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HPR-QP: A dual Halpern Peaceman-Rachford method for solving large-scale convex composite quadratic programming

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that a dual Halpern Peaceman–Rachford method, HPR-QP, solves large-scale convex quadratic programs up to 18x faster than the compared solvers on a GPU while keeping a worst-case $O(1/k)$ residual bound.

desk verdict A credible dual HPR solver for large-scale convex QP with real numerical reach, but the O(1/k) guarantee is tied to uncertified power-method eigenvalue estimates and the abstract overstates the benchmark wins. read the letter →

arxiv 2507.02470 v1 pith:74J3FFDG submitted 2025-07-03 math.OC

Kaihuang Chen , Defeng Sun , Yancheng Yuan , Guojun Zhang , Xinyuan Zhao This is my paper · ORCID
classification math.OC MSC 90C2090C0690C2565Y20
keywords convexcompositequadraticprogrammingHalpernPeaceman–RachfordmethodrestrictedWolfedualsymmetricGauss–Seidelshadowsequencematrix-freeoptimizationGPU-acceleratedsolver
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 claims that a dual Halpern Peaceman–Rachford method named HPR-QP solves large-scale convex composite quadratic programs far faster on GPUs than the currently compared first-order and interior-point solvers. The central move is to work with the restricted Wolfe dual rather than the primal, restricting all updates to the range space of the quadratic Hessian $Q$ so that the proximal operators have smaller spectral norms, and using shadow sequences so that no explicit projection onto that subspace is ever computed. The authors prove an $O(1/k)$ bound on the KKT residual and objective error, and they report GPU benchmark results in which their Julia implementation solves more instances than PDQP, SCS, CuClarabel, and Gurobi across standard QP benchmarks, including matrix-free quadratic-assignment relaxations with up to 8,192 locations.

What carries the argument

The load-bearing object is the restricted Wolfe dual (1.5), whose constraint $w \in W = \mathrm{Range}(Q)$ keeps the quadratic term positive definite on the subspace the algorithm actually updates. Three mechanisms carry the argument: the symmetric Gauss–Seidel operator $\hat{S}_{\mathrm{sGS}}$ of (2.7), which decomposes the coupled $(y,w)$ subproblem into three explicit block updates; the shadow-sequence identity of Proposition 2.4, which says that solving $Q\bar{w}^+ = QR/(1 + \sigma\lambda_Q)$ for $\bar{w}^+ \in W$ is faithfully represented by the full-space vector $R/(1 + \sigma\lambda_Q)$ whenever every later use of $\bar{w}^+$ is through $Q\bar{w}^+$; and the Halpern averaging step $u^{k+1} = \tfrac{1}{k+2}u^0 + \tfrac{k+1}{k+2}\hat{u}^{k+1}$, which converts the Peaceman–Rachford sweep into the accelerated iteration with the $O(1/k)$ rate. The proximal choices $S_y = \lambda_A I - AA^*$ and $S_w = Q(\lambda_Q I - Q)$, with $\lambda_A \ge \lambda_1(AA^*)$ and $\lambda_Q \ge \lambda_1(Q)$, make each block update a closed-form or projection step.

What would settle it

On a small instance whose spectra can be computed exactly, compare the power-method outputs $\lambda_A$, $\lambda_Q$ to the true largest eigenvalues, form $S_y = \lambda_A I - AA^*$ and $S_w = Q(\lambda_Q I - Q)$, and check whether either operator has a negative eigenvalue; if so, the implementation ran with an invalid proximal operator. A second check: replace the power-method estimates with certified upper bounds and measure whether the KKT residual still decays like $1/k$ as Theorem 2.2 predicts.

