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arxiv: 2603.26503 · v2 · submitted 2026-03-27 · 🧮 math.OC · cs.NA· math.NA

The adjoint state method for parametric definable optimization without smoothness or uniqueness

J\'er\^ome Bolte , Edouard Pauwels , Cheik Traor\'e This is my paper

Pith reviewed 2026-05-14 22:24 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords adjoint state methodparametric optimizationdefinable optimizationconservative fieldnonsmooth optimizationvalue functionqualification condition
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The pith

Nonconvex definable parametric optimization problems admit an adjoint state formula under a qualification condition without smoothness or unique solutions.

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

The paper shows that parametric optimization problems definable in o-minimal structures can have their value function's conservative field computed via an adjoint state, even if the objectives are nonsmooth and solutions are not unique. This holds under a qualification condition and applies to inequality and conic constraints. The construction avoids direct differentiation of the solution mapping and can be used with primal-dual solvers. It highlights that without definability, even smooth problems may fail to have a proper adjoint.

Core claim

We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping.

What carries the argument

Adjoint state formula selecting a conservative field for the value function in definable parametric optimization.

Load-bearing premise

The optimization problems must be definable in an o-minimal structure and satisfy a qualification condition.

What would settle it

A counterexample of a smooth parametric optimization problem that is not definable where the adjoint state does not provide a conservative field.

Figures

Figures reproduced from arXiv: 2603.26503 by Cheik Traor\'e, Edouard Pauwels, J\'er\^ome Bolte.

Figure 1
Figure 1. Figure 1: −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 1.5 x y [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
read the original abstract

We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool which can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.

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

0 major / 3 minor

Summary. The manuscript establishes that nonconvex parametric optimization problems definable in an o-minimal structure admit an adjoint-state formula that selects a conservative field for the value function. The result applies to problems with nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions, provided only a qualification condition holds. The construction avoids explicit differentiation of the solution mapping and is shown to be compatible with existing primal-dual solvers; counterexamples illustrate that definability and conservativity are essential.

Significance. If the central theorem holds, the work supplies a first-order sensitivity tool for a wide family of non-smooth, non-unique parametric problems that previously lacked rigorous adjoint formulas. The explicit separation of definability from smoothness, together with the conservative-field guarantee, fills a documented gap and directly supports numerical sensitivity analysis without additional regularity assumptions.

minor comments (3)
  1. [Abstract] The abstract states that examples demonstrate failure of the formal adjoint without definability or conservativity; please add a brief parenthetical remark indicating the dimension or constraint type of the counterexamples so readers can locate them quickly.
  2. [Section 2] Notation for the conservative field (e.g., the symbol used for the selection of the limiting subdifferential) should be introduced once in the preliminaries and used consistently thereafter; occasional re-definition in later sections disrupts readability.
  3. [Theorem 3.1] The statement of the main theorem would benefit from an explicit remark that the qualification condition is independent of uniqueness of the primal-dual pair; this is already implicit but worth highlighting for readers familiar with classical KKT theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an existence result for an adjoint-state formula in nonconvex definable parametric optimization problems under a qualification condition. The derivation chain relies on o-minimal definability to guarantee conservative fields for the value function, without any reduction of the claimed formula to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The abstract and stated claims isolate definability as an essential external assumption (with counter-examples for non-definable smooth cases) and position the result as compatible with existing solvers. No step in the provided derivation chain equates the output to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definability of the problem data and a qualification condition; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption The parametric optimization problem is definable in an o-minimal structure
    Invoked to guarantee the existence of the adjoint formula and conservative field selection.
  • domain assumption A qualification condition holds for the constraint system
    Required for the adjoint state formula to be valid.

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes.Ingénieurs de l’Automobile, 836: 33–36, 2015

    Grégoire Allaire. A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes.Ingénieurs de l’Automobile, 836: 33–36, 2015

    work page 2015

  2. [2]

    Optnet: Differentiable optimization as a layer in neural networks

    Brandon Amos and J Zico Kolter. Optnet: Differentiable optimization as a layer in neural networks. InInternational conference on machine learning, pages 136–145. PMLR, 2017

    work page 2017

  3. [3]

    Auslender.Differentiable stability in non convex and non differentiable program- ming, pages 29–41

    A. Auslender.Differentiable stability in non convex and non differentiable program- ming, pages 29–41. Springer Berlin Heidelberg, Berlin, Heidelberg, 1979. ISBN 978-3-642-00798-9. doi: 10.1007/BFb0120841. URL https://doi.org/10.1007/ BFb0120841

    work page doi:10.1007/bfb0120841 1979

  4. [4]

