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arxiv: 2604.11124 · v1 · submitted 2026-04-13 · 🧮 math.OC

Polyconvexity with Moments and Sums of Squares

Giovanni Fantuzzi , Didier Henrion (LAAS-POP) , Martin Kru{\v{z}}\'ik (UTIA / CAS) , Ajay Murali , Stephan Weis This is my paper

Pith reviewed 2026-05-10 15:50 UTC · model grok-4.3

classification 🧮 math.OC
keywords polyconvexitysum of squaresmoment relaxationspolynomial optimizationnonlinear elasticitycalculus of variationsconvex envelopes
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The pith

Polynomial matrix functions admit sum-of-squares certificates for polyconvexity and moment-based computations of their envelopes.

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

A function is polyconvex if it depends convexly on the minors of its matrix argument. This property guarantees existence of energy minimizers in nonlinear elasticity. For polynomial functions, verifying polyconvexity or finding the largest polyconvex lower bound has been hard. The paper develops sum-of-squares conditions that give sufficient convex-optimization tests for polyconvexity and a moment hierarchy that approximates the polyconvex envelope at any given matrix point. These tools matter because they turn theoretical regularity requirements into computable checks that can be used when designing or analyzing material models.

Core claim

For a polynomial function of a matrix, polyconvexity can be certified by checking whether a certain polynomial expression in the minors is a sum of squares, which is decided by semidefinite programming. The polyconvex envelope at a point is found as the solution of a polynomial optimization problem over the set of possible minor values, which is relaxed using the moment-sum-of-squares hierarchy to yield a sequence of lower bounds that converge to the envelope value.

What carries the argument

Sum-of-squares decompositions for certifying nonnegativity of polynomials in the matrix minors, combined with the moment relaxations of the polynomial optimization problem that defines the polyconvex envelope.

If this is right

  • Low-degree polynomial energies in elasticity can now be checked numerically for polyconvexity.
  • The polyconvex envelope can be evaluated pointwise to obtain a relaxed energy that still satisfies the existence theorem.
  • These relaxations provide a systematic way to handle non-polyconvex but physically motivated energies.
  • The hierarchy gives a sequence of improving approximations whose quality can be assessed by the duality gap or rank conditions.

Where Pith is reading between the lines

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

  • Numerical experiments on standard test functions would show how often the SOS conditions succeed in certifying polyconvexity.
  • The computed envelopes could be used to create globally polyconvex approximations by interpolation for practical use.
  • The framework allows optimization over the set of certified polyconvex polynomials to find closest fits to a target energy.

Load-bearing premise

The degree of the polynomial matrix function is low enough for the sum-of-squares and moment relaxations to be computationally tractable while also being tight enough to be useful.

What would settle it

Apply the moment hierarchy to compute the polyconvex envelope of a polynomial function whose exact polyconvex envelope is known analytically, and check whether the numerical result matches the exact value within solver tolerance.

Figures

Figures reproduced from arXiv: 2604.11124 by Ajay Murali, Didier Henrion (LAAS-POP), Giovanni Fantuzzi, Martin Kru{\v{z}}\'ik (UTIA / CAS), Stephan Weis.

Figure 1
Figure 1. Figure 1: Rank of the moment matrix for the example in Section [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Histogram of computational times (in seconds, on a standard laptop) for computing [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

A function of a matrix is polyconvex when it can be expressed as a convex function of the matrix minors. Polyconvexity is a regularity condition ensuring existence of minimizers in nonlinear elasticity and, more broadly, in vectorial problems of the calculus of variations, when minimizing integral gradient functionals. The polyconvex envelope of a function is the largest polyconvex lower bound. Yet deciding whether a given energy is polyconvex, or computing the polyconvex envelope, are generally difficult problems. This paper focuses on polynomial matrix functions. We propose (i) tractable convex-optimization based sufficient conditions to certify polyconvexity via sum-of-squares (SOS) technology, and (ii) a principled numerical method to compute the polyconvex envelope pointwise, based on the moment-SOS hierarchy from polynomial optimization.

