":{"variants":["Optimal scaling proven for ill-conditioned quadratic cone programs","Theorem reveals optimal step-size ratio for first-order conic solves","Hypersphere preconditioner: closed-form optimum for QCPs","Closed-form scaling minimizes KKT condition number"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2822,"prompt_tokens":952,"completion_tokens":1870,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":1802}},"tokens_in":568,"tokens_out":1870,"duration_ms":13816,"temperature":1.0,"reasoning_tokens":1802,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:15:32.026746+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test Theorem 1 directly, take a small full-rank $H$ (for example $m=2,n=3$), evaluate $\\kappa(\\lambda)$ from Corollary 1 across a fine grid of $\\lambda$, and check whether the minimum occurs at $\\lambda^*=\\sqrt{\\sigma_{\\min}/2}$; a counterexample would disprove the theorem. To test the practical claim, construct a second-order cone program whose block row-normalization step provably leaves $H$ ill-conditioned and compare PIPG iterations with and without the full preconditioner; if iteration counts do not improve, the heuristic step fails in that setting.","supporting_citations":[{"cited_title":"Customized real-time first-order methods for onboard dual quaternion-based 6-DoF powered-descent guidance,","cited_arxiv_id":null,"evidence_quote":"Introduces the hypersphere preconditioning idea for problems where the cone contains only the zero cone; this paper extends it to general cones."},{"cited_title":"Proportional-integral projected gradient method for model predictive control,","cited_arxiv_id":null,"evidence_quote":"Defines the PIPG first-order primal-dual method and the convex optimal control problem used as a test case."},{"cited_title":"Real-time sequential conic optimization for multi-phase rocket landing guidance,","cited_arxiv_id":null,"evidence_quote":"Presents the multi-phase rocket landing guidance problem solved by sequential conic optimization, used here for the nonconvex demonstration."},{"cited_title":"Conic optimization via operator splitting and homogeneous self-dual embedding,","cited_arxiv_id":null,"evidence_quote":"Specifies the block row-normalization requirement for cone programs that the second step of the preconditioner follows."},{"cited_title":"OSQP: an operator splitting solver for quadratic programs,","cited_arxiv_id":null,"evidence_quote":"Provides the modified Ruiz equilibration baseline against which the hypersphere preconditioner is compared."},{"cited_title":"Constraint preconditioning and parameter selection for a first-order primal-dual method applied to model predictive control,","cited_arxiv_id":null,"evidence_quote":"Supplies the QR factorization-based constraint preconditioner used as the second baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The shifted power iteration algorithm used to estimate the smallest singular value $\\sigma_{\\min}$ without factorizations."},{"cited_title":"Extrapolated proportional-integral projected gradient method for conic optimization,","cited_arxiv_id":null,"evidence_quote":"Defines the extrapolated PIPG variant and the step-size ratio $\\omega$ that Theorem 2 equates with objective scaling."}],"review_version":1}