Watch

Extended reading notes

Core claim

The paper's central claim is that for convex composite quadratic programs, the restricted Wolfe dual is a better substrate for splitting methods than any primal reformulation. In the primal, decoupling the quadratic operator $Q$ from the constraint operator $A$ forces a proximal operator built from $\lambda_Q I - Q$ plus $\sigma(\lambda_A I - A^*A)$, whose large spectral norm slows progress; the dual treatment decouples the two operators with a symmetric Gauss–Seidel pass, and by confining the $w$-updates to $W = \mathrm{Range}(Q)$ it replaces the costly term with $S_w = Q(\lambda_Q I - Q)$, which acts only on the range space. Proposition 2.4 supplies the shadow-sequence identity $Q\bar{w}^+ = Q(R/(1 + \sigma\lambda_Q))$, so the projection onto $W$ is never computed explicitly. On this foundation the paper states an $O(1/k)$ iteration complexity for the KKT residual and the objective error (Theorem 2.2) and derives adaptive restart and penalty-parameter rules from the same bound. The numerical experiments, run on an NVIDIA A100 GPU, report speedups over the compared solvers of roughly 2x to 18x on standard benchmarks and successful solution of QAP relaxations up to $d = 8192$ with $Q$ given only implicitly.

Load-bearing premise

The convergence guarantee requires $\lambda_A \ge \lambda_1(AA^*)$ and $\lambda_Q \ge \lambda_1(Q)$ so that $S_y = \lambda_A I - AA^*$ and $S_w = Q(\lambda_Q I - Q)$ stay positive semidefinite, but Section 4.1 states only that the implementation estimates these constants with the power method, which can return values below the true largest eigenvalue; an underestimate makes the proximal operators indefinite and voids the $O(1/k)$ guarantee.

Editorial extensions

If this is right

  • Large quadratic programs that currently require interior-point or active-set solvers become approachable by cheap GPU splitting iterations, including problems where $Q$ exists only as an implicit linear operator (the paper demonstrates this on quadratic assignment relaxations).
  • The dual framework covers linear programming as the special case $Q = 0$, so the method unifies the LP and QP solvers built on the same HPR machinery (Remark 2.1).
  • The $O(1/k)$ KKT-residual bound gives the restart and penalty-parameter heuristics a theoretical anchor: the surrogate merit function they monitor is the quantity the complexity theorem controls.
  • At a 1e-8 tolerance the paper reports that HPR-QP solves 128 of 137 Maros–Mészáros instances and all 30 synthetic and 36 QAP-relaxation instances, with runtime speedups over the strongest first-order competitor between roughly 2x and 18x depending on the problem class.

Reading between the lines

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

  • Guarded eigenvalue estimation is the natural robustness upgrade: replacing the power-method bounds with certified upper bounds, or adding a small relative margin, would make the $O(1/k)$ guarantee unconditional at negligible cost.
  • The shadow-sequence identity is a transferable template for rank-deficient or matrix-free settings beyond QP: any proximal term of the form $Q(\lambda I - Q)$ can be implemented without projections whenever the algorithm consumes its iterates only through $Q\cdot$, which may extend to semidefinite and other composite problems.
  • Because the $\sigma$-update rule minimizes a bound that depends on the unknown solution and the implementation uses surrogate coefficients, an ablation that compares the adaptive rule against a fixed well-tuned $\sigma$ would reveal how much of the reported speedup actually comes from the adaptation.
  • The matrix-free QAP results suggest the practical ceiling is set by memory for the iterates and by the sparsity of $A$, not by forming $Q$; problems with dense $Q$ but structured $A$, such as convolution or graph-Laplacian operators, are untested but plausible targets.
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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

3 major / 5 minor

Summary. The paper introduces HPR-QP, a Halpern Peaceman-Rachford method applied to the restricted Wolfe dual of large-scale convex composite quadratic programming. The method decouples the operators A and Q through the symmetric Gauss-Seidel technique, uses a shadow sequence to avoid explicit projection onto Range(Q), and adds adaptive restart and penalty-update heuristics. The authors state an O(1/k) iteration complexity in the KKT residual (Theorem 2.2), and report GPU-based numerical experiments on Maros-Meszaros QPs, synthetic CQPs, Lasso problems, and QAP relaxations up to d=8192. The theoretical core is inherited from the published HPR framework of [43], with the sGS substitution verified in Proposition 2.3.

