Pith. sign in

REVIEW 2 major objections 4 minor 46 references

A Curvature-Aware Rank-Adaptive Distributed Augmented-Lagrangian Solver for Large-Scale SDPs

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper presents CARDAL, a rank-adaptive distributed augmented-Lagrangian solver for large-scale semidefinite programs, and claims it can stop rank growth near the Barvinok–Pataki scale while certifying finite outputs.

desk verdict Substantial theory and systems work; the practical certificate claim overreaches because the shifted power iteration is not a verified lower bound. read the letter →

arxiv 2607.17933 v1 pith:MXRVQ5YQ submitted 2026-07-20 math.OC cs.DC

Hongpei Li , Huikang Liu , Dongdong Ge , Yinyu Ye This is my paper · ORCID
classification math.OCcs.DC MSC 90C2290C0690C30
keywords semidefiniteprogrammingBurer–MonteirofactorizationaugmentedLagrangianrankadaptationsecond-orderstationarityBarvinok–Patakiboundmulti-GPUdistributedoptimizationapproximateKKTcertificate
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

Large semidefinite programs (SDPs) are usually solved either by interior-point methods that cannot scale, or by low-rank factorizations whose rank and optimality are hard to control. This paper claims that a curvature-aware rank-adaptive augmented-Lagrangian method can stop factor-rank growth near the Barvinok–Pataki bound for almost every objective, while retaining deterministic certificates of approximate optimality at finite accuracy. The argument rests on a reverse multiplier shift that converts a negative dual-slack direction into exact negative curvature after rank expansion, and on a joint rank-lift problem that couples several such directions across PSD blocks. The same algebraic primitives are then distributed over a Constraint x Rank x Cone GPU mesh, yielding exact distributed evaluations of residuals, gradients, Hessian actions, and slack products; experiments report faster time-to-target on robotics, chemistry, and Max-Cut relaxations, and up to 4x speedups on four GPUs.

What carries the argument

The load-bearing identity is the expression of the factored augmented Lagrangian gradient and Hessian in terms of the shifted dual slack S_c(y−ρc(F)): ∇_F Φ_{ρ,y}(F)=2(S_c(ŷ)F_c)_c and the quadratic form ⟨U,∇²Φ[U]⟩=2∑⟨U_c,S_c(ŷ)U_c⟩+ρ∥D_c(F)[U]∥². This makes negative slack visible as negative curvature only when the penalty term is controlled. The reverse multiplier shift ŷ=y−ρc(F) makes the zero-padded direction U_v=[0; v] a null direction of the constraint derivative, leaving pure curvature 2q_v; the exact quartic line search has closed-form minimizer t* = sqrt(−q_v/(ρ∥a_v∥²)) when the recession direction is nonzero. The joint rank-lift problem then couples selected negative directions acr

What would settle it

Run CARDAL on an SDP instance at nominal costs with default stopping, then compute the exact smallest eigenvalue of each terminal slack block (e.g., with a dense eigensolver or verified interval method) on the returned multiplier. If any slack eigenvalue is below −τ_dual while the solver reported success, the a posteriori certificate is invalid and the rank-adaptation test could have missed a rank-increasing direction.

Watch

Extended reading notes

Core claim

The paper's central claim is that a Burer–Monteiro factorization of a semidefinite program, solved by an augmented Lagrangian method that targets approximate second-order stationary points, can be made rank-adaptive without sacrificing global optimality. At fixed ranks, the inner L-BFGS–NC solve drives the gradient to zero and checks the Hessian; when the returned dual slack has a negative direction, a reverse multiplier shift makes that direction a strict negative-curvature direction in a zero-padded column, and an exact quartic line search (or a small joint rank-lift SDP) selects the step. The theory says that under constant-rank smoothness, every accumulation point is second-order critica

Load-bearing premise

The finite-output certificate assumes a validated lower bound S_c(y_out) ⪰ −τ_dual I for every block, but the implementation obtains the terminal slack estimate from a shifted power iteration, which does not by itself prove a rigorous eigenvalue lower bound.

