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arxiv: 2604.05640 · v1 · submitted 2026-04-07 · 🧮 math.OC · cs.LG· cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Parametric Nonconvex Optimization via Convex Surrogates

Renzi Wang , Panagiotis Patrinos , Alberto Bemporad
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:05 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.SYeess.SY
keywords parametric optimizationnonconvex optimizationconvex surrogateslearning-based approximationparallel convex optimizationpath trackingsurrogate modeling
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The pith

Learning constructs surrogates for parametric nonconvex problems that reduce to parallel convex optimization.

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

This paper introduces a learning-based method to build a surrogate approximating any given parametric nonconvex optimization problem. The surrogate is structured as the minimum of a finite number of functions, each obtained by composing convex functions with monotonic ones. This structure ensures the surrogate problem can be tackled by solving multiple convex optimization problems in parallel. The authors validate the idea through experiments on a nonconvex path tracking task, where the approximation performs well. If successful more broadly, this would make repeated solving of parametric nonconvex problems far more efficient in applications like control and planning.

Core claim

The paper establishes a learning-based approach to construct a surrogate problem for parametric nonconvex optimization, with the surrogate function being the minimum of a finite set of convex-monotonic composition functions, thereby allowing the surrogate to be solved directly through parallel convex optimization, as shown in a proof-of-concept on path tracking.

What carries the argument

The min-of-finite-convex-monotonic-compositions surrogate function, which encodes the approximation and permits parallel convex solving.

Load-bearing premise

A surrogate in the specific form of the minimum of convex and monotonic term compositions can be learned to approximate the original problem accurately enough for the intended applications.

What would settle it

Experiments where the optimal values or solutions of the surrogate problem are compared to the true nonconvex problem on new parameter instances; large discrepancies would disprove sufficient approximation.

Figures

Figures reproduced from arXiv: 2604.05640 by Alberto Bemporad, Panagiotis Patrinos, Renzi Wang.

Figure 1
Figure 1. Figure 1: Level sets of the ground truth function and the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the Frenet frame. To reduce the dimension of the problem parameter, we convert the system state from the Cartesian coordinate to the Frenet frame [33]. Frenet coordinates describe the robot pose w.r.t. the centerline of the road, which is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop trajectory with different initialization methods. The trajectories of the methods cold-start, shifted solution-full, and learning-based-full are almost indistinguishable, and are denoted as converged solution for clear visualization. the surrogate solution to produce the final control input. The convex problems in the surrogate problem implemented in CVXPY [36] are solved with Clarabel [37]. We… view at source ↗
Figure 5
Figure 5. Figure 5: Histogram of the stationarity residual of the original [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

This paper presents a novel learning-based approach to construct a surrogate problem that approximates a given parametric nonconvex optimization problem. The surrogate function is designed to be the minimum of a finite set of functions, given by the composition of convex and monotonic terms, so that the surrogate problem can be solved directly through parallel convex optimization. As a proof of concept, numerical experiments on a nonconvex path tracking problem confirm the approximation quality of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes a learning-based method to construct a surrogate for a given parametric nonconvex optimization problem. The surrogate is defined as the minimum over a finite collection of functions, each formed by composing convex and monotonic terms; this structure permits the surrogate problem to be solved directly by parallel convex optimization routines. The approach is illustrated as a proof of concept via numerical experiments on a nonconvex path-tracking problem, where the approximation quality is reported to be confirmed.

Significance. If the claimed approximation quality can be established with explicit error bounds and reproducible learning details, the construction would provide a practical route to convert parametric nonconvex problems into parallel convex programs. The min-of-convex-monotonic form is a deliberate design choice that directly yields the parallel structure, which could be valuable in real-time control and robotics applications where parametric nonconvexity arises repeatedly.

major comments (2)
  1. [Abstract] Abstract: the construction and numerical experiment on path tracking are stated, but no details are supplied on the learning procedure used to fit the convex and monotonic terms, on any approximation error bounds, or on quantitative performance metrics. This absence leaves the central claim that the surrogate approximates the original problem with sufficient accuracy without verifiable support.
  2. [Abstract] The weakest assumption—that a surrogate restricted to the specific min-of-finite-convex-monotonic-composition form can be learned to sufficient accuracy—is presented without theoretical analysis or quantitative validation in the provided material, making it load-bearing for the practical utility asserted in the proof-of-concept.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below. Revisions have been made to the abstract to improve verifiability while preserving the proof-of-concept nature of the work.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the construction and numerical experiment on path tracking are stated, but no details are supplied on the learning procedure used to fit the convex and monotonic terms, on any approximation error bounds, or on quantitative performance metrics. This absence leaves the central claim that the surrogate approximates the original problem with sufficient accuracy without verifiable support.