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 / 2 minor

Summary. The paper proposes tractable sufficient conditions, based on sum-of-squares (SOS) certificates, to certify that a polynomial matrix function is polyconvex (i.e., convex in its minors). It also develops a moment-SOS hierarchy from polynomial optimization to compute the polyconvex envelope of such a function at a given point.

Significance. If the SOS certificates are valid and the hierarchy converges as claimed, the work supplies practical convex-optimization tools for verifying polyconvexity and approximating envelopes. These are directly relevant to existence questions in nonlinear elasticity and vectorial calculus of variations, where polyconvexity is a key regularity condition. The approach reuses established SOS/moment technology rather than inventing new relaxations, which is a methodological strength.

major comments (2)
  1. [§3] §3 (SOS certificate for polyconvexity): the formulation appears to treat the minors as free variables when writing the SOS program. Because the minors satisfy nontrivial algebraic relations (Plücker relations, rank constraints, etc.), an SOS certificate in the free variables may certify a stronger property than polyconvexity or, conversely, may miss valid polyconvex functions. The manuscript must either work on the variety or prove that the free relaxation remains valid for the polyconvexity claim; neither is obvious from the abstract and both become fragile for matrix sizes or degrees beyond the smallest cases.
  2. [§4] §4 (moment-SOS hierarchy): the claim that the hierarchy computes the polyconvex envelope pointwise requires a convergence theorem. It is unclear whether the proof accounts for the algebraic dependencies among the minors or whether the relaxation is performed in the ambient space of all minors. Without this, the numerical method may converge to the convex envelope of the lifted function rather than the true polyconvex envelope.
minor comments (2)
  1. Notation for the minor map and the lifted function should be introduced once and used consistently; several passages switch between M and the vector of all minors without explicit redefinition.
  2. The numerical examples would benefit from a table reporting both the SOS certificate degree and the moment relaxation order used, together with the resulting bound and CPU time.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our manuscript. The concerns raised regarding the treatment of algebraic relations among minors in both the SOS certificates and the moment-SOS hierarchy are important, and we will revise the paper to address them explicitly. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [§3] §3 (SOS certificate for polyconvexity): the formulation appears to treat the minors as free variables when writing the SOS program. Because the minors satisfy nontrivial algebraic relations (Plücker relations, rank constraints, etc.), an SOS certificate in the free variables may certify a stronger property than polyconvexity or, conversely, may miss valid polyconvex functions. The manuscript must either work on the variety or prove that the free relaxation remains valid for the polyconvexity claim; neither is obvious from the abstract and both become fragile for matrix sizes or degrees beyond the smallest cases.

    Authors: We thank the referee for this observation. As stated in the abstract, our SOS-based conditions are sufficient (not necessary) for certifying polyconvexity. By constructing an SOS-convex polynomial g in the free minor variables such that g composed with the minor map recovers the original polynomial function, we obtain a valid sufficient certificate for polyconvexity, since the definition requires convexity of g over the entire ambient space. This approach does not claim to characterize all polyconvex functions, only to provide a tractable way to certify some of them. To address the potential for missing cases, we will revise the manuscript to include a discussion clarifying that the free-variable SOS provides a sufficient condition and, where possible, note how the linear constraints for matching the function account for the relations implicitly through the composition. We will also consider adding an optional formulation using the quotient algebra for cases where a tighter certificate is desired. This revision will make the scope and limitations clearer, particularly for larger matrix sizes. revision: partial

  2. Referee: [§4] §4 (moment-SOS hierarchy): the claim that the hierarchy computes the polyconvex envelope pointwise requires a convergence theorem. It is unclear whether the proof accounts for the algebraic dependencies among the minors or whether the relaxation is performed in the ambient space of all minors. Without this, the numerical method may converge to the convex envelope of the lifted function rather than the true polyconvex envelope.