Editorial extensions

If this is right

  • If the claims hold, a low-rank solver can terminate with an a posteriori approximate KKT certificate rather than relying on the final rank being high.
  • Blockwise rank growth is finite for almost every cost once per-block rank exceeds the visible affine dimension, so heterogeneous blocks can have different final ranks.
  • The distributed operators are exact rearrangements of single-device operators, so the theory and practice use the same algebraic primitives on multi-GPU meshes.
  • On a 75-instance sparse SDP benchmark, the solver passes a uniform accuracy test on 65 instances; on robotics, chemistry, and Max-Cut relaxations it reports up to 4x wall-clock speedups with four GPUs.

Reading between the lines

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

  • The a posteriori certificate in Proposition 4 requires a verified lower bound on each slack block's smallest eigenvalue; the implemented shifted-power-iteration estimate is only a Ritz-value estimate, so replacing it with a rigorous bound would close the gap between the proven certificate and the production stopping test.
  • Because the generic landscape theorem is stated per block (τ(k_c)>r_c), one can expect that in mixed-size moment/SOS relaxations, small blocks may certify at tiny ranks while large blocks need ranks tied to their own visible constraint count; this suggests cone-aware rank policies rather than a single global rank.
  • The smoothing result covers independently perturbed cost blocks used throughout a run; extending it to arbitrary nominal costs would require either a deterministic covering argument over tolerance envelopes or exact verification of slack bounds, which may be the next natural step toward fully rigorous practical certificates.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes CARDAL, a rank-adaptive distributed augmented-Lagrangian solver for large-scale SDPs with heterogeneous products of PSD cones. The algorithm combines a Burer–Monteiro low-rank factorization with an ALM outer loop; at fixed rank, an L-BFGS method with negative-curvature corrections targets an approximate Euclidean second-order stationary point of the factored augmented Lagrangian. Rank growth is driven by negative directions in the dual slack, with a reverse multiplier shift that converts a negative slack direction into exact negative curvature after zero-padding, and a joint rank-lift problem for batched corrections. The theoretical contributions include a product-cone generic landscape theorem with per-block rank conditions, a finite-accuracy smoothed guarantee via a blockwise extension of Cifuentes–Moitra AFAC, and a deterministic a posteriori approximate KKT certificate. System contributions include a three-axis (Constraint x Rank x Cone) GPU decomposition with exact distributed operator identities. Numerical experiments report Mittelmann benchmark results and scaling studies on moment relaxations, electronic-structure SOS relaxations, and Max-Cut instances.

Significance. If the claims hold, the paper makes several useful contributions: a clean product-cone extension of the Boumal–Voroninski–Bandeira landscape theory with blockwise Barvinok–Pataki rank conditions; a finite-accuracy AFAC-style guarantee under cost smoothing; a deterministic finite-output certificate; and a composable multi-GPU decomposition whose distributed operators are exact rearrangements of the single-device operators. The paper is also commendably explicit about its limitations: no polynomial iteration bound is claimed for the L-BFGS–NC inner solver, the smoothed guarantee requires prespecified rank profiles and cost perturbations, and the generic results exclude measure-zero cost sets. The key gap is implementation-level: the promised 'verified slack lower bound' is not actually verified by the described shifted-power-iteration procedure, and the pseudocode of the rank-lift step appears inconsistent with the reverse-shift theory. These issues are local and fixable, but they bear on the paper's central algorithmic claims.