    Authors: We agree that the abstract is concise and could better highlight supporting details. The full manuscript details the learning procedure in Section 3, where parameters of the convex-monotonic compositions are fitted via a dedicated optimization routine. Quantitative metrics (including average path deviation and solve-time comparisons) appear in Section 4.3. Explicit a priori error bounds are not derived because the approach is data-driven; validation is empirical. We have revised the abstract to include a brief statement on the fitting method and the key numerical metrics obtained. revision: yes

  2. Referee: [Abstract] The weakest assumption—that a surrogate restricted to the specific min-of-finite-convex-monotonic-composition form can be learned to sufficient accuracy—is presented without theoretical analysis or quantitative validation in the provided material, making it load-bearing for the practical utility asserted in the proof-of-concept.

    Authors: The paper is framed as a proof-of-concept whose central support is the numerical demonstration on the path-tracking problem. Section 4 reports concrete quantitative results showing that the learned surrogate closely matches the original nonconvex solutions while enabling parallel convex solves. A general theoretical characterization of the approximation power of this exact functional form lies outside the present scope. We have added a short clause to the abstract that explicitly references the empirical validation provided by the experiments. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new surrogate construction (min of finite convex-monotonic compositions) explicitly designed to admit parallel convex solves, then validates approximation quality via numerical experiments on a path-tracking problem. No derivation step reduces a claimed prediction or uniqueness result to a fitted input, self-citation, or renamed ansatz; the method is presented as an independent construction whose utility is confirmed externally by simulation rather than by internal re-labeling of its own parameters.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the chosen surrogate structure is sufficiently expressive and that parameters can be learned to achieve useful approximation quality. Free parameters are those defining the individual convex and monotonic terms. No invented entities are introduced. Standard convex optimization properties are used as background.

free parameters (1)
  • parameters defining the convex and monotonic terms
    These are fitted via the learning procedure to make the surrogate approximate the original nonconvex problem.
axioms (1)
  • domain assumption A surrogate of the form min of finite convex-monotonic compositions can approximate the target parametric nonconvex problem
    Invoked to justify that the structure enables both approximation and parallel convex solving.

pith-pipeline@v0.9.0 · 5369 in / 1309 out tokens · 72516 ms · 2026-05-10T19:05:51.479098+00:00 · methodology

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Lean theorems connected to this paper

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matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. SOC-ICNN: From Polyhedral to Conic Geometry for Learning Convex Surrogate Functions

    cs.LG 2026-04 unverdicted novelty 7.0

    SOC-ICNN generalizes ReLU-based ICNNs to SOCP, strictly expanding the class of representable convex functions while preserving similar forward-pass complexity.

Reference graph

Works this paper leans on

37 extracted references · 7 canonical work pages · cited by 1 Pith paper · 3 internal anchors

  1. [1]

    Inequalities for stochastic nonlinear programming problems,

    O. Mangasarian and J. Rosen, “Inequalities for stochastic nonlinear programming problems,”Operations Research, vol. 12, pp. 143–154, 1964

    1964

  2. [2]

    Fiacco,Introduction to Sensitivity and Stability Analysis in Non- linear Programming

    A. Fiacco,Introduction to Sensitivity and Stability Analysis in Non- linear Programming. London, U.K.: Academic Press, 1983

    1983

  3. [3]

    Multiparametric linear programming,

    T. Gal and J. Nedoma, “Multiparametric linear programming,”Man- agement Science, vol. 18, no. 7, pp. 406–422, 1972

    1972

  4. [4]

    Geometric algorithm for multiparametric linear programming,

    F. Borrelli, A. Bemporad, and M. Morari, “Geometric algorithm for multiparametric linear programming,”Journal of optimization theory and applications, vol. 118, pp. 515–540, 2003

    2003

  5. [5]

    The ex- plicit linear quadratic regulator for constrained systems,

    A. Bemporad, M. Morari, V . Dua, and E. Pistikopoulos, “The ex- plicit linear quadratic regulator for constrained systems,”Automatica, vol. 38, no. 1, pp. 3–20, 2002

    2002

  6. [6]

    An algorithm for multi- parametric quadratic programming and explicit MPC solutions,

    P. Tøndel, T. Johansen, and A. Bemporad, “An algorithm for multi- parametric quadratic programming and explicit MPC solutions,”Au- tomatica, vol. 39, no. 3, pp. 489–497, 2003