    Authors: We agree that the convergence of the moment-SOS hierarchy to the polyconvex envelope needs to be rigorously established, taking into account the semialgebraic structure of the set of realizable minor vectors. In the current manuscript, the hierarchy is set up in the ambient minor space with moment constraints derived from the original matrix variables, which implicitly respects the dependencies. However, we will strengthen §4 by providing a detailed convergence theorem in the revised version. This theorem will adapt standard results on the convergence of moment-SOS hierarchies for polynomial optimization over semialgebraic sets (accounting for the Plücker relations and rank conditions via additional polynomial constraints if needed) to show that the hierarchy converges to the value of the polyconvex envelope. We believe this addresses the concern and ensures the method computes the desired quantity rather than a looser convex relaxation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard external SOS/moment relaxations applied to polyconvexity

full rationale

The paper's central claims rest on applying established sum-of-squares certificates and the Lasserre-type moment hierarchy (standard tools from the polynomial optimization literature) to certify polyconvexity of polynomial matrix functions and to compute their polyconvex envelopes pointwise. These relaxations are not defined in terms of the target polyconvex envelope or minor map within the paper; they are imported as externally validated convex-optimization primitives. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the derivation to a tautology appear in the described approach. The algebraic dependencies among minors are a practical implementation detail rather than a source of circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on the standard definition of polyconvexity, the known equivalence between SOS and semidefinite programming, and the convergence properties of the moment-SOS hierarchy; no new free parameters or invented entities are introduced.

axioms (3)
  • standard math A function is polyconvex if it is a convex function of the minors of its argument matrix.
    Invoked in the first sentence of the abstract as the definition used throughout.
  • standard math Sum-of-squares polynomials can be certified via semidefinite programming.
    Basis for the proposed sufficient conditions.
  • domain assumption The moment-SOS hierarchy converges to the true infimum for the polynomial optimization problems that arise when evaluating the polyconvex envelope.
    Required for the numerical method to be exact in the limit.

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    A. A. Ahmadi and P. A. Parrilo. A complete characterization of the gap between convexity and sos-convexity. SIAM J. Optim. , 23(2):811–833, 2013

    work page 2013

  2. [2]

    Alibert and B

    J.-J. Alibert and B. Dacorogna. An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal., 117(2):155–166, 1992

    work page 1992

  3. [3]

    M. F. Anjos and J. B. Lasserre, editors. Handbook on Semidefinite, Conic and Polynomial Optimization, volume 166 of International Series in Operations Research & Management Science. Springer, N2012

    work page

  4. [4]

    J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77. 22

    work page 1976

  5. [5]

    S. Bartels. Reliable and efficient approximation of polyconvex envelopes. SIAM J. Numer. Anal., 43(1):363–385, 2005

    work page 2005

  6. [6]

    Bartels and M

    S. Bartels and M. Kruˇ z´ ık. An efficient approach to the numerical solution of rate-independent problems with nonconvex energies. Multiscale Model. Simul., 9(3):1276–1300, 2011

    work page 2011

  7. [7]

    Bartels, C

    S. Bartels, C. Carstensen, K. Hackl, and U. Hoppe. Effective relaxation for microstructure simulations: algorithms and applications. Comput. Methods Appl. Mech. Engrg. , 193(48-51): 5143–5175, 2004

    work page 2004

  8. [8]

    Bartels, C

    S. Bartels, C. Carstensen, S. Conti, K. Hackl, U. Hoppe, and A. Orlando. Relaxation and the computation of effective energies and microstructures in solid mechanics. In Analysis, modeling and simulation of multiscale problems , pages 197–224. Springer, 2006

    work page 2006

  9. [9]

    Ben-Tal and A

    A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications . MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001

    work page 2001

  10. [10]

    Billingsley

    P. Billingsley. Convergence of Probability Measures . Wiley Series in Probability and Statistics. Wiley, 1999

    work page 1999

  11. [11]

    Blekherman, P

    G. Blekherman, P. A. Parrilo, and R. R. Thomas, editors. Semidefinite Optimization and Convex Algebraic Geometry, volume 13 of MOS-SIAM Series on Optimization . Society for Industrial and Applied Mathematics, Philadelphia, 2013

    work page 2013

  12. [12]