    Deep equilibrium models.Advances in neural information processing systems, 32, 2019

    Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models.Advances in neural information processing systems, 32, 2019

    work page 2019

  5. [5]

    Robust optimization–methodology and applications.Mathematical programming, 92(3):453–480, 2002

    Aharon Ben-Tal and Arkadi Nemirovski. Robust optimization–methodology and applications.Mathematical programming, 92(3):453–480, 2002

    work page 2002

  6. [6]

    A mathematical model for automatic differenti- ation in machine learning.Advances in Neural Information Processing Systems, 33: 10809–10819, 2020

    Jérôme Bolte and Edouard Pauwels. A mathematical model for automatic differenti- ation in machine learning.Advances in Neural Information Processing Systems, 33: 10809–10819, 2020

    work page 2020

  7. [7]

    Conservative set valued fields, automatic differ- entiation, stochastic gradient methods and deep learning.Mathematical Programming, 188(1):19–51, 2021

    Jérôme Bolte and Edouard Pauwels. Conservative set valued fields, automatic differ- entiation, stochastic gradient methods and deep learning.Mathematical Programming, 188(1):19–51, 2021

    work page 2021

  8. [8]

    Qualification conditions in semialgebraic programming.SIAM Journal on Optimization, 28(2):1867–1891, 2018

    Jérôme Bolte, Antoine Hochart, and Edouard Pauwels. Qualification conditions in semialgebraic programming.SIAM Journal on Optimization, 28(2):1867–1891, 2018

    work page 2018

  9. [9]

    Subgradient sampling for nonsmooth nonconvex minimization.SIAM Journal on Optimization, 33(4):2542–2569, 2023

    Jérôme Bolte, Tam Le, and Edouard Pauwels. Subgradient sampling for nonsmooth nonconvex minimization.SIAM Journal on Optimization, 33(4):2542–2569, 2023

    work page 2023

  10. [10]

    Optimization problems with perturba- tions: A guided tour.SIAM review, 40(2):228–264, 1998

    J Frédéric Bonnans and Alexander Shapiro. Optimization problems with perturba- tions: A guided tour.SIAM review, 40(2):228–264, 1998

    work page 1998

  11. [11]

    Springer Science & Business Media, 2000

    J Frédéric Bonnans and Alexander Shapiro.Perturbation analysis of optimization problems. Springer Science & Business Media, 2000

    work page 2000

  12. [12]

    An optimal engineering design method with failure rate constraints and sensitivity analysis

    C Castillo, R Mínguez, E Castillo, and MA Losada. An optimal engineering design method with failure rate constraints and sensitivity analysis. application to composite breakwaters.Coastal Engineering, 53(1):1–25, 2006. 24

    work page 2006

  13. [13]

    Jean Céa. Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût.ESAIM: Modélisation mathématique et analyse numérique, 20(3):371–402, 1986

    work page 1986

  14. [14]

    Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018

    Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018

    work page 2018

  15. [15]

    Clarke.Optimization and Nonsmooth Analysis

    Frank H. Clarke.Optimization and Nonsmooth Analysis. Canadian Mathematical Society series of monographs and advanced texts. John Wiley & Sons, 1983. ISBN 9780471875048

    work page 1983

  16. [16]

    Directional derivative of a minimax function

    Rafael Correa and Alberto Seeger. Directional derivative of a minimax function. Nonlinear Analysis: Theory, Methods & Applications, 9(1):13–22, 1985

    work page 1985

  17. [17]

    Istituti editoriali e poligrafici internazionali Pisa, 2000

    Michel Coste.An introduction to o-minimal geometry. Istituti editoriali e poligrafici internazionali Pisa, 2000

    work page 2000

  18. [18]

    Stochastic subgradient method converges on tame functions.Foundations of computational mathematics, 20(1):119–154, 2020

    Damek Davis, Dmitriy Drusvyatskiy, Sham Kakade, and Jason D Lee. Stochastic subgradient method converges on tame functions.Foundations of computational mathematics, 20(1):119–154, 2020

    work page 2020

  19. [19]

    Shape sensitivity analysis via min max differentiability

    MC Delfour and J-P Zolésio. Shape sensitivity analysis via min max differentiability. SIAM Journal on Control and Optimization, 26(4):834–862, 1988

    work page 1988

  20. [20]

    Control, shape, and topological derivatives via minimax differen- tiability of lagrangians

    Michel C Delfour. Control, shape, and topological derivatives via minimax differen- tiability of lagrangians. InNumerical Methods for Optimal Control Problems, pages 137–164. Springer, 2019

    work page 2019

  21. [21]

