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arxiv: 2205.14132 · v4 · submitted 2022-05-27 · 🧮 math.OC · math.AP

The gap between a variational problem and its occupation measure relaxation

Milan Korda , Rodolfo Rios-Zertuche This is my paper

Pith reviewed 2026-05-24 11:52 UTC · model grok-4.3

classification 🧮 math.OC math.AP
keywords occupation measurescalculus of variationsoptimal controlrelaxation gapnormal currentsHardt-Pitts decomposition
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The pith

The classical and relaxed minima coincide for variational problems when the codomain has dimension one.

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

This paper establishes that the lowest value of a variational or optimal control problem equals the lowest value of its occupation measure relaxation whenever the unknown function takes values in a one-dimensional space. The equality holds for both calculus of variations problems and optimal control problems. The result is proved by establishing a generalization of the Hardt-Pitts decomposition of normal currents that applies to the currents generated by these problems. A counterexample shows that a positive gap can appear when both the domain and codomain have dimension greater than one, and that integral constraints can produce a gap at any dimension.

Core claim

We prove that the classical and relaxed minima coincide when the dimension of the codomain of the unknown function equals one, both for calculus of variations and for optimal control problems. This is shown by proving a generalization of the Hardt-Pitts decomposition of normal currents applicable in our setting. We also show by means of a counterexample that if both the dimensions of the domain and of the codomain are greater than one, there may be a positive gap. Finally, we show that in the presence of integral constraints, a positive gap may occur at any dimension of the domain and of the codomain.

What carries the argument

Generalization of the Hardt-Pitts decomposition of normal currents, which equates the infimum over classical functions with the infimum over occupation measures when the codomain dimension is one.

If this is right

  • The occupation measure linear program recovers the exact classical minimum for every problem whose unknown function is scalar-valued.
  • Semidefinite-programming approximations of the relaxed problem are tight and solve the original problem without loss when the codomain dimension is one.
  • A gap between classical and relaxed values can appear for vector-valued unknown functions whose domain and codomain both have dimension greater than one.
  • Additional integral constraints can create a gap even when the domain or codomain has dimension one.

Where Pith is reading between the lines

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

  • The counterexample indicates that relaxed occupation measures can sometimes describe averaged or distributed solutions that no single classical function achieves.
  • If the decomposition result extends beyond the stated hypotheses, the equality may hold for a wider class of differential constraints.
  • Numerical methods that optimize over occupation measures remain useful even in cases where a gap exists, because they may locate the conceptually better relaxed solution.

Load-bearing premise

The generalized Hardt-Pitts decomposition of normal currents holds for the currents that arise from the variational problems under consideration.

What would settle it

A concrete variational problem with one-dimensional codomain in which the classical minimum is strictly larger than the relaxed minimum would falsify the equality claim.

Figures

Figures reproduced from arXiv: 2205.14132 by Milan Korda, Rodolfo Rios-Zertuche.

Figure 1
Figure 1. Figure 1: The left-hand side diagram illustrates the values of [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Translating a rectangle in the proof of Lemma 2.9. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The sets ρ −1 (−∞, r] and their exterior unit normal ηr, as in the situation of Lemma 2.15. De Giorgi’s Structure Theorem ( [6, Th. 5.15 and 5.16] or [16, Th. 15.9]) then implies that νr is supported on the boundary ∂ρ−1 (−∞, r], that this boundary is of Hausdorff dimension n, and that the unit normal ηr to the boundary of ρ −1 (−∞, r] is well defined for almost every point (x, y) on the boundary with resp… view at source ↗
Figure 4
Figure 4. Figure 4: When there is a vertical segment {x}×[a, b] in the boundary ∂ρ−1 (−∞, r], the normal vector is horizontal, that is, of the form (z, 0), z ∈ R n. The proof of Lemma 2.17 shows that the n-dimensional volume of the union of these segments is zero. Equality (30) follows from Z I Z {y≥ϕr(x)} div X dx dy dν(r) = Z I Z ∂ρ−1(−∞,r] hX, ηridHn dν(r) = Z I Z Ω hX(x, ϕr(x)),(ζr(x), −1)idx dν(r), which is Lemma 2.19(ii… view at source ↗
Figure 5
Figure 5. Figure 5: Illustrating a step in the proof of Lemma 2.19, we see that the difference of integrals [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: This very rough scheme captures only the topological aspect of the situation to [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: For x ∈ Ω, this is the plane {x} × Y . We have shaded the region ∆, and indicated the vectors u0(x) and u1(x) = −u0(x), together with their length, kxk 3 , and the distance from ∆ to the origin, kxk 3/10. We have also indicated what the values of ψ, U, and V are on each of the connected components of ∆ ∩ ({x} × Y ). We have also included a reminder that g (defined just after Lemma 3.3) is positive only out… view at source ↗
Figure 8
Figure 8. Figure 8: Radial scheme of the graph of f, made up of those of u0 and u1, and of ∆ (shaded, with dashed boundary). Also, by Lemma 3.3(ii), S(x, y) = O(kxk 3 ) as x → 0. Thus in order to get a function g that complies with inequality (51), it suffices to take g equal to S in a small neighborhood of ∆ while ensuring that it remains ≥ S everywhere. The function g will force the minimizers to be supported within ∆. Rema… view at source ↗
Figure 9
Figure 9. Figure 9: This 3-dimensional projection of the graph of [PITH_FULL_IMAGE:figures/full_fig_p042_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: In polar coordinates (θ, r), the corona Γ can be parameterized by the rectangle [α, α + 2π] × [ 1 2 , 1], and its image under ¯uα and ¯uα+2π is a double-covering. In the picture, we illustrate the definition of the disjoint sets Bα and Bα+2π for a given function h; these sets are the subsets of Γ in which h is E-close to ¯uα(Γ) and ¯uα+2π(Γ), respectively. Changing α translates the picture in the θ direct… view at source ↗
read the original abstract