    Bosse, L

    T. Bosse, L. Eneya, and A. Griewank. An algorithm for pointwise evaluation of polyconvex envelopes II: generalization and numerical results. Afr. Mat., 26(1-2):31–52, 2015

    work page 2015

  13. [13]

    Br´ ezis, L

    H. Br´ ezis, L. Nirenberg, and G. Stampacchia. A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital. (9) , 1(2):257–264, 2008

    work page 2008

  14. [14]

    M. D. Choi, T. Y. Lam, and B. Reznick. Sums of squares of real polynomials. Proceedings of Symposia in Pure Mathematics , 58.2:103–126, 1995

    work page 1995

  15. [15]

    P. G. Ciarlet. Mathematical elasticity. Volume I. Three-dimensional elasticity , volume 84 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM),

    work page

  16. [16]

    Reprint of the 1988 edition

    work page 1988

  17. [17]

    Dacorogna

    B. Dacorogna. Direct methods in the calculus of variations , volume 78 of Applied Mathe- matical Sciences. Springer, second edition, 2008

    work page 2008

  18. [18]

    Dacorogna and P

    B. Dacorogna and P. Marcellini. A counterexample in the vectorial calculus of variations. In Material instabilities in continuum mechanics (Edinburgh, 1985–1986) , Oxford Sci. Publ., pages 77–83. Oxford Univ. Press, 1988

    work page 1985

  19. [19]

    Eneya, T

    L. Eneya, T. Bosse, and A. Griewank. A method for pointwise evaluation of polyconvex envelopes. Afr. Mat., 24(1):1–24, 2013

    work page 2013

  20. [20]

    N. B. Firoozye. Optimal use of the translation method and relaxations of variational problems. Comm. Pure Appl. Math. , 44(6):643–678, 1991

    work page 1991

  21. [21]

    Gamertsfelder and B

    L. Gamertsfelder and B. Mourrain. The effective countable generalized moment problem. arXiv 2501.09385, 2025

    work page arXiv 2025

  22. [22]

    Gatermann and P

    K. Gatermann and P. A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra , 192(1-3):95–128, 2004. 23

    work page 2004

  23. [23]

    H. Hartwig. A polyconvexity condition in dimension two. Proc. Roy. Soc. Edinburgh Sect. A, 125(5):901–910, 1995

    work page 1995

  24. [24]

    J. W. Helton and J. Nie. Semidefinite representation of convex sets. Mathematical Pro- gramming, 122(1):21–64, 2010

    work page 2010

  25. [25]

    Henrion and M

    D. Henrion and M. Kruˇ z´ ık. Polyconvex double well functions.Calc. Var. Partial Differential Equations, 65:88, 2026

    work page 2026

  26. [26]

    D. Hilbert. Ueber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann., 32(3):342–350, 1888

    work page

  27. [27]

    Kristensen

    J. Kristensen. On the non-locality of quasiconvexity. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 16(1):1–13, 1999

    work page 1999

  28. [28]

    Kruˇ z´ ık and T

    M. Kruˇ z´ ık and T. Roub´ ıˇ cek.Mathematical Methods in Continuum Mechanics of Solids . Springer, 2019

    work page 2019

  29. [29]

    J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J. Optim., 11(3):796–817, 2001

    work page 2001

  30. [30]

    J. B. Lasserre. Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim., 17(3):822–843, 2006

    work page 2006

  31. [31]

    J. B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim., 19(4):1995–2014, 2008

    work page 1995

  32. [32]

    J. B. Lasserre. Moments, positive polynomials and their applications , volume 1 of Imperial College Press Optimization Series . Imperial College Press, 2010

    work page 2010

  33. [33]

    J. B. Lasserre. An introduction to polynomial and semi-algebraic optimization . Cambridge Texts in Applied Mathematics. Cambridge University Press, 2015

    work page 2015

  34. [34]

    M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry , volume 149 of IMA Vol. Math. Appl. , pages 157–270. Springer, 2009

    work page 2009

  35. [35]

    C. B. Morrey, Jr. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25–53, 1952

    work page 1952

  36. [36]