    Semidifferential of minimax of lagrangians with respect to parame- ters, controls, or geometric variables.SIAM Journal on Control and Optimization, 63(2):936–965, 2025

    Michel C Delfour. Semidifferential of minimax of lagrangians with respect to parame- ters, controls, or geometric variables.SIAM Journal on Control and Optimization, 63(2):936–965, 2025

    work page 2025

  22. [22]

    Parametric semidifferentiability of minimax of lagrangians: averaged adjoint approach.J

    Michel C Delfour and Kevin Sturm. Parametric semidifferentiability of minimax of lagrangians: averaged adjoint approach.J. Convex Anal, 24(4):1117–1142, 2017

    work page 2017

  23. [23]

    On decomposition methods for a class of partially separable nonlinear programs.Mathematics of Operations Research, 33(1): 119–139, 2008

    Victor DeMiguel and Francisco J Nogales. On decomposition methods for a class of partially separable nonlinear programs.Mathematics of Operations Research, 33(1): 119–139, 2008

    work page 2008

  24. [24]

    Stochastic bregman subgradient methods for nonsmooth nonconvex optimization problems: K

    Kuangyu Ding and Kim-Chuan Toh. Stochastic bregman subgradient methods for nonsmooth nonconvex optimization problems: K. ding, k.-c. toh.Journal of Optimization Theory and Applications, 206(3):67, 2025

    work page 2025

  25. [25]

    Ekeland and R

    I. Ekeland and R. Temam.Analyse convexe et problèmes variationnels. Etudes mathématiques. Dunod, 1974. ISBN 9782040073688

    work page 1974

  26. [26]

    Extensions of the gauvin-tolle optimal value differential stability results to general mathematical programs (no

    Anthony V Fiacco and William P Hutzler. Extensions of the gauvin-tolle optimal value differential stability results to general mathematical programs (no. serialt393). Technical report, Institute for Management Science and Engineering, School of Engineering and Applied Science, The George Washington University, 1979. 25

    work page 1979

  27. [27]

    Differential stability in nonlinear programming

    Jacques Gauvin and Jon W Tolle. Differential stability in nonlinear programming. SIAM Journal on Control and Optimization, 15(2):294–311, 1977

    work page 1977

  28. [28]

    The methodof parametric decomposition in mathematical programming: The nonconvex case

    Jean Gauvin. The methodof parametric decomposition in mathematical programming: The nonconvex case. InIIASA Proceedings Series, page 131, 1978

    work page 1978

  29. [29]

    Snopt: An sqp algorithm for large-scale constrained optimization.SIAM review, 47(1):99–131, 2005

    Philip E Gill, Walter Murray, and Michael A Saunders. Snopt: An sqp algorithm for large-scale constrained optimization.SIAM review, 47(1):99–131, 2005

    work page 2005

  30. [30]

    SIAM, 2008

    Andreas Griewank and Andrea Walther.Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, 2008

    work page 2008

  31. [31]

    SpringerBriefs on PDEs and Data Science

    Frédéric Hecht, Gontran Lance, and Emmanuel Trélat.PDE-Constrained Optimiza- tion within FreeFEM. SpringerBriefs on PDEs and Data Science. Springer Singapore, Singapore, 2026. ISBN 978-981-95-7048-5. eBook ISBN: 978-981-95-7049-2

    work page 2026

  32. [32]

    Gradients généralisés de fonctions marginales.SIAM Journal on Control and Optimization, 16(2):301–316, 1978

    JB Hiriart-Urruty. Gradients généralisés de fonctions marginales.SIAM Journal on Control and Optimization, 16(2):301–316, 1978

    work page 1978

  33. [33]

    Springer Berlin Heidelberg, Berlin, Heidelberg, 1984

    Robert Janin.Directional derivative of the marginal function in nonlinear program- ming, pages 110–126. Springer Berlin Heidelberg, Berlin, Heidelberg, 1984. ISBN 978-3-642-00913-6. doi: 10.1007/BFb0121214. URL https://doi.org/10.1007/ BFb0121214

    work page doi:10.1007/bfb0121214 1984

  34. [34]

    Constraint qualifications and lagrange multipliers in nondifferentiable programming problems.Journal of Optimization Theory and Applications, 81(3): 533–548, 1994

    A Jourani. Constraint qualifications and lagrange multipliers in nondifferentiable programming problems.Journal of Optimization Theory and Applications, 81(3): 533–548, 1994

    work page 1994

  35. [35]

    Nonsmooth nonconvex stochastic heavy ball.Journal of Optimization Theory and Applications, 201(2):699–719, 2024

