Pith. sign in

REVIEW 3 major objections 5 minor 143 references

Quadratic sum-of-squares programs can be solved directly, without conic lifting, by regularizing the SOS variables and maximizing a smooth concave dual.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 04:02 UTC pith:M2EQOUYG

load-bearing objection A useful regularized-dual trick with mostly checkable bounds, but the claims outrun the analysis (restart mismatch, 'unconditional' line) and the benchmarks need better baselines. the 3 major comments →

arxiv 2607.13701 v1 pith:M2EQOUYG submitted 2026-07-15 math.OC cs.SYeess.SY

Lifting-Free Quadratic Sum-Of-Squares Programming

Gabriel F. Machado , Ross Drummond , Morgan Jones This is my paper
classification math.OC cs.SYeess.SY MSC 90C2290C2565K05
keywords quadratic sum-of-squaresconic programmingdual gradient ascentregularizationsemidefinite projectionnon-asymptotic convergencefirst-order methodsconstrained regression
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper aims to show that quadratic sum-of-squares programs—optimization problems with a convex quadratic objective and constraints expressed through sum-of-squares polynomials—can be solved without the usual trick of translating the quadratic objective into a second-order cone or semidefinite lifting that inflates the conic dimension. The proposed route adds a tunable norm penalty to the sum-of-squares variables, a small regularization that does not change the cone structure, and then maximizes the resulting concave dual with an accelerated gradient method. The payoff is a solver whose per-iteration work is dominated by one projection onto a positive semidefinite cone per SOS block, with memory that scales only with the number of equality constraints, and with a proven non-asymptotic guarantee: after N iterations the objective gap to the true optimum is O(1/N) plus a bias term that vanishes as the regularization shrinks. A sympathetic reader would care because this removes a computational bottleneck for large problems in system identification and machine learning, and the theory converts a heuristic regularization into a certified algorithm.

Core claim

The paper's central discovery is that adding the quadratic penalty (rho/2) sum_j ||xi_sos^(j)||^2 to the objective of a quadratic conic program does not merely stabilize the problem—it makes it solvable in closed form at each dual step and smooth in the dual. The partial Lagrangian separates into a free-variable piece, minimized by Q^{-1}(A_f^T lambda - w_f), and independent SOS pieces, each minimized by projecting (1/rho)(A_sos^(j)^T lambda - w_sos^(j)) onto the positive semidefinite cone. By the standard polar-cone decomposition, the dual function collapses to a concave quadratic minus squared norms of those projections, giving a globally Lipschitz gradient whose value is exactly the negat

What carries the argument

The load-bearing object is the regularization term (rho/2) sum_j ||xi_sos^(j)||_2^2 added to the objective. This single modification preserves the original cone structure (no lifting), makes each SOS block's Lagrangian subproblem a Euclidean projection onto the positive semidefinite cone—a closed-form operation via eigenvalue thresholding—and makes the dual function concave, differentiable, and with Lipschitz gradient constant L = ||A_f Q^{-1} A_f^T||_2 + (1/rho) sum_j ||A_sos^(j) A_sos^(j)^T||_2. The same gradient equals the negative of the affine constraint residual, so monitoring it yields a combined primal-feasibility and optimality check. The polar-cone identity that rewrites ||P_{S_+}(

Load-bearing premise

The central proofs require every SOS block to individually span all equality constraints (each A_sos^(j) A_sos^(j)^T invertible) and Q to be strictly positive definite, so the method's certified regime excludes sparse multi-block SOS problems and the linear SOS case Q=0.