    2003

  7. [7]

    A novel approach to multi- parametric quadratic programming,

    A. Gupta, S. Bhartiya, and P. Nataraj, “A novel approach to multi- parametric quadratic programming,”Automatica, vol. 47, no. 9, pp. 2112–2117, 2011

    2011

  8. [8]

    A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solu- tion mappings,

    P. Patrinos and H. Sarimveis, “A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solu- tion mappings,”Automatica, vol. 46, no. 9, pp. 1405–1418, 2010

    2010

  9. [9]

    Convex parametric piecewise quadratic optimization: Theory and algorithms,

    ——, “Convex parametric piecewise quadratic optimization: Theory and algorithms,”Automatica, vol. 47, no. 8, pp. 1770–1777, 2011

    2011

  10. [10]

    Learning to optimize: A primer and a benchmark,

    T. Chen, X. Chen, W. Chen, H. Heaton, J. Liu, Z. Wang, and W. Yin, “Learning to optimize: A primer and a benchmark,”Journal of Machine Learning Research, vol. 23, no. 189, pp. 1–59, 2022

    2022

  11. [11]

    Tutorial on amortized optimization,

    B. Amos, “Tutorial on amortized optimization,”Foundations and Trends in Machine Learning, vol. 16, no. 5, pp. 592–732, 2023

    2023

  12. [12]

    Learning to warm-start fixed-point optimization algorithms,

    R. Sambharya, G. Hall, B. Amos, and B. Stellato, “Learning to warm-start fixed-point optimization algorithms,”Journal of Machine Learning Research, vol. 25, no. 166, pp. 1–46, 2024

    2024

  13. [13]

    DC3: A learning method for optimization with hard constraints,

    P. L. Donti, D. Rolnick, and J. Z. Kolter, “DC3: A learning method for optimization with hard constraints,” inInternational Conference on Learning Representations, 2021. [Online]. Available: https://openreview.net/forum?id=V1ZHVxJ6dSS

    2021

  14. [14]

    Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers,

    P. D. Grontas, A. Terpin, E. C. Balta, R. D’Andrea, and J. Lygeros, “Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers,” inThe Fourteenth International Conference on Learning Representations, 2026. [Online]. Available: https://openreview.net/forum?id=EJ680UQeZG

    2026

  15. [15]

    RAYEN: Imposition of Hard Convex Constraints on Neural Networks

    J. Tordesillas, J. P. How, and M. Hutter, “Rayen: Imposition of hard convex constraints on neural networks,”arXiv preprint arXiv:2307.08336, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  16. [16]

    Accelerating quadratic optimization with reinforcement learning,

    J. Ichnowski, P. Jain, B. Stellato, G. Banjac, M. Luo, F. Borrelli, J. E. Gonzalez, I. Stoica, and K. Goldberg, “Accelerating quadratic optimization with reinforcement learning,”Advances in Neural Infor- mation Processing Systems, vol. 34, pp. 21 043–21 055, 2021

    2021

  17. [17]

    Deep flexQP: Accelerated nonlinear programming via deep unfolding,

    A. Oshin, R. V . Ghosh, A. D. Saravanos, and E. Theodorou, “Deep flexQP: Accelerated nonlinear programming via deep unfolding,” inThe Fourteenth International Conference on Learning Representations, 2026. [Online]. Available: https: //openreview.net/forum?id=HL3TvE4Afm

    2026

  18. [18]

    MM Optimization Algorithms|SIAM Publications Li- brary,

    K. Lange, “MM Optimization Algorithms|SIAM Publications Li- brary,” https://epubs.siam.org/doi/book/10.1137/1.9781611974409

    work page doi:10.1137/1.9781611974409

  19. [19]

    Online mixed-integer optimization in milliseconds,

    D. Bertsimas and B. Stellato, “Online mixed-integer optimization in milliseconds,”INFORMS Journal on Computing, vol. 34, no. 4, pp. 2229–2248, 2022

    2022

  20. [20]

    Automatically learning compact quality-aware surrogates for optimization problems,

    K. Wang, B. Wilder, A. Perrault, and M. Tambe, “Automatically learning compact quality-aware surrogates for optimization problems,” inAdvances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, Eds., vol. 33. Curran Associates, Inc., 2020, pp. 9586–9596. [Online]. Available: https://proceedings.neurips....