major comments (2)
  1. [§5.4, Table 1, and Proposition 4 (Eq. (48))] The abstract and conclusion promise a 'verified slack lower bound' supporting an a posteriori approximate KKT certificate. Proposition 4 requires S_c(y_out) ⪰ -τ_dual I for every block. However, Section 5.4 and Table 1 state that the terminal slack estimate is 'a shifted power iteration per cone, used in the reported dual residual.' A shifted power iteration (or any Lanczos/Rayleigh-quotient procedure) returns an eigenvalue estimate, not a rigorous lower bound. If the estimated value is less negative than the true smallest eigenvalue, dres can be reported as zero and Algorithm 1 can terminate while λ_min(S_c(y_out)) < -τ_dual, invalidating Proposition 4's certificate and potentially missing rank-growth directions. The paper should either describe an actual certification procedure that produces a mathematically valid lower bound, or explicitly downgrade the implementation's slack test to
  2. [Algorithm 1, lines 10–11, and Lemma 1] There is an inconsistency between the pseudocode and the reverse-shift theory. Lemma 1 shows that the exact quartic line search and the negative-curvature result hold for Φ_{ρ,y_base} where y_base is the multiplier used in the subproblem that produced F. In Algorithm 1, line 10 correctly recovers that base multiplier via y_base ← y + ρ c(F), but line 11 then calls JOINT RANKLIFT(y, ρ, (E_c,A_c)_c), passing the shifted multiplier y rather than the recovered y_base. As written, the joint rank-lift step would use the wrong multiplier in the quartic objective and would not enjoy the identity (38) on which the exact line search and the closed-form step (37)/(41) depend. The pseudocode should pass y_base to JOINT RANKLIFT, or the paper should clarify that JOINT RANKLIFT internally reconstructs y_base from y and c(F).
minor comments (4)
  1. [Appendix A.2, proof of Theorem 2] The proof invokes 'the tube-volume estimate used in [10, Theorem 6]' without stating the estimate or verifying that the constants (4e, the exponent r_c, and the (4n_c^3 δ_c/σ_c)^{τ(k_c)} factor) transfer exactly to the blockwise product-cone setting. Since Theorem 2 is a central finite-accuracy result, including the lemma statement in an appendix would make the paper more self-contained and easier to verify.
  2. [Section 5.4] The sentence 'A shifted power iteration supplies the terminal slack estimate used in the reported dual residual' is easy to misread as a certified bound. Please add an explicit statement that this estimate is a heuristic eigenvalue estimate unless accompanied by a rigorous error bound.
  3. [Algorithm 2 and Section 3.2] Assumption A3 assumes every inner solve satisfies (34), but no convergence analysis is given for the L-BFGS–NC algorithm itself. The paper correctly disclaims an iteration bound, but it would help to state clearly that Assumption A3 is a conditional assumption about the inner solver's behavior, not a proved property.
  4. [Table 1] The row 'ALM penalty ρ0 = 2/sqrt(Σ n_c + n_lp); multiplier 3.33; cap 5×10^5' should identify 'multiplier' as the penalty growth factor γ_ρ in Eq. (42), to avoid ambiguity.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; central theorems are conditional implications built on external landscape and smoothing results; self-citations are contextual, and the main practical gap is a shifted-power-iteration estimate being described as a verified slack lower bound, which is a correctness risk rather than a derivation-from-inputs.

full rationale

The paper's central derivation chain is not circular. Theorem 1 is a product-cone extension of the external Boumal–Voroninski–Bandeira landscape argument, proved by the same determinantal-variety/codimension technique; it does not use the paper's algorithm or its fitted parameters as inputs. Proposition 4 is a direct algebraic implication: assuming the bounds in (48), including S_c(y_out) ⪰ -τ_dual I, the definitions in (25) are verified by elementary inequalities; this is a conditional certificate, not a prediction obtained from fitting. Theorem 2 extends Cifuentes–Moitra's external tube-volume argument to independent block costs, with the probability bound stated explicitly and the event uniform over bounded AFAC pairs; it is not constructed from the solver's own outputs. Self-citations—D-PDLP [25], cuLoRADS [17,18], COPT [13]—are used for architectural inspiration, benchmark baselines, or solver listing, and do not carry the load-bearing landscape, smoothing, or ALM convergence proofs. No uniqueness theorem is imported from the authors' prior work, and no fitted constant is renamed as a prediction. The notable weakness is in Section 5.4 and Table 1: the abstract and Proposition 4 promise 'a verified slack lower bound,' but the implementation says 'A shifted power iteration supplies the terminal slack estimate used in the reported dual residual.' A shifted power iteration gives a Ritz-value estimate, not a certified lower bound on the smallest eigenvalue, so the reported dres could be optimistic and rank-growth directions could be missed. This is a genuine validation gap and a caveat on the practical certificate, but it is not circularity: the theorem remains a valid conditional implication, and no input is being relabeled as its own output. Overall the paper's results are self-contained against external theoretical foundations and external benchmarks, so the appropriate circularity score is low.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