What would settle it

Take a QSOS problem with two SOS blocks where one block's coupling matrix is rank-deficient while the combined constraint matrix is still surjective, run the algorithm, and check whether the affine feasibility residual decays as O(1/N) or stalls; the paper's coercivity theorem predicts the bound should not hold, so this separates the assumption from the method's practical behavior.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For each iteration, the dominant computation is a single PSD projection per SOS block; no auxiliary conic variables or factorizations are introduced, so memory grows only with the number of equality constraints, not with the SOS block dimensions.
  • The dual gradient is the negative primal affine residual, so an O(1/N) feasibility certificate is obtained for free from the dual convergence rate; termination can be monitored without any additional solves.
  • Setting rho small makes the recovered solution arbitrarily close to the original optimum, with the trade-off that the Lipschitz constant and hence the iteration count grow as O(1/rho); the O(1/N) convergence of the accelerated scheme compensates for that growth.
  • The method applies to any QSOS problem with positive definite Q and a Slater point, including constrained regression and system identification examples; numerical evidence shows it can solve all benchmark instances where lifted interior-point solvers time out or exhaust memory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The regularization is effectively a Moreau-Yosida smoothing of the SOS constraints in the dual, suggesting the method can be recast as a proximal point algorithm; that view would justify adaptive rho schedules (homotopy) that the paper leaves to future work.
  • The coercivity proof requires each SOS block's coupling Gram matrix A_sos^(j) A_sos^(j)^T to be invertible, a condition that fails in common sparse multi-block SOS problems; an extension using block-coordinate ascent or a shared regularizer across blocks would be needed to cover those instances—this is a boundary of the current claim, not a defect of the experiments, where a single surjective bloc
  • The same lifting-free dualization should transfer to quadratic conic programs over other cones with cheap projections (for instance, a second-order cone or a positive orthant), giving an immediate family of algorithms; the paper only demonstrates the PSD/SOS case.
  • A concrete testable extension is to decay rho geometrically across outer iterations and analyze the total iteration count needed to reach a target bias; the paper's fixed-rho bound suggests total complexity roughly O(1/epsilon) with a log factor, but that is not proven here.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a lifting-free regularization method for Quadratic Sum-of-Squares (QSOS) programs. A quadratic penalty is added to the SOS variables, only the affine constraints are dualized, and the resulting concave dual is maximized by Nesterov accelerated gradient. The authors provide closed-form primal updates (Lemmas 3.1–3.2), an explicit dual gradient and Lipschitz constant (Theorem 3.6), coercivity under a Slater and invertibility assumption (Theorem 4.3), asymptotic primal feasibility (Theorem 4.2), and a non-asymptotic O(1/N)+O(ρ) suboptimality bound (Theorem 4.5). Numerical experiments on 240 constrained regression problems report that the method solves all instances, is about 40% faster on average than SCS, and is more reliable than MOSEK on lifted formulations.

Significance. If the claims are fully established, the method would be a valuable scalable alternative to lifting-based solvers for large QSOS/QCP problems, with per-iteration cost dominated by PSD projections and no slack variables or augmented Lagrangians. The core derivations—closed-form minimizers, dual gradient, Lipschitz constant, and the suboptimality proof—are internally consistent and self-contained. The benchmark study is extensive and the empirical advantage is promising. However, the theoretical guarantees as stated do not cover the algorithm as implemented or the benchmark instances, which weakens the paper’s central message. With the gaps identified below addressed, the contribution could be suitable for publication.