Recent works have proposed linear programming relaxations of variational optimization problems subject to nonlinear PDE constraints based on the occupation measure formalism. The main appeal of these methods is the fact that they rely on convex optimization, typically semidefinite programming. In this work we close an open question related to this approach. We prove that the classical and relaxed minima coincide when the dimension of the codomain of the unknown function equals one, both for calculus of variations and for optimal control problems, thereby complementing analogous results that existed for the case when the dimension of the domain equals one. In order to do so, we prove a generalization of the Hardt-Pitts decomposition of normal currents applicable in our setting. We also show by means of a counterexample that, if both the dimensions of the domain and of the codomain are greater than one, there may be a positive gap. The example we construct to show the latter serves also to show that sometimes relaxed occupation measures may represent a more conceptually-satisfactory "solution" than their classical counterparts, so that -- even though they may not be equivalent -- algorithms rendering accessible the minimum in the larger space of relaxed occupation measures remain extremely valuable. Finally, we show that in the presence of integral constraints, a positive gap may occur at any dimension of the domain and of the codomain.

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

1 major / 2 minor

Summary. The manuscript proves that the classical minimum of a variational problem or optimal control problem coincides with its occupation-measure relaxation when the codomain dimension of the unknown function is one. The proof proceeds by reducing the problem to a statement about normal currents and invoking a generalization of the Hardt-Pitts decomposition theorem, which the authors establish under the relevant hypotheses. The paper also constructs an explicit counter-example showing a positive gap between classical and relaxed values when both domain and codomain dimensions exceed one, and demonstrates that integral constraints can produce a gap at arbitrary dimensions. The counter-example is further used to argue that the relaxed formulation can sometimes be conceptually preferable even when the values differ.

Significance. If the central equivalence holds, the result closes an open question on the tightness of occupation-measure LP relaxations for nonlinear PDE-constrained problems, complementing the known case of domain dimension one. The explicit counter-example supplies a concrete, falsifiable illustration of when the relaxation is not tight and simultaneously shows that the larger relaxed space can yield more satisfactory solutions, thereby justifying continued algorithmic development of the relaxed formulation. The work consists of direct mathematical arguments and a reproducible counter-example construction with no free parameters or fitted quantities.

major comments (1)
  1. [Section containing the generalized Hardt-Pitts statement and its application to occupation measures] The equivalence result for codomain dimension one rests entirely on the claimed generalization of the Hardt-Pitts decomposition being applicable to the normal currents obtained from the occupation-measure lift of the original variational problem. The manuscript asserts that the required rectifiability, boundary, and mass-bound conditions hold under the problem hypotheses, but does not supply an explicit verification that the lifted measures satisfy these conditions (in particular, that the support and boundary operator remain compatible with the generalized statement). This verification is load-bearing for Theorem 1 (or the main equivalence statement) and for the subsequent reduction in both the calculus-of-variations and optimal-control settings.
minor comments (2)
  1. [Abstract] The abstract states that the counter-example 'serves also to show that sometimes relaxed occupation measures may represent a more conceptually-satisfactory solution'; a brief parenthetical remark indicating the precise sense in which the relaxed solution is preferable (e.g., lower cost or smoother support) would improve readability.
  2. [Preliminaries / notation section] Notation for the occupation measures and the associated currents is introduced without an explicit comparison table; adding a short table that lists the classical objects, their measure lifts, and the corresponding current notation would aid readers who are not already familiar with the occupation-measure formalism.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point where the exposition of the proof can be strengthened. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section containing the generalized Hardt-Pitts statement and its application to occupation measures] The equivalence result for codomain dimension one rests entirely on the claimed generalization of the Hardt-Pitts decomposition being applicable to the normal currents obtained from the occupation-measure lift of the original variational problem. The manuscript asserts that the required rectifiability, boundary, and mass-bound conditions hold under the problem hypotheses, but does not supply an explicit verification that the lifted measures satisfy these conditions (in particular, that the support and boundary operator remain compatible with the generalized statement). This verification is load-bearing for Theorem 1 (or the main equivalence statement) and for the subsequent reduction in both the calculus-of-variations and optimal-control settings.