CARDAL introduces no new physical or mathematical entities. The theoretical results depend on explicit regularity, compactness, generic-position, and cost-smoothing assumptions, and the fixed-rank convergence theorem depends on the unproven premise that the L-BFGS-NC inner solver delivers the required SOSP. The implementation also relies on heuristic caps that are not guaranteed by the theory.

free parameters (4)
  • Initial ALM penalty ρ0 = 2 / sqrt(Σ n_c + n_lp)
    Hand-chosen default in Table 1. Affects the penalty trajectory and conditioning but is not part of the theoretical claims.
  • Penalty growth factor and cap = 3.33, cap 5×10^5
    Table 1 heuristic; no tuning or sensitivity analysis is given.
  • Lanczos step caps and retained directions = 15 Hessian-Lanczos steps; 30 slack Lanczos steps per cone; at most r_inc,c directions
    Heuristic caps in Table 1. If negative eigenvalues are missed, rank growth and the terminal slack certificate can fail.
  • Per-cone rank cap = (sqrt(8 m_c + 1) - 1) / 2
    Heuristic safeguard in Table 1, not derived from the theory; it can stop rank growth below the Barvinok–Pataki scale for some instances.
assumptions (6)
  • domain assumption Assumption 1: Dc(F) has constant rank on a neighborhood of each nonempty product-cone BM feasible set at visited rank profiles.
    Used in Section 2.1 to make M_k an embedded submanifold and to define smooth tangent/normal projectors. Not guaranteed by problem data.
  • domain assumption Assumption 2: compactness/boundedness of accepted factors and inner level sets, bounded multipliers, and inner solves satisfying (34) with tolerances tending to zero.
    Invoked in Section 3.2 for fixed-rank ALM convergence. Whether the implemented L-BFGS-NC satisfies (34) is assumed, not proved.
  • domain assumption The primal SDP feasible set is nonempty and compact, and no nonzero PSD recession direction lies in the nullspace of the affine map.
    Used in Lemma 1 to ensure a_v ≠ 0 for nonzero v and to make the exact quartic line-search minimizer well-defined.
  • domain assumption The cost tuple C lies outside the measure-zero exceptional set in Theorem 1 for every visited rank profile.
    Required for the generic global-optimality conclusion; the guarantee is almost-everywhere, not worst-case.
  • ad hoc to paper For the finite-accuracy theorem, the block costs are independently perturbed with full-dimensional uniform distributions, and rank profiles/tolerances are predeclared before sampling.
    Theorem 2 applies only to sampled costs used throughout the run; the transfer back to nominal costs incurs additive σ_max accuracy floors.
  • standard math The tube-volume estimate for determinantal varieties from Cifuentes–Moitra [10, Theorem 6] is valid in the blockwise product-cone setting.
    The proof of Theorem 2 in Appendix A.2 defers the probability estimate to [10] rather than reproving it here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Curvature-Aware Rank-Adaptive Distributed Augmented-Lagrangian Solver for Large-Scale SDPs." pith.science (2026). https://pith.science/paper/MXRVQ5YQ

@misc{pith2026260717933,
  author       = {Pith},
  title        = {Pith review of: A Curvature-Aware Rank-Adaptive Distributed Augmented-Lagrangian Solver for Large-Scale SDPs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXRVQ5YQ}},
  note         = {Machine review of arXiv:2607.17933}
}
read the original abstract