major comments (3)
  1. [§4, Theorem 4.3; §5] The coercivity assumption (4.5), requiring each Z^(j)=A_sos^(j)A_sos^(j)T to be invertible, is violated in the constrained regression benchmarks. There m = n_f, while each SOS block has only h = (n_x+d)!/(n_x! d!) columns; for d>0, h<n_f, so rank(A_sos) ≤ h < m and Z is singular. Therefore Lemma 4.4 and Theorem 4.5 do not apply to the experiments. Please either prove coercivity under a weaker sufficient condition (e.g., A_f Q^{-1} A_f^T ≻ 0, which holds here) or explicitly state that the experiments lie outside the proven guarantees.
  2. [§3, Algorithm 3.1; §4, Theorem 4.1] The convergence analysis treats the standard accelerated gradient sequence λ^(k) without restart, but Algorithm 3.1 includes an adaptive restart (lines 7–10) and outputs the extrapolated point y^(k) (line 22). Theorem 4.1 bounds g(λ*)-g(λ^(N)), and Theorems 4.2/4.5 use λ^(N), not y^(k). No proof is given that the restarted sequence or the output y^(k) satisfies the same bounds. Please align the algorithm and the theory—either analyze the implemented variant, or change the output to λ^(k) and state that the bounds apply to that output.
  3. [§6, Conclusion] The statement “These guarantees are unconditional and hold for all problems without any additional assumptions” is incorrect. Theorem 4.5 requires Q≻0, a Slater point, and the invertibility condition (4.5) used in Theorem 4.3. Furthermore, the Q=0 linear SOS case highlighted in the introduction is outside the class covered by the closed-form update in (3.6). The assumptions should be clearly stated in the abstract and conclusion to avoid overclaiming.
minor comments (5)
  1. [§4, Theorem 4.3 proof] The proof switches between treating ξ_sos as a vector in R^{h_j(h_j+1)/2} and as a symmetric matrix in S^{h_j}. Using svec/smat consistently throughout would improve readability.
  2. [§5, Algorithm implementation] The paper does not describe how the Lipschitz constant L in (3.15) is computed or estimated for problems with m up to 20,000. Forming Γ = Σ A_sos^(j)A_sos^(j)T requires O(m^2) memory, so an implementation detail is missing. Please add the method used to obtain η=1/L.
  3. [§5, Fig. 1] The x-axis label is garbled (“Relativetimetosolve,=50”); it should be “Relative time to solve, τ”.
  4. [§4, Theorem 4.1 proof] The proof is deferred to “[40]” with “similar arguments.” Since Nesterov’s rate is central to the paper, a more complete proof or a precise theorem statement with assumptions would be helpful.
  5. [§5, Memory claim] The abstract and conclusions claim memory scales only with the number of equality constraints, but the per-iteration PSD projection requires storing and eigendecomposing an h_j×h_j matrix. The claim should be qualified accordingly.

Circularity Check

0 steps flagged

No significant circularity: the dual function, gradient, Lipschitz constant, and convergence bounds are derived from problem data under explicitly stated assumptions; the single self-citation is not load-bearing.

full rationale

The derivation chain is self-contained. (Reg-QCP) is defined with a fixed regularization parameter ρ = 1e-6 (Section 5), not fitted to the target. The dual function g(λ) is derived from the Lagrangian (Eqs. 3.2-3.13), the closed-form primal updates are derived in Lemma 3.1 and Lemma 3.2 using only Q positive definiteness and projection properties, and the gradient formula (3.14) and Lipschitz constant (3.15) are computed analytically from the data. The convergence theorems (Theorems 4.1, 4.2, 4.5) rely on standard convex analysis: Nesterov's accelerated gradient (cited externally [40]) and strong duality. The feasibility residual is exactly the negative dual gradient (Eq. 4.1), so Theorem 4.2 is a corollary of the dual convergence, not a fitted prediction. The bound in Theorem 4.5 contains the exact regularization bias ρ Σ||ξ*_sos||², which is a consequence of the method's definition rather than an empirical fit. Numerical benchmarks fix ρ across 240 problems and measure actual suboptimality and feasibility against known ground truth, so there is no fitted-parameter-called-prediction. The only self-citation [32] appears in the introduction as an application example and is not load-bearing. One non-circular caveat: the Conclusion states the guarantees are 'unconditional and hold for all problems without any additional assumptions,' whereas Theorems 4.3-4.5 explicitly require Q positive definite, Slater's condition, and invertibility of each Z⁽ʲ⁾ = A_sos⁽ʲ⁾ A_sos⁽ʲ⁾ᵀ; this is a correctness/overclaim issue, not a circularity issue.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No new physical entities are introduced. The method rests on standard convex-analysis facts (Moreau decomposition, nonexpansive projections), standard duality (strong duality under Slater), and three structural assumptions on the problem data: Q positive definite, each SOS block's coupling matrix Z^(j) invertible, and strict feasibility. The free parameters are the regularization rho and the benchmark design choices (D = 2n_f, sampling/scaling) used to force Q positive definite. The heavy lifting the paper adds is the regularization-penalty construction and the non-asymptotic analysis; everything else is inherited from Malick 2004 and standard convex optimization.