    The MOSEK Optimization Toolbox for MATLAB manual

    MOSEK ApS. The MOSEK Optimization Toolbox for MATLAB manual. Version 11.1 , 2025

    work page 2025

  37. [37]

    Nesterov

    Y. Nesterov. Squared functional systems and optimization problems. In High performance optimization, volume 33 of Appl. Optim., pages 405–440. Kluwer Acad. Publ., 2000

    work page 2000

  38. [38]

    Neumeier, M

    T. Neumeier, M. A. Peter, D. Peterseim, and D. Wiedemann. Computational polyconvexifi- cation of isotropic functions. Multiscale Model. Simul., 22(4):1402–1420, 2024

    work page 2024

  39. [39]

    J. Nie. Moment and polynomial optimization , volume 31 of MOS-SIAM Series on Op- timization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, 2023

    work page 2023

  40. [40]

    Nirenberg

    L. Nirenberg. Topics in nonlinear functional analysis , volume 6 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New York; American Mathe- matical Society, 2001. 24

    work page 2001

  41. [41]

    P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Math. Program., 96(2, Ser. B):293–320, 2003

    work page 2003

  42. [42]

    Pedregal

    P. Pedregal. Variational methods in nonlinear elasticity . Society for Industrial and Applied Mathematics (SIAM), 2000

    work page 2000

  43. [43]

    M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. , 42(3):969–984, 1993

    work page 1993

  44. [44]

    Riener, T

    C. Riener, T. Theobald, L. J. Andr´ en, and J. B. Lasserre. Exploiting symmetries in SDP-relaxations for polynomial optimization. Math. Oper. Res., 38(1):122–141, 2013

    work page 2013

  45. [45]

    F. Rindler. Calculus of variations . Universitext. Springer, 2018

    work page 2018

  46. [46]

    Roub´ ıˇ cek.Relaxation in Optimization Theory and Variational Calculus

    T. Roub´ ıˇ cek.Relaxation in Optimization Theory and Variational Calculus . De Gruyter, 2 edition, 2020

    work page 2020

  47. [47]

    Schlosser, M

    C. Schlosser, M. Tacchi-B´ enard, and A. Lazarev. Convergence rates for the moment-SoS hierarchy. Numerical Algebra, Control and Optimization , 16:105–156, 2026

    work page 2026

  48. [48]

    H. v. Stackelberg. The theory of the market economy . Oxford University Press, 1952

    work page 1952

  49. [49]

    Theobald

    T. Theobald. Real Algebraic Geometry and Optimization , volume 241 of Graduate Studies in Mathematics. American Mathematical Society, 2024

    work page 2024

  50. [50]

    F. Topsøe. Topology and Measure, volume 133 of Lecture Notes in Mathematics . Springer, 1970

    work page 1970

  51. [51]

    V. S. Varadarajan. Weak convergence of measures on separable metric spaces. Sankhy¯ a: The Indian Journal of Statistics , 19(1/2):15–22, 1958

    work page 1958

  52. [52]

    J. Wang, V. Magron, and J.-B. Lasserre. Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension. SIAM J. Optim. , 31(1):114–141, 2021

    work page 2021

  53. [53]

    J. Wang, V. Magron, and J.-B. Lasserre. TSSOS: a moment-SOS hierarchy that exploits term sparsity. SIAM J. Optim. , 31(1):30–58, 2021

    work page 2021

  54. [54]

    J. Wang, V. Magron, J. B. Lasserre, and N. H. A. Mai. CS-TSSOS: correlative and term sparsity for large-scale polynomial optimization. ACM Trans. Math. Software, 48(4):Art. 42, 26, 2022

    work page 2022

  55. [55]

    Wolkowicz, R

    H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Pro- gramming: Theory, Algorithms, and Applications , volume 27 of International Series in Operations Research & Management Science. Kluwer Academic Publishers, 2000

    work page 2000

  56. [56]

    Zheng, G

    Y. Zheng, G. Fantuzzi, and A. Papachristodoulou. Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization. Annu. Rev. Control , 52:243–279, 2021. 25

    work page 2021