We present CARDAL (Curvature-Aware Rank-Adaptive Distributed Augmented Lagrangian), a distributed multi-GPU solver for large-scale semidefinite programs (SDPs) based on a rank-adaptive Burer-Monteiro factorization and an augmented Lagrangian method. At fixed ranks, a matrix-free L-BFGS method with negative-curvature corrections targets an approximate Euclidean second-order stationary point of the factored augmented Lagrangian. A reverse multiplier shift turns a negative dual-slack direction into exact negative curvature after rank expansion, and a small joint rank-lift problem selects a batched low-rank correction. A verified slack lower bound provides an a posteriori approximate KKT certificate. Our analysis establishes generic global-optimality guarantees for heterogeneous products of PSD cones at per-block ranks near the Barvinok-Pataki scale, together with a finite-accuracy counterpart under blockwise cost smoothing. For scalable execution, CARDAL distributes constraint rows, factor columns, and PSD blocks over a Constraint x Rank x Cone device mesh. The primal residual, gradient, Hessian-vector products, and slack matrix-vector products are evaluated using device-local operations and axis-wise collectives. On the Mittelmann benchmark, CARDAL exhibits stronger robustness than existing low-rank GPU approaches under a uniform accuracy standard. Experiments on large-scale SDP relaxations from robotics, electronic structure, and Max-Cut demonstrate the complementary scaling regimes of the three distribution axes, with observed wall-clock speedups of up to 4x on four H100 GPUs.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 5 linked inside Pith

  1. [1]

    Aguirre, Diego Cifuentes, Vincent Guigues, Renato D

    Jacob M. Aguirre, Diego Cifuentes, Vincent Guigues, Renato D. C. Monteiro, Victor Hugo Nascimento, and Arnesh Sujanani. cuHALLaR: A GPU-accelerated low-rank augmented Lagrangian method for large-scale semidefinite programming.arXiv preprint arXiv:2505.13719, 2025

    arXiv 2025

  2. [2]

    Barvinok

    Alexander I. Barvinok. Problems of distance geometry and convex properties of quadratic maps.Discrete & Computational Geometry, 13(2):189–202, 1995

    1995

  3. [3]

    Bertsekas.Constrained Optimization and Lagrange Multiplier Methods

    Dimitri P. Bertsekas.Constrained Optimization and Lagrange Multiplier Methods. Computer Science and Applied Mathe- matics. Academic Press, New York, 1982. ISBN 0-12-093480-9

    1982

  4. [4]

    Bandeira

    Nicolas Boumal, Vladislav V oroninski, and Afonso S. Bandeira. Deterministic guarantees for Burer–Monteiro factorizations of smooth semidefinite programs.Communications on Pure and Applied Mathematics, 73(3):581–608, 2020

    2020

  5. [5]

    Cambridge University Press, 2004

    Stephen Boyd and Lieven Vandenberghe.Convex Optimization. Cambridge University Press, 2004

    2004

  6. [6]

    Computational enhancements in low-rank semidefinite programming.Optimisation Meth- ods and Software, 21(3):493–512, 2006

    Samuel Burer and Changhui Choi. Computational enhancements in low-rank semidefinite programming.Optimisation Meth- ods and Software, 21(3):493–512, 2006

    2006

  7. [7]

    Samuel Burer and Renato D. C. Monteiro. A projected gradient algorithm for solving the Max-Cut SDP relaxation.Opti- mization Methods and Software, 15(3-4):175–200, 2001. 20 PREPRINT. UNDER REVIEW

    2001

  8. [8]

    Samuel Burer and Renato D. C. Monteiro. A nonlinear programming algorithm for solving semidefinite programs via low- rank factorization.Mathematical Programming, 95(2):329–357, 2003

    2003

Show all 46 references
  1. [9]

    Quantum algorithms for the ordered search problem via semidef- inite programming.Physical Review A: Atomic, Molecular, and Optical Physics, 75(3):032335, 2007