free parameters (2)
  • rho (regularization parameter) = 1e-6 (all benchmarks)
    Added norm penalty on the SOS blocks. Makes the SOS subproblem strongly convex, giving the projection minimizer (3.8), but also sets the accuracy floor rho*sum||xi*_sos||^2 in Theorem 4.5 and inflates the Lipschitz constant as 1/rho (3.15). The choice 1e-6 is not justified by a sensitivity study.
  • Benchmark design: D = 2 n_f data points; x-samples in [-1,1]^n_x; xi*_f in [-1,1]^n_f = D = 2 n_f
    Chosen so that Q = 2 sum z_2d(x) z_2d(x)^T is positive definite, which Lemma 3.1 requires for Q^{-1}. For D < n_f the method is undefined on these problems; this is a design choice of the experiments, not a constant fitted to make the competition lose.
axioms (5)
  • domain assumption Q is positive definite (Q in S^{n_f}_{++})
    Lemma 3.1 and all subsequent results require Q^{-1}. The paper's own introduction presents Q = 0 (linear SOS) as a special case of (QSOS), but the method cannot handle Q = 0 or rank-deficient Q. No extension for singular Q is given anywhere in the paper.
  • domain assumption Each Z^(j) = A_sos^(j) A_sos^(j)^T is invertible (Eq. 4.5)
    Required in Theorem 4.3 for coercivity of the dual, hence for bounded dual iterates (Lemma 4.4) and the central non-asymptotic bound (Theorem 4.5). Fails whenever a single SOS block's coupling rows are linearly dependent or the block dimension is smaller than m; in multi-block SOS with m > 1 this is the typical case. The single-block benchmarks satisfy it automatically because the Gram map is surjective.
  • domain assumption Slater's condition for (Reg-QCP) (Eq. 4.4): a strictly feasible point with each SOS block positive definite
    Underpins strong duality used in Step 2 of Theorem 4.5 and the strict positivity used in the coercivity construction of Theorem 4.3. The paper verifies it only implicitly for the regression benchmarks.
  • standard math Moreau decomposition, nonexpansiveness of projections, and the eigendecomposition formula for PSD projection (Theorem 2.4 in [34])
    Used in Corollary 3.4 and Lemma 3.2 to simplify the dual function and obtain closed-form primal updates; standard results treated as external inputs.
  • domain assumption The SOSTOOLS parse of a QSOS problem yields a (QCP) with the assumed constraint matrix A, b, and one SOS block per benchmark instance
    All numerical comparisons assume the parser faithfully produces the (QCP) form and that the benchmark structure (single Gram block, m = n_f coefficient-matching equalities) is representative of QSOS problems generally.

pith-pipeline@v1.3.0-alltime-deepseek · 22581 in / 36482 out tokens · 339194 ms · 2026-08-02T04:02:08.561989+00:00 · methodology

0 comments
read the original abstract

Quadratic Sum-Of-Squares (QSOS) optimization problems appear in system identification and machine learning, but standard Schur-complement and second-order cone liftings enlarge conic dimensions and create computational bottlenecks for interior-point methods. This paper introduces a lifting-free regularization that preserves the original conic structure by adding a norm penalty to SOS variables, yielding closed-form primal updates and an unconstrained, concave dual with Lipschitz-continuous gradient. Accelerated first-order methods efficiently maximize this dual, and convergence analysis shows non-asymptotic recovery of the solution. Numerical experiments on constrained regression problems show the proposed method can be 40\% faster than existing solvers such as SCS and handle larger problems than MOSEK, with memory scaling only in the number of equality constraints.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

143 extracted references · 3 canonical work pages

  1. [1]