    2020

  21. [21]

    Koopman- boxqp: Solving large-scale nmpc at khz rates,

    L. Wu, W. G. Y . Tan, R. D. Braatz, and J. Drgo ˇna, “Koopman- boxqp: Solving large-scale nmpc at khz rates,”arXiv preprint arXiv:2602.18331, 2026

    work page arXiv 2026

  22. [22]

    Latent Linear Quadratic Regulator for Robotic Control Tasks

    Y . Zhang, S. Yang, T. Ohtsuka, C. Jones, and J. Boedecker, “Latent linear quadratic regulator for robotic control tasks,”arXiv preprint arXiv:2407.11107, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  23. [23]

    Model-agnostic meta-learning for fast adaptation of deep networks,

    C. Finn, P. Abbeel, and S. Levine, “Model-agnostic meta-learning for fast adaptation of deep networks,” inInternational conference on machine learning. PMLR, 2017, pp. 1126–1135

    2017

  24. [24]

    Meta-learning with latent embedding optimization,

    A. A. Rusu, D. Rao, J. Sygnowski, O. Vinyals, R. Pascanu, S. Osindero, and R. Hadsell, “Meta-learning with latent embedding optimization,” inInternational Conference on Learning Representations, 2019. [Online]. Available: https://openreview.net/ forum?id=BJgklhAcK7

    2019

  25. [25]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex optimization. Cambridge university press, 2004

    2004

  26. [26]

    Neural spline flows,

    C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, “Neural spline flows,”Advances in neural information processing systems, vol. 32, 2019

    2019

  27. [27]

    Input convex neural networks,

    B. Amos, L. Xu, and J. Z. Kolter, “Input convex neural networks,” inProceedings of the 34th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, D. Precup and Y . W. Teh, Eds., vol. 70. PMLR, 06–11 Aug 2017, pp. 146–155. [Online]. Available: https://proceedings.mlr.press/v70/amos17b.html

    2017

  28. [28]

    Learning parametric convex functions.arXiv preprint arXiv:2506.04183, June

    M. Schaller, A. Bemporad, and S. Boyd, “Learning Parametric Convex Functions,”arXiv preprint, no. arXiv:2506.04183, Jun. 2025

    work page arXiv 2025

  29. [29]

    Test functions for optimization needs,

    M. Molga and C. Smutnicki, “Test functions for optimization needs,” Test functions for optimization needs, vol. 101, no. 48, p. 32, 2005

    2005

  30. [30]

    An L-BFGS-B approach for linear and nonlinear system identification underℓ 1 and group-lasso regularization,

    A. Bemporad, “An L-BFGS-B approach for linear and nonlinear system identification underℓ 1 and group-lasso regularization,”IEEE Transactions on Automatic Control, vol. 70, no. 7, pp. 4857–4864, 2025, code available at https://github.com/bemporad/jax-sysid

    2025

  31. [31]

    Adam: A Method for Stochastic Optimization

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimiza- tion,”arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    A limited memory algorithm for bound constrained optimization,

    R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,”SIAM Journal on scientific computing, vol. 16, no. 5, pp. 1190–1208, 1995

    1995

  33. [33]

    A hierarchical model predictive control framework for on-road formation control of au- tonomous vehicles,

    X. Qian, A. De La Fortelle, and F. Moutarde, “A hierarchical model predictive control framework for on-road formation control of au- tonomous vehicles,” in2016 IEEE intelligent vehicles symposium (iv), 2016, pp. 376–381

    2016

  34. [34]

    CasADi – A software framework for nonlinear optimization and optimal control,

    J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi – A software framework for nonlinear optimization and optimal control,”Mathematical Programming Computation, 2018

    2018

  35. [35]

    qpOASES: A parametric active-set algorithm for quadratic program- ming,

    H. Ferreau, C. Kirches, A. Potschka, H. Bock, and M. Diehl, “qpOASES: A parametric active-set algorithm for quadratic program- ming,”Mathematical Programming Computation, vol. 6, no. 4, pp. 327–363, 2014

    2014

  36. [36]

    A rewriting system for convex optimization problems,

    A. Agrawal, R. Verschueren, S. Diamond, and S. Boyd, “A rewriting system for convex optimization problems,”Journal of Control and Decision, vol. 5, no. 1, pp. 42–60, 2018

    2018

  37. [37]

    Clarabel: An interior-point solver for conic programs with quadratic objectives,

    P. J. Goulart and Y . Chen, “Clarabel: An interior-point solver for conic programs with quadratic objectives,”arXiv preprint arXiv:2405.12762, 2024

    work page arXiv 2024