    Andrew M Childs, Andrew J Landahl, and Pablo A Parrilo. Quantum algorithms for the ordered search problem via semidef- inite programming.Physical Review A: Atomic, Molecular, and Optical Physics, 75(3):032335, 2007

    2007

  2. [10]

    Polynomial time guarantees for the Burer–Monteiro method.Advances in Neural Infor- mation Processing Systems, 35:23923–23935, 2022

    Diego Cifuentes and Ankur Moitra. Polynomial time guarantees for the Burer–Monteiro method.Advances in Neural Infor- mation Processing Systems, 35:23923–23935, 2022

    2022

  3. [11]

    Revisiting spectral bundle methods: Primal-dual (sub)linear convergence rates.SIAM Journal on Optimization, 33(2):1305–1332, 2023

    Lijun Ding and Benjamin Grimmer. Revisiting spectral bundle methods: Primal-dual (sub)linear convergence rates.SIAM Journal on Optimization, 33(2):1305–1332, 2023

    2023

  4. [12]

    New understandings and computation on augmented Lagrangian methods for low-rank semidefinite programming.arXiv preprint arXiv:2505.15775, 2025

    Lijun Ding, Haihao Lu, and Jinwen Yang. New understandings and computation on augmented Lagrangian methods for low-rank semidefinite programming.arXiv preprint arXiv:2505.15775, 2025

    2025 arXiv

  5. [13]

    Cardinal optimizer (COPT) user guide.arXiv preprint arXiv:2208.14314, 2022

    Dongdong Ge, Qi Huangfu, Zizhuo Wang, Jian Wu, and Yinyu Ye. Cardinal optimizer (COPT) user guide.arXiv preprint arXiv:2208.14314, 2022

    2022

  6. [14]

    Goemans and David P

    Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability prob- lems using semidefinite programming.Journal of the ACM, 42(6):1115–1145, 1995

    1995

  7. [15]

    Clarabel: An interior-point solver for conic programs with quadratic objectives.Mathemat- ical Programming Computation, pages 1–83, 2026

    Paul J Goulart and Yuwen Chen. Clarabel: An interior-point solver for conic programs with quadratic objectives.Mathemat- ical Programming Computation, pages 1–83, 2026

    2026

  8. [16]

    An improved rounding method and semidefinite programming relaxation for graph partition.Mathematical Programming, 92(3):509–535, 2002

    Qiaoming Han, Yinyu Ye, and Jiawei Zhang. An improved rounding method and semidefinite programming relaxation for graph partition.Mathematical Programming, 92(3):509–535, 2002

    2002

  9. [17]

    A low-rank ADMM splitting approach for semidefinite programming.arXiv preprint arXiv:2403.09133, 2024

    Qiushi Han, Chenxi Li, Zhenwei Lin, Caihua Chen, Qi Deng, Dongdong Ge, Huikang Liu, and Yinyu Ye. A low-rank ADMM splitting approach for semidefinite programming.arXiv preprint arXiv:2403.09133, 2024

    2024 arXiv

  10. [18]

    Accelerating low-rank factorization-based semidefinite programming algorithms on GPU.arXiv preprint arXiv:2407.15049, 2024

    Qiushi Han, Zhenwei Lin, Hanwen Liu, Caihua Chen, Qi Deng, Dongdong Ge, and Yinyu Ye. Accelerating low-rank factorization-based semidefinite programming algorithms on GPU.arXiv preprint arXiv:2407.15049, 2024

    2024 arXiv

  11. [19]

    Fast and certifiable trajectory optimization

    Shucheng Kang, Xiaoyang Xu, Jay Sarva, Ling Liang, and Heng Yang. Fast and certifiable trajectory optimization. In International Workshop on the Algorithmic Foundations of Robotics, pages 43–65. Springer, 2024

    2024

  12. [20]

    Global contact-rich planning with sparsity-rich semidefinite relaxations.arXiv preprint arXiv:2502.02829, 2025