    SIAM Journal on Optimization , volume=

    Global optimization with polynomials and the problem of moments , author=. SIAM Journal on Optimization , volume=. 2001 , publisher=

    2001

  2. [2]

    Mathematical programming , volume=

    Semidefinite programming relaxations for semialgebraic problems , author=. Mathematical programming , volume=. 2003 , publisher=

    2003

  3. [3]

    Mathematics , volume=

    Review on system identification, control, and optimization based on artificial intelligence , author=. Mathematics , volume=. 2025 , publisher=

    2025

  4. [4]

    Emerging applications of algebraic geometry , pages=

    Sums of squares, moment matrices and optimization over polynomials , author=. Emerging applications of algebraic geometry , pages=. 2008 , publisher=

    2008

  5. [5]

    2013 , publisher=

    Convex analysis and minimization algorithms I: Fundamentals , author=. 2013 , publisher=

    2013

  6. [6]

    Journal of Optimization Theory and Applications , volume=

    Conic optimization via operator splitting and homogeneous self-dual embedding , author=. Journal of Optimization Theory and Applications , volume=. 2016 , publisher=

    2016

  7. [7]

    2021 , publisher=

    Garstka, Michael and Cannon, Mark and Goulart, Paul , journal=. 2021 , publisher=

    2021

  8. [8]

    2006 , publisher=

    Numerical optimization , author=. 2006 , publisher=

    2006

  9. [9]

    Acta Numerica , volume=

    Interior-point methods for optimization , author=. Acta Numerica , volume=. 2008 , publisher=

    2008

  10. [10]

    Clarabel:

    Goulart, Paul J and Chen, Yuwen , journal=. Clarabel:. 2026 , publisher=

    2026

  11. [11]

    Ye, Yinyu and Todd, Michael J and Mizuno, Shinji , journal=. An. 1994 , publisher=

    1994

  12. [12]

    1997 , publisher=

    Primal-dual interior-point methods , author=. 1997 , publisher=

    1997

  13. [13]

    Foundations and Trends

    Distributed optimization and statistical learning via the alternating direction method of multipliers , author=. Foundations and Trends. 2011 , publisher=

    2011

  14. [14]

    Acta Numerica , volume=

    Semidefinite optimization , author=. Acta Numerica , volume=. 2001 , publisher=

    2001

  15. [15]

    SIAM review , volume=

    Semidefinite programming , author=. SIAM review , volume=. 1996 , publisher=

    1996

  16. [16]

    SIAM Journal on Optimization , volume=

    An interior-point method for semidefinite programming , author=. SIAM Journal on Optimization , volume=. 1996 , publisher=

    1996

  17. [17]

    Primal-dual interior-point methods for semidefinite programming:

    Alizadeh, Farid and Haeberly, Jean-Pierre A and Overton, Michael L , journal=. Primal-dual interior-point methods for semidefinite programming:. 1998 , publisher=

    1998

  18. [18]

    2013 , publisher=

    Practical methods of optimization , author=. 2013 , publisher=

    2013

  19. [19]

    , author=

    On a homogeneous algorithm for the monotone complementarity problem. , author=. Mathematical Programming , volume=. 1999 , publisher=

    1999

  20. [20]

    IMA journal of Numerical Analysis , volume=

    Computing the nearest correlation matrix—a problem from finance , author=. IMA journal of Numerical Analysis , volume=. 2002 , publisher=

    2002

  21. [21]

    Linear algebra and its applications , volume=

    Computing a nearest symmetric positive semidefinite matrix , author=. Linear algebra and its applications , volume=. 1988 , publisher=

    1988

  22. [22]

    SIAM Journal on Mathematics of Data Science , volume=

    Scalable semidefinite programming , author=. SIAM Journal on Mathematics of Data Science , volume=. 2021 , publisher=

    2021

  23. [23]

    Improving efficiency and scalability of sum of squares optimization:

    Ahmadi, Amir Ali and Hall, Georgina and Papachristodoulou, Antonis and Saunderson, James and Zheng, Yang , booktitle=. Improving efficiency and scalability of sum of squares optimization:. 2017 , organization=

    2017

  24. [24]

    Annual Review of Control, Robotics, and Autonomous Systems , volume=

    Recent scalability improvements for semidefinite programming with applications in machine learning, control, and robotics , author=. Annual Review of Control, Robotics, and Autonomous Systems , volume=. 2020 , publisher=

    2020

  25. [25]

    Optimization methods and software , volume=

    Toh, Kim-Chuan and Todd, Michael J and T. Optimization methods and software , volume=. 1999 , publisher=

    1999

  26. [26]

    Solving semidefinite-quadratic-linear programs using

    T. Solving semidefinite-quadratic-linear programs using. Mathematical programming , volume=. 2003 , publisher=

    2003

  27. [27]

    2020 , publisher=

    Koopman operator in systems and control , author=. 2020 , publisher=

    2020

  28. [28]

    Formulas for data-driven control:

    De Persis, Claudio and Tesi, Pietro , journal=. Formulas for data-driven control:. 2019 , publisher=

    2019

  29. [29]

    Proceedings of the 2005, American Control Conference, 2005

    A tutorial on sum of squares techniques for systems analysis , author=. Proceedings of the 2005, American Control Conference, 2005. , pages=. 2005 , organization=

    2005

  30. [30]

    Annual Symposium on Foundations of Computer Science (FOCS) , pages=

    The power of sum-of-squares for detecting hidden structures , author=. Annual Symposium on Foundations of Computer Science (FOCS) , pages=. 2017 , organization=

    2017

  31. [31]

    IEEE Access , volume=

    Application of sum-of-squares method in estimation of region of attraction for nonlinear polynomial systems , author=. IEEE Access , volume=. 2020 , publisher=

    2020

  32. [32]

    IEEE Robotics and Automation Letters , volume=

    Data-driven MPC for quadrotors , author=. IEEE Robotics and Automation Letters , volume=. 2021 , publisher=

    2021

  33. [33]

    Data Science and Model Predictive Control:

    Morato, Marcelo M and Felix, Monica S , journal=. Data Science and Model Predictive Control:. 2024 , publisher=

    2024

  34. [34]

    IEEE Transactions on Automatic Control , volume=

    Data-driven model predictive control with stability and robustness guarantees , author=. IEEE Transactions on Automatic Control , volume=. 2020 , publisher=

    2020

  35. [35]

    Automatica , volume=

    A data-driven approach to robust control of multivariable systems by convex optimization , author=. Automatica , volume=. 2017 , publisher=

    2017

  36. [36]

    IEEE control systems letters , volume=

    Data-driven LQR control design , author=. IEEE control systems letters , volume=. 2018 , publisher=

    2018

  37. [37]

    , title =

    Sturm, Jos F. , title =. Optimization Methods and Software , volume =. 1999 , doi =

    1999

  38. [38]

    Moreau, Jean Jacques , journal=. D

  39. [39]

    Studies in mathematics and its applications , volume=

    Chapter ix applications of the method of multipliers to variational inequalities , author=. Studies in mathematics and its applications , volume=. 1983 , publisher=

    1983

  40. [40]

    Regularization methods for

    Nie, Jiawang and Wang, Li , journal=. Regularization methods for. 2012 , publisher=

    2012

  41. [41]

    An accelerated first-order method for solving

    Bertsimas, Dimitris and Freund, Robert M and Sun, Xu Andy , journal=. An accelerated first-order method for solving. 2013 , publisher=

    2013

  42. [42]

    Mathematical Programming , volume=

    Accelerated first-order methods for hyperbolic programming , author=. Mathematical Programming , volume=. 2019 , publisher=

    2019

  43. [43]

    SIAM Journal on Optimization , volume=

    An optimal-storage approach to semidefinite programming using approximate complementarity , author=. SIAM Journal on Optimization , volume=. 2021 , publisher=

    2021

  44. [44]

