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REVIEW 2 major objections 6 minor 47 references

Optimal Preconditioning for Online Quadratic Cone Programming

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.14191 v2 pith:2UWJVNFV submitted 2025-01-24 math.OC

Abhinav G. Kamath , Purnanand Elango , Behçet Aç{i}kmeşe This is my paper · ORCID
classification math.OC MSC 90C2590C4665K05
keywords optimalpreconditioningquadraticconeprogrammingfirst-ordermethodsKKTmatrixconditionnumberhyperspherepreconditionersequentialconicoptimizationshiftedpoweriteration
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 6 minor

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.

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 (2)
  1. [Section II-C / Lemma 1] 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.
  2. [Section II-C / Lemma 1] 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.
minor comments (6)
  1. [Section II-B, Eq. (4)] 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.
  2. [Section III, Fig. 3] 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.
  3. [Abstract and Conclusion] 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.
  4. [Section II-C, after Lemma 1] 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.
  5. [Table II and Algorithm 1] 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.
  6. [Appendix B, Algorithm 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal scaling factor is derived from the KKT spectrum, and the numerical claims are benchmarked against interior-point ground truth rather than fitted to the reported results.

full rationale

The derivation chain is self-contained. In Section II-C, Lemma 1 derives the spectrum of K = [λI Hᵀ; H 0], Corollary 1 gives κ(λ), and Theorem 1 minimizes κ(λ) in closed form, obtaining λ* = sqrt(σ_min / 2) by equating the increasing and decreasing branches of the max expression; this is a mathematical derivation, not a fit to numerical outcomes. The only estimated quantity, σ_min, is obtained from H via shifted power iteration (Algorithm 1) and is part of the preconditioner's construction, not a parameter tuned to match the iteration counts in Figures 3 or 4. The experimental claims are validated against an off-the-shelf interior-point 'ground truth' with a fixed 0.5% relative-error threshold, and the reported metric is measured PIPG iteration count and solve time, so the performance claim is externally benchmarked. The paper explicitly flags its one heuristic component: block row-normalization is described in Section I-A as 'a simple, customization-friendly heuristic... although not guaranteed to reduce the condition number of the constraint matrix,' and Section III notes the shifted power iteration limitation of slow convergence when the smallest eigenvalues of H Hᵀ are clustered. These are acknowledged limitations, not circular steps. The hypersphere transform is attributed to the authors' prior work [4], but the present paper restates it elementarily in Section II-A, and the new optimal-scaling theorem does not depend on [4] for its validity. The PIPG solver is also from prior work, but it is the algorithm under test rather than evidence for the preconditioner's correctness. No load-bearing self-citation chain, by-construction equivalence, or fitted-input-called-prediction was found.

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

The central theoretical result is self-contained once the full-rank condition and the KKT-condition-number proxy are accepted. The row-normalization step is a heuristic with no guarantee. No free parameters are fitted to data; the only tuning values are numerical tolerances for power iteration.

assumptions (4)
  • domain assumption The constraint matrix H has full row rank with n > m.
    Lemma 1 requires this; if rank H < m, the KKT matrix is singular, the spectrum formula in Eq. (6) does not apply, and lambda* is undefined.
  • domain assumption Cone equivalence under block row normalization can be preserved by scaling each separable cone block by a single positive scalar.
    Section II-B imposes this to keep K unchanged; for SOC and PSD cones this limits normalization to uniform scaling within each block, which may be too weak to improve conditioning.
  • domain assumption The shifted power iteration has a strictly dominant eigenvalue in the shifted matrix and converges within the set budget.
    Appendix B states this convergence assumption; the paper notes that clustering of small eigenvalues slows convergence, e.g. for small gamma in Table II.
  • domain assumption The KKT condition number is a reliable proxy for PIPG convergence speed.
    The optimal scaling minimizes kappa(K), and the experiments show fewer iterations, but the paper provides no theorem linking kappa(K) to PIPG's iteration count.

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Cite this review

Pith. "Pith review of Optimal Preconditioning for Online Quadratic Cone Programming." pith.science (2026). https://pith.science/paper/2UWJVNFV

@misc{pith2026250114191,
  author       = {Pith},
  title        = {Pith review of: Optimal Preconditioning for Online Quadratic Cone Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2UWJVNFV}},
  note         = {Machine review of arXiv:2501.14191}
}
read the original abstract

First-order conic optimization solvers are sensitive to problem conditioning and typically perform poorly in the face of ill-conditioned problem data. To mitigate this, we propose an approach to preconditioning--the hypersphere preconditioner--for a class of quadratic cone programs (QCPs), i.e., conic optimization problems with a quadratic objective function, wherein the objective function is strongly convex and possesses a certain structure. This approach lends itself to factorization-free, customizable, first-order conic optimization for online applications wherein the solver is called repeatedly to solve problems of the same size/structure, but with changing problem data. We demonstrate the efficacy of our approach on numerical convex and nonconvex trajectory optimization examples, using a first-order conic optimizer under the hood.

Figures

Figures reproduced from arXiv: 2501.14191 by the authors.

Figure 1
Figure 1. ∴ λ ⋆ = √ λ⋆2+4 σmin−λ ⋆ 2 =⇒ λ ⋆ = pσmin 2 . ■ x f(x) g(x) x ⋆ f(x ⋆) = g(x ⋆) max{f(x), g(x)} min x max{f(x), g(x)} [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The condition number of the preconditioned KKT matrix as a function of the terminal state cost matrix scaling factor, γ. “None” refers to the case where no preconditioning is applied. 100 101 102 103 104 105 106 101 102 103 Performance Robustness Terminal State Cost Matrix Scaling Factor Combined Solve Time [ms] [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The combined solve time (preconditioning and convex solve) as a function of γ. The combined solve time is only plotted if PIPG was able to converge (within 105 iterations). None 3081 11209 105 105 105 105 105 Ruiz 1101 237 1262 10774 105 105 105 QR 2240 2098 2014 3894 28588 105 105 Hypersphere 684 672 558 529 525 524 524 γ 1 10 102 103 104 105 106 TABLE I THE NUMBER OF PIPG ITERATIONS AS A FUNCTION OF γ. THE GREEN V… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The effect of the optimal objective function scaling factor, λ⋆—on (i) the condition number, κ(λ), of the KKT matrix, and (ii) the number of PIPG iterations—at each iteration of SeCO for the multi-phase rocket landing guidance problem [6]. IV. CONCLUSION We propose a t…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.