    Tam Le. Nonsmooth nonconvex stochastic heavy ball.Journal of Optimization Theory and Applications, 201(2):699–719, 2024

    work page 2024

  36. [36]

    doi: https: //doi.org/10.1007/978-1-4419-9982-5

    John M. Lee.Introduction to Smooth Manifolds, volume 218 ofGraduate Texts in Mathematics. Springer, New York, NY, 2 edition, 2012. ISBN 978-1-4419-9981-8. doi: 10.1007/978-1-4419-9982-5

    work page doi:10.1007/978-1-4419-9982-5 2012

  37. [37]

    A parametric optimal control problem applied to daily irrigation.Mathematical Modelling of Natural Phenomena, 20:2, 2025

    Ana P Lemos-Paião, Sofia O Lopes, and MDR de Pinho. A parametric optimal control problem applied to daily irrigation.Mathematical Modelling of Natural Phenomena, 20:2, 2025

    work page 2025

  38. [38]

    Lipschitzian stability of constraint systems and generalized equations.Nonlinear Analysis: Theory, Methods & Applications, 22(2):173–206, 1994

    Boris Mordukhovich. Lipschitzian stability of constraint systems and generalized equations.Nonlinear Analysis: Theory, Methods & Applications, 22(2):173–206, 1994

    work page 1994

  39. [39]

    Sensitivity analysis for parametric nonlinear programming: A tutorial.arXiv preprint arXiv:2504.15851, 2025

    François Pacaud. Sensitivity analysis for parametric nonlinear programming: A tutorial.arXiv preprint arXiv:2504.15851, 2025

    work page arXiv 2025

  40. [40]

    Conservative parametric optimality and the ridge method for tame min-max problems.Set-Valued and Variational Analysis, 31(3):19, 2023

    Edouard Pauwels. Conservative parametric optimality and the ridge method for tame min-max problems.Set-Valued and Variational Analysis, 31(3):19, 2023

    work page 2023

  41. [41]

    Hyperparameter optimization with approximate gradient

    Fabian Pedregosa. Hyperparameter optimization with approximate gradient. In International conference on machine learning, pages 737–746. PMLR, 2016

    work page 2016

  42. [42]

    Princeton university press, 1970

    R Tyrrell Rockafellar.Convex analysis, volume 28. Princeton university press, 1970. 26

    work page 1970

  43. [43]

    SIAM, 1974

    R Tyrrell Rockafellar.Conjugate duality and optimization. SIAM, 1974

    work page 1974

  44. [44]

    Springer, 1998

    R Tyrrell Rockafellar and Roger JB Wets.Variational analysis. Springer, 1998

    work page 1998

  45. [45]

    Sdpt3—a matlab software packageforsemidefiniteprogramming, version1.3.Optimization methods and software, 11(1-4):545–581, 1999

    Kim-Chuan Toh, Michael J Todd, and Reha H Tütüncü. Sdpt3—a matlab software packageforsemidefiniteprogramming, version1.3.Optimization methods and software, 11(1-4):545–581, 1999

    work page 1999

  46. [46]

    Geometric categories and o-minimal structures

    Lou Van den Dries and Chris Miller. Geometric categories and o-minimal structures. Duke Mathematical Journal, 84(2):497–540, 1996

    work page 1996

  47. [47]

    On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming.Mathematical programming, 106(1):25–57, 2006

    Andreas Wächter and Lorenz T Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming.Mathematical programming, 106(1):25–57, 2006

    work page 2006

  48. [48]

    Adam-family methods for nonsmooth optimization with convergence guarantees.Journal of Machine Learning Research, 25(48):1–53, 2024

    Nachuan Xiao, Xiaoyin Hu, Xin Liu, and Kim-Chuan Toh. Adam-family methods for nonsmooth optimization with convergence guarantees.Journal of Machine Learning Research, 25(48):1–53, 2024

    work page 2024

  49. [49]

    Implementation and evaluation of sdpa 6.0 (semidefinite programming algorithm 6.0).Optimization Methods and Software, 18(4):491–505, 2003

    Makoto Yamashita, Katsuki Fujisawa, and Masakazu Kojima. Implementation and evaluation of sdpa 6.0 (semidefinite programming algorithm 6.0).Optimization Methods and Software, 18(4):491–505, 2003. doi: 10.1080/1055678031000118482. URLhttps://doi.org/10.1080/1055678031000118482

    work page doi:10.1080/1055678031000118482 2003

  50. [50]

    Shapes and geometries: Metrics, analysis, differential calculus, and optimization, 2011

    Jean-Paul Zolésio and MC Delfour. Shapes and geometries: Metrics, analysis, differential calculus, and optimization, 2011. 27

    work page 2011