    Projection methods for conic feasibility problems:

    Henrion, Didier and Malick, J. Projection methods for conic feasibility problems:. Optimization Methods & Software , volume=. 2011 , publisher=

    2011

  45. [45]

    SIAM Journal on Matrix Analysis and Applications , volume=

    A dual approach to semidefinite least-squares problems , author=. SIAM Journal on Matrix Analysis and Applications , volume=. 2004 , publisher=

    2004

  46. [46]

    Grundlehren der mathematischen Wissenschaften , year=

    Convex Analysis and Minimization Algorithms II , author=. Grundlehren der mathematischen Wissenschaften , year=

  47. [47]

    arXiv preprint arXiv:0709.2556 , year=

    Henrion, Didier and Malick, J. arXiv preprint arXiv:0709.2556 , year=

    Pith/arXiv arXiv

  48. [48]

    SIAM Journal on Optimization , volume =

    Jarre, Florian and Rendl, Franz , title =. SIAM Journal on Optimization , volume =. 2008 , doi =

    2008

  49. [49]

    SIAM Journal on Optimization , volume=

    Regularization methods for semidefinite programming , author=. SIAM Journal on Optimization , volume=. 2009 , publisher=

    2009

  50. [50]

    Peet and Declan Jagt , year=

    Antonis Papachristodoulou and James Anderson and Giorgio Valmorbida and Stephen Prajna and Pete Seiler and Pablo Parrilo and Matthew M. Peet and Declan Jagt , year=. 1310.4716 , archivePrefix=

    Pith/arXiv arXiv

  51. [51]

    Learning for Dynamics and Control , pages=

    Learning dynamical systems with side information , author=. Learning for Dynamics and Control , pages=. 2020 , organization=

    2020

  52. [52]

    SIAM Review , volume =

    Ahmadi, Amir Ali and Khadir, Bachir El , title =. SIAM Review , volume =

  53. [53]

    2019 18th European Control Conference (ECC) , pages=

    Grammar-based representation and identification of dynamical systems , author=. 2019 18th European Control Conference (ECC) , pages=. 2019 , organization=

    2019

  54. [54]

    and Pillonetto, G

    Chiuso, A. and Pillonetto, G. , title =. Annual Review of Control, Robotics, and Autonomous Systems , volume =

  55. [55]

    Brunton and Joshua L

    Steven L. Brunton and Joshua L. Proctor and J. Nathan Kutz , title =. Proceedings of the National Academy of Sciences , volume =. 2016 , doi =

    2016

  56. [56]

    Nature Reviews Physics , volume=

    Physics-informed machine learning , author=. Nature Reviews Physics , volume=. 2021 , publisher=

    2021

  57. [57]

    Statistics Surveys , number =

    Cynthia Rudin and Chaofan Chen and Zhi Chen and Haiyang Huang and Lesia Semenova and Chudi Zhong , title =. Statistics Surveys , number =

  58. [58]

    Wild and Karen Willcox , institution =

    Nathan Baker and Frank Alexander and Timo Bremer and Aric Hagberg and Yannis Kevrekidis and Habib Najm and Manish Parashar and Abani Patra and James Sethian and Stefan M. Wild and Karen Willcox , institution =. Workshop Report on Basic Research Needs for Scientific Machine Learning: Core Technologies for Artificial Intelligence , doi =

  59. [59]

    2015 , issn =

    Bayesian system identification of dynamical systems using highly informative training data , journal =. 2015 , issn =. doi:https://doi.org/10.1016/j.ymssp.2014.10.003 , author =

    work page doi:10.1016/j.ymssp.2014.10.003 2015

  60. [60]

    Frontiers of Chemical Science and Engineering , volume=

    An integrated approach for machine-learning-based system identification of dynamical systems under control: application towards the model predictive control of a highly nonlinear reactor system , author=. Frontiers of Chemical Science and Engineering , volume=. 2022 , publisher=

    2022

  61. [61]