    Shucheng Kang, Guorui Liu, and Heng Yang. Global contact-rich planning with sparsity-rich semidefinite relaxations.arXiv preprint arXiv:2502.02829, 2025

    2025 arXiv

  13. [21]

    Guanghui Lan and Renato D. C. Monteiro. Iteration-complexity of first-order augmented Lagrangian methods for convex programming.Mathematical Programming, 155(1):511–547, 2016

    2016

  14. [22]

    Global optimization with polynomials and the problem of moments.SIAM Journal on Optimization, 11(3): 796–817, 2001

    Jean B Lasserre. Global optimization with polynomials and the problem of moments.SIAM Journal on Optimization, 11(3): 796–817, 2001

    2001

  15. [23]

    Convergent SDP-relaxations in polynomial optimization with sparsity.SIAM Journal on Optimization, 17 (3):822–843, 2006

    Jean B Lasserre. Convergent SDP-relaxations in polynomial optimization with sparsity.SIAM Journal on Optimization, 17 (3):822–843, 2006

    2006

  16. [24]

    SNAP Datasets: Stanford large network dataset collection.http://snap.stanford

    Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection.http://snap.stanford. edu/data, 2014

    2014

  17. [25]

    D-PDLP: Scaling PDLP to distributed multi-GPU systems.arXiv preprint arXiv:2601.07628, 2026

    Hongpei Li, Yicheng Huang, Huikang Liu, Dongdong Ge, and Yinyu Ye. D-PDLP: Scaling PDLP to distributed multi-GPU systems.arXiv preprint arXiv:2601.07628, 2026

    2026 arXiv

  18. [26]

    An overview and comparison of spectral bundle methods for primal and dual semidefinite programs.Computational Optimization and Applications, 93(2):523–566, 2026

    Feng-Yi Liao, Lijun Ding, and Yang Zheng. An overview and comparison of spectral bundle methods for primal and dual semidefinite programs.Computational Optimization and Applications, 93(2):523–566, 2026

    2026

  19. [27]

    Fast quantum simulation of electronic structure by spectral amplification.Physical Review X, 15(4):041016, 2025

    Guang Hao Low, Robbie King, Dominic W Berry, Qiushi Han, A Eugene DePrince III, Alec F White, Ryan Babbush, Rolando D Somma, and Nicholas C Rubin. Fast quantum simulation of electronic structure by spectral amplification.Physical Review X, 15(4):041016, 2025

    2025

  20. [28]

    Mittelmann

    Hans D. Mittelmann. Benchmarks for optimization software: Sparse SDP.https://plato.asu.edu/ftp/sparse_sdp. html, 2026. Accessed 2026

    2026

  21. [29]

    Renato D. C. Monteiro, Arnesh Sujanani, and Diego Cifuentes. A low-rank augmented Lagrangian method for large-scale semidefinite programming based on a hybrid convex–nonconvex approach.Mathematical Programming, pages 1–57, 2026

    2026

  22. [30]

    MOSEK ApS, 2019

    MOSEK ApS.MOSEK Optimization Suite. MOSEK ApS, 2019

    2019

  23. [31]

    Conic optimization via operator splitting and homogeneous self-dual embedding.Journal of Optimization Theory and Applications, 169(3):1042–1068, 2016

    Brendan O’Donoghue, Eric Chu, Neal Parikh, and Stephen Boyd. Conic optimization via operator splitting and homogeneous self-dual embedding.Journal of Optimization Theory and Applications, 169(3):1042–1068, 2016

    2016

  24. [32]

    On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues.Mathe- matics of Operations Research, 23(2):339–358, 1998

    Gábor Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues.Mathe- matics of Operations Research, 23(2):339–358, 1998

    1998

  25. [33]

    Diagonal preconditioning for first-order primal-dual algorithms in convex optimiza- tion

    Thomas Pock and Antonin Chambolle. Diagonal preconditioning for first-order primal-dual algorithms in convex optimiza- tion. In2011 International Conference on Computer Vision (ICCV), pages 1762–1769, 2011