Significance. If the theoretical and numerical claims hold, HPR-QP would be a substantial practical contribution: it offers a matrix-free, GPU-friendly solver for large-scale convex QPs with an O(1/k) KKT-residual guarantee, and the reported speedups over PDQP, SCS, CuClarabel, and Gurobi on synthetic and QAP instances are impressive. The shadow-sequence construction in Proposition 2.4 is an elegant way to handle the Range(Q) constraint, and the paper is honest in importing the convergence machinery from [43] rather than re-deriving it. The main weakness is a gap between the analyzed Algorithm 3 and the implemented Algorithm 4, and an eigenvalue-estimation issue in Section 4.1 that can invalidate the assumptions on which the O(1/k) guarantee rests. These issues are fixable, so the paper is a credible candidate for publication after major revision.

major comments (3)
  1. [Section 4.1; Eq. (3.4); Eq. (2.8)] The O(1/k) guarantee of Theorem 2.2 requires Assumption 2.2, which in turn requires Sy = lambda_A I - AA* and Sw = Q(lambda_Q I - Q) to be positive semidefinite, i.e., lambda_A >= lambda_1(AA*) and lambda_Q >= lambda_1(Q). Section 4.1 states only that 'we estimate lambda_A and lambda_Q using the power method.' A power-method Rayleigh quotient is a lower bound on the largest eigenvalue, not a certified upper bound; without a safety margin, a block-Lanczos/Gershgorin upper bound, or some other certificate, the shipped implementation can violate the assumptions on clustered-spectrum matrices or with unlucky initialization. In that case Sy or Sw becomes indefinite, the sGS equivalence in Proposition 2.3 no longer applies, and Algorithm 4 is no longer covered by Theorem 2.2. The manuscript gives no power-method iteration count, no margin policy, and no code, so the reader cannot verify that all reported runs used valid lambda values. Please either use certified upper bounds, add an explicit margin with a proof that the margin preserves PSD-ness, and report the resulting values, or clearly state that the practical solver is a heuristic variant whose theoretical guarantee requires external verification.
  2. [Sections 3.2-3.3; Algorithm 4; Algorithm 5] Algorithm 4, which is the solver benchmarked in Section 4, incorporates adaptive restarts (conditions (3.6)-(3.8)) and a dynamic penalty update (Algorithm 5). No convergence or complexity statement is proved for this algorithm. Theorem 2.2 applies to Algorithm 3 with a fixed sigma and no restarts; the restart and sigma-update steps can break the invariant used in the HPR argument. The paper should either prove a convergence result for Algorithm 4 (or a restricted version of it), or explicitly present Algorithm 4 as a heuristic acceleration layer whose theoretical properties are open, while making clear which numerical claims depend on the heuristic part. The surrogate merit function in Section 3.2 also depends on the unknown u*, so the restart criterion is not shown to control the true distance to the solution set.
  3. [Section 4.4; Tables 4 and 5] The Lasso comparison is not apples-to-apples: HPR-QP solves the original Lasso formulation (4.1), while PDQP, SCS, CuClarabel, and Gurobi solve the lifted CQP reformulation (4.2) with additional variables s in R^p and t in R^q. The reported speedups over PDQP and CuClarabel on these instances may therefore reflect the smaller problem size and different constraint structure rather than algorithmic superiority. The authors acknowledge the difference in the table caption, but the central claim of 'significantly outperforming state-of-the-art solvers' would be more convincing if HPR-QP were also run on the reformulation (4.2), or if the comparison were restricted to solvers applied to the same formulation.
minor comments (5)
  1. [Section 4.1] The text 'within the range [10 −16, 1016]' has a typographical issue; it should be [10^{-16}, 10^{16}]. Also, the paper should state the number of power-method iterations and the stopping criterion used for the eigenvalue estimates.
  2. [Section 2.3, Eq. (2.8)] The notation w_Q is introduced in Algorithm 3, but Theorem 2.1 and Theorem 2.2 use w^k_Q without explicitly defining the space it belongs to; defining w_Q as an element of R^n whose Q-image equals the corresponding W-vector would improve readability.
  3. [Section 3.3, Eq. (3.10)] The strict convexity argument assumes theta_2 > 0 and theta_3 > 0. The numerical stabilization in Algorithm 5 enforces this, but the theoretical discussion could mention what happens if theta_2 or theta_3 is zero.
  4. [General] No code or software availability statement is provided. Given that the paper's practical claims rest on an implementation whose internal parameter choices (eigenvalue estimates, restart thresholds, sigma updates) are not all documented, making the code available would substantially strengthen reproducibility.
  5. [References] Reference [46] is listed with 'XXX, in print'; please update with the full publication data if available.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the O(1/k) convergence theory is imported from published prior work [43] by overlapping authors, but that is a dependence on externally published mathematics, not a circular reduction.