    2010 , issn =

    Perspectives on system identification , journal =. 2010 , issn =. doi:https://doi.org/10.1016/j.arcontrol.2009.12.001 , author =

    doi:10.1016/j.arcontrol.2009.12.001 2010

  62. [62]

    2019 , issn =

    Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior , journal =. 2019 , issn =. doi:https://doi.org/10.1016/j.ymssp.2018.08.033 , author =

    doi:10.1016/j.ymssp.2018.08.033 2019

  63. [63]

    2020 , publisher=

    Modeling, identification and simulation of dynamical systems , author=. 2020 , publisher=

    2020

  64. [64]

    International Conference on Application of Information and Communication Technologies (AICT) , pages=

    Model structures and fitting criteria for system identification with neural networks , author=. International Conference on Application of Information and Communication Technologies (AICT) , pages=. 2020 , organization=

    2020

  65. [65]

    2019 , publisher=

    Neural network modeling and identification of dynamical systems , author=. 2019 , publisher=

    2019

  66. [66]

    Computer-Aided Civil and Infrastructure Engineering , volume=

    Real-time system identification: an algorithm for simultaneous model class selection and parametric identification , author=. Computer-Aided Civil and Infrastructure Engineering , volume=. 2015 , publisher=

    2015

  67. [67]

    2016 , publisher=

    Dynamic mode decomposition: data-driven modeling of complex systems , author=. 2016 , publisher=

    2016

  68. [68]

    Physical review letters , volume=

    Discovering physical concepts with neural networks , author=. Physical review letters , volume=. 2020 , publisher=

    2020

  69. [69]

    Proceedings of the National Academy of Sciences , volume=

    Data-driven discovery of coordinates and governing equations , author=. Proceedings of the National Academy of Sciences , volume=. 2019 , publisher=

    2019

  70. [70]

    Nature communications , volume=

    Physics-informed learning of governing equations from scarce data , author=. Nature communications , volume=. 2021 , publisher=

    2021

  71. [71]

    Proceedings of the Royal Society A , volume=

    Sparse identification of nonlinear dynamics for model predictive control in the low-data limit , author=. Proceedings of the Royal Society A , volume=. 2018 , publisher=

    2018

  72. [72]

    ACM Computing Surveys , volume=

    Integrating scientific knowledge with machine learning for engineering and environmental systems , author=. ACM Computing Surveys , volume=. 2022 , publisher=

    2022

  73. [73]

    Annual Reviews in Control , volume=

    Behavioral systems theory in data-driven analysis, signal processing, and control , author=. Annual Reviews in Control , volume=. 2021 , publisher=

    2021

  74. [74]

    Journal of fluid mechanics , volume=

    Dynamic mode decomposition of numerical and experimental data , author=. Journal of fluid mechanics , volume=. 2010 , publisher=

    2010

  75. [75]

    2020 , publisher=

    Kaheman, Kadierdan and Kutz, J Nathan and Brunton, Steven L , journal=. 2020 , publisher=

    2020

  76. [76]

    Positive Polynomials on Compact Semi-algebraic Sets , urldate =

    Mihai Putinar , journal =. Positive Polynomials on Compact Semi-algebraic Sets , urldate =

  77. [77]

    and El Ghaoui, Laurent , year=

    Calafiore, Giuseppe C. and El Ghaoui, Laurent , year=. Linear equations and least squares , DOI=. Optimization Models , publisher=

  78. [78]

    MOSEK Optimization Toolbox for MATLAB Release 10.1.11

    MOSEK ApS. MOSEK Optimization Toolbox for MATLAB Release 10.1.11

  79. [79]

    1989 , issn =

    Computational chaos - a prelude to computational instability , journal =. 1989 , issn =. doi:https://doi.org/10.1016/0167-2789(89)90072-9 , author =

    work page doi:10.1016/0167-2789(89)90072-9 1989

  80. [80]

    2004 , organization=

    Lofberg, Johan , booktitle=. 2004 , organization=

    2004

Showing first 80 references.