    2011

  26. [34]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Mathematics of Operations Research, 1(2):97–116, 1976

    1976

  27. [35]

    Stephen B. Seidman. Network structure and minimum degree.Social Networks, 5(3):269–287, 1983

    1983

  28. [36]

    Jos F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.Optimization Methods and Software, 11(1-4):625–653, 1999

    1999

  29. [37]

    SDPNAL+: A MATLAB software for semidefinite programming with bound constraints (version 1.0).Optimization Methods and Software, 35(1):87–115, 2020

    Defeng Sun, Kim-Chuan Toh, Yancheng Yuan, and Xin-Yuan Zhao. SDPNAL+: A MATLAB software for semidefinite programming with bound constraints (version 1.0).Optimization Methods and Software, 35(1):87–115, 2020

    2020

  30. [38]

    Michael J. Todd. Semidefinite optimization.Acta Numerica, 10:515–560, 2001

    2001

  31. [39]

    On the implementation and usage of SDPT3–a MATLAB software package for semidefinite-quadratic-linear programming, version 4.0

    Kim-Chuan Toh, Michael J Todd, and Reha H Tütüncü. On the implementation and usage of SDPT3–a MATLAB software package for semidefinite-quadratic-linear programming, version 4.0. InHandbook on semidefinite, conic and polynomial optimization, pages 715–754. Springer, 2012. 21 PR...

    2012

  32. [40]

    Solving low-rank semidefinite programs via manifold optimization.Journal of Scientific Computing, 104(1):33, 2025

    Jie Wang and Liangbing Hu. Solving low-rank semidefinite programs via manifold optimization.Journal of Scientific Computing, 104(1):33, 2025

    2025

  33. [41]

    A decomposition augmented Lagrangian method for low-rank semidefinite programming.SIAM Journal on Optimization, 33(3):1361–1390, 2023

    Yifei Wang, Kangkang Deng, Haoyang Liu, and Zaiwen Wen. A decomposition augmented Lagrangian method for low-rank semidefinite programming.SIAM Journal on Optimization, 33(3):1361–1390, 2023

    2023

  34. [42]

    Liuqin Yang, Defeng Sun, and Kim-Chuan Toh. SDPNAL+: a majorized semismooth Newton–CG augmented Lagrangian method for semidefinite programming with nonnegative constraints.Mathematical Programming Computation, 7(3):331– 366, 2015

    2015

  35. [43]

    A conditional gradient framework for composite convex minimization with applications to semidefinite programming

    Alp Yurtsever, Olivier Fercoq, Francesco Locatello, and V olkan Cevher. A conditional gradient framework for composite convex minimization with applications to semidefinite programming. InInternational Conference on Machine Learning, pages 5727–5736. PMLR, 2018

    2018

  36. [44]

    A conditional-gradient-based augmented Lagrangian framework

    Alp Yurtsever, Olivier Fercoq, and V olkan Cevher. A conditional-gradient-based augmented Lagrangian framework. In International Conference on Machine Learning, pages 7272–7281. PMLR, 2019

    2019

  37. [45]

    Scalable semidefinite programming.SIAM Journal on Mathematics of Data Science, 3(1):171–200, 2021

    Alp Yurtsever, Joel A Tropp, Olivier Fercoq, Madeleine Udell, and V olkan Cevher. Scalable semidefinite programming.SIAM Journal on Mathematics of Data Science, 3(1):171–200, 2021

    2021

  38. [46]

    A Newton–CG augmented Lagrangian method for semidefinite program- ming.SIAM Journal on Optimization, 20(4):1737–1765, 2010

    Xin-Yuan Zhao, Defeng Sun, and Kim-Chuan Toh. A Newton–CG augmented Lagrangian method for semidefinite program- ming.SIAM Journal on Optimization, 20(4):1737–1765, 2010. A Additional Proofs A.1 Product-cone stationarity and landscape Proof of Proposition 1.LetF∈ M k be a feasi...

    2010

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.