full rationale

The paper's convergence claims are explicitly inherited rather than re-derived: Proposition 2.1 is stated "Based on Corollary 3.5 in [43]" and Proposition 2.2 "Based on the iteration complexity results in Proposition 2.9, Theorem 3.7, and Remark 3.8 of [43]." Reference [43] (Sun, Yuan, Zhang, Zhao, SIAM J. Optim. 2025) shares four authors with this paper, so the O(1/k) KKT-residual complexity of HPR-QP does rest on a self-citation chain. However, this is not a circular step under the review rules: [43] is a peer-reviewed, published theorem with stated assumptions that do not include the present target, and the paper verifies in-system the conditions needed to apply it. The substitution T1 = S + S_sGS is checked in Proposition 2.3; the shadow-sequence equivalence in Theorem 2.1 is an algebraic identity proved directly; and Theorem 2.2 correctly chains Propositions 2.2, 2.3 and Theorem 2.1. No fitted value is repackaged as a prediction: the penalty update (3.12) and the restart criteria (3.6)-(3.8) are presented as heuristics "to further enhance its performance and robustness," explicitly derived from the complexity bound rather than validated against it. The restriction to W and the choice Sw = Q(lambda_Q I - Q) are justified analytically in Remark 2.2. The genuine weakness is a correctness risk, not circularity: Section 4.1 states only "we estimate lambda_A and lambda_Q using the power method," while Assumption 2.2 and Remark 2.2 require lambda_A >= lambda_1(AA*) and lambda_Q >= lambda_1(Q) for Sy and Sw to be positive semidefinite; a power-method Rayleigh quotient is a lower bound, so the shipped solver's O(1/k) guarantee can silently fail if the estimates land below the true largest eigenvalues. That is an implementation/theory gap, not a derivation that reduces to its own inputs. Accordingly, the honest finding is no significant circularity (score 0-2); 1 reflects the load-bearing but legitimately published self-citation.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central theoretical claim rests on assumptions about existence of solutions and positive definiteness of the sGS-preconditioned operator, all imported from prior work. The practical performance depends on power-method eigenvalue estimates and several hand-chosen restart/penalty hyperparameters. There are no physical or empirical entities introduced; the shadow sequence is a proven mathematical device rather than an unexplained postulate.

free parameters (5)
  • lambda_Q (power-method estimate of lambda_1(Q)) = not reported, instance-dependent
    Used in Sw = Q(lambda_Q I - Q) (Eq. 2.8). The theoretical guarantee requires lambda_Q >= lambda_1(Q); the implementation estimates it with the power method (Section 4.1), which can underestimate.
  • lambda_A (power-method estimate of lambda_1(AA*)) = not reported, instance-dependent
    Used in Sy = lambda_A I - AA* (Eq. 3.4). Requires lambda_A >= lambda_1(AA*) for the subproblems to be well posed and for convergence.
  • initial penalty sigma_0 = ||b||/||c||, or 1 when norms are outside [1e-16, 1e16]
    Chosen heuristically in Section 4.1 after preconditioning. It affects practical convergence speed but not the stated complexity bound.
  • restart thresholds alpha_1, alpha_2, alpha_3 = 0.2, 0.8, 0.5 (alpha_3 tightened to 0.2 when residual ratio <= 0.1)
    User-defined parameters in the adaptive restart criteria (3.6)-(3.8), Section 3.2. They are chosen by hand and are not covered by the convergence theorem.
  • sigma update smoothing factor beta = exp(-eR_{r,tau_r-1}/eR_{0,tau_0-1})
    Heuristic in Algorithm 5 to stabilize sigma updates. No theoretical backing is provided.
assumptions (5)
  • domain assumption Assumption 2.1: There exists a KKT point (y*, w*, z*, x*) satisfying the KKT system (2.1).
    The convergence and O(1/k) results are stated only under this existence/feasibility assumption. If strong duality fails or a constraint qualification is violated, the restricted Wolfe dual may not provide a valid solution.
  • domain assumption Assumption 2.2: T1 is self-adjoint positive semidefinite and T1+H is positive definite on Rm x W.
    Needed for unique solvability of the subproblems in Algorithm 1 and for the metric M to define R0. Proposition 2.3 is invoked to satisfy this through the sGS construction.
  • standard math The convergence and complexity results of Sun, Yuan, Zhang, and Zhao [43] (SIAM J. Optim. 2025) are taken as given.
    Propositions 2.1 and 2.2 are established by direct citation of [43]; the full chain of lemmas in that prior paper is not reproved here.
  • domain assumption The restricted Wolfe dual (1.5) is strongly dual to the primal CCQP (1.1), and primal recovery via x* is valid.
    The algorithm solves the dual and recovers a primal solution through the KKT system. This duality is imported from Li, Sun, and Toh [25].
  • ad hoc to paper The power method returns lambda_A and lambda_Q satisfying lambda_A >= lambda_1(AA*) and lambda_Q >= lambda_1(Q).
    Section 4.1 states only that lambda_A and lambda_Q are estimated by the power method. The proximal operators Sw and Sy and the convergence guarantee require these upper bounds exactly; no safeguard or verification is provided.
invented entities (1)
  • Shadow sequence w_Q (lifted surrogate for range-space iterates)
    purpose: Enables computing range-space (W) updates of Algorithm 2 without explicit projection onto W; the algorithm tracks Q w_Q instead of w.
    Introduced in Section 2.3, Algorithm 3, and Proposition 2.4. It is a mathematical construction, not a physical object; its correctness is established by Theorem 2.1 (equivalence to the projected sequence), so no external falsifiable handle exists.

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Pith. "Pith review of HPR-QP: A dual Halpern Peaceman-Rachford method for solving large-scale convex composite quadratic programming." pith.science (2026). https://pith.science/paper/74J3FFDG

@misc{pith2026250702470,
  author       = {Pith},
  title        = {Pith review of: HPR-QP: A dual Halpern Peaceman-Rachford method for solving large-scale convex composite quadratic programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/74J3FFDG}},
  note         = {Machine review of arXiv:2507.02470}
}
abstract

In this paper, we introduce HPR-QP, a dual Halpern Peaceman-Rachford (HPR) method designed for solving large-scale convex composite quadratic programming. One distinctive feature of HPR-QP is that, instead of working with the primal formulations, it builds on the novel restricted Wolfe dual introduced in recent years. It also leverages the symmetric Gauss-Seidel technique to simplify subproblem updates without introducing auxiliary slack variables that typically lead to slow convergence. By restricting updates to the range space of the Hessian of the quadratic objective function, HPR-QP employs proximal operators of smaller spectral norms to speed up the convergence. Shadow sequences are elaborately constructed to deal with the range space constraints. Additionally, HPR-QP incorporates adaptive restart and penalty parameter update strategies, derived from the HPR method's $O(1/k)$ convergence in terms of the Karush-Kuhn-Tucker residual, to further enhance its performance and robustness. Extensive numerical experiments on benchmark data sets using a GPU demonstrate that our Julia implementation of HPR-QP significantly outperforms state-of-the-art solvers in both speed and scalability.

Figures

Figures reproduced from arXiv: 2507.02470 by the authors.

Figure 1
Figure 1. Absolute performance profiles of solvers on the 137 Maros-M´esz´aros QP instances [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Absolute performance profiles of tested solvers on 30 synthetic CQP instances. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Absolute performance profiles of tested solvers on 36 instances of QAP relaxations. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Numerical performance of tested solvers on 137 instances of the Maros-M´esz´aros data [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Numerical performance of tested solvers on 30 synthetic CQP problems (Tol. 10 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Numerical performance of tested solvers on 36 QAP relaxations (Tol. 10 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

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Cited by 1 Pith paper

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