    Authors: We agree that an explicit verification of the rectifiability, boundary, and mass-bound conditions for the normal currents arising from the occupation-measure lift would improve the clarity and self-contained nature of the argument. In the revised version we will insert a short dedicated paragraph (or appendix subsection) immediately following the statement of the generalized Hardt-Pitts result. This paragraph will verify, under the standing hypotheses on the integrand and the admissible set, that (i) the lifted current is rectifiable, (ii) its boundary operator is compatible with the decomposition hypotheses, and (iii) the mass is locally bounded. The verification relies only on the definition of the occupation-measure lift and the compactness properties already used elsewhere in the paper; no new technical machinery is required. We believe this addition will fully address the concern while leaving the logical structure of Theorem 1 unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity; direct proof via independently derived auxiliary theorem

full rationale

The central claim (coincidence of classical and relaxed minima for codomain dimension 1) is obtained by proving a generalization of the Hardt-Pitts decomposition for normal currents under the problem hypotheses and then applying the result to the occupation-measure relaxation. This is a standard self-contained mathematical argument with no equations or claims that reduce by construction to fitted parameters, self-definitions, or load-bearing prior self-citations. The paper supplies an explicit counterexample for the higher-dimensional case and discusses integral constraints separately, confirming the derivation chain does not collapse into its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on standard results from geometric measure theory and optimal control; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard properties of normal currents and occupation measures in the calculus of variations and optimal control
    Invoked throughout the proof of the generalized Hardt-Pitts decomposition and the equivalence statements.

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Forward citations

Cited by 1 Pith paper

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

  1. Semidefinite relaxations for nonlinear elasticity with energies convex in the Cauchy-Green strain tensor

    math.OC 2026-04 unverdicted novelty 7.0

    For frame-indifferent energies convex in the Cauchy-Green tensor, the quasiconvex envelope equals the occupation-measure relaxation, making the first Lasserre SDP exact under linear boundary conditions and SOS-convexity.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    On some geometric properties of currents and frobenius theorem

    Giovanni Alberti and Annalisa Massaccesi. On some geometric properties of currents and frobenius theorem. Rendiconti Lincei-Matematica e Applicazioni, 28(4):861–869, 2017

    work page 2017

  2. [2]

    Transport equation and cauchy problem for non-smooth vector fields

    Luigi Ambrosio. Transport equation and cauchy problem for non-smooth vector fields. In Calculus of variations and nonlinear partial differential equations , pages 1–41. Springer, 2008

    work page 2008

  3. [3]

    Young measures, superposition and transport

    Patrick Bernard. Young measures, superposition and transport. Indiana Univ. Math. J. , 57(1):247–275, 2008

    work page 2008

  4. [4]

    Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of- squares optimization

    Alexander Chernyavsky, Jason J Bramburger, Giovanni Fantuzzi, and David Goluskin. Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of- squares optimization. arXiv preprint arXiv:2110.03079 , 2021

    work page arXiv 2021

  5. [5]

    Regularity of area minimizing currents i: gradi- ent l p estimates

    Camillo De Lellis and Emanuele Spadaro. Regularity of area minimizing currents i: gradi- ent l p estimates. Geometric and Functional Analysis , 24(6):1831–1884, 2014

    work page 2014

  6. [6]

    Evans and Ronald F

    Lawrence C. Evans and Ronald F. Gariepy. Measure theory and fine properties of functions. CRC Press, 2015

    work page 2015

  7. [7]

    Weak kam theorem in lagrangian dynamics, preliminary version number 10

    Albert Fathi. Weak kam theorem in lagrangian dynamics, preliminary version number 10. Available online, 2008

    work page 2008

  8. [8]

    Nonlinear analysis, volume 9 of ser

    L Gasinski and N Papageorgiou. Nonlinear analysis, volume 9 of ser. Math. Anal. Appl. Chapman & Hall/CRC, Boca Raton, FL , 2006

    work page 2006

  9. [9]

    Solving the plateau’s problem for hypersurfaces without the compact- ness theorem for integral currents.Geometric measure theory and the calculus of variations, 44:255–295, 1996

    R Hardt and J Pitts. Solving the plateau’s problem for hypersurfaces without the compact- ness theorem for integral currents.Geometric measure theory and the calculus of variations, 44:255–295, 1996

    work page 1996

  10. [10]

    Convex computation of the region of attraction of polynomial control systems

    Didier Henrion and Milan Korda. Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control , 59(2):297–312, 2014

    work page 2014

  11. [11]

    Moment-sos Hierarchy, The: Lectures In Probability, Statistics, Computational Geometry, Control And Nonlinear Pdes , volume 4

    Didier Henrion, Milan Korda, and Jean Bernard Lasserre. Moment-sos Hierarchy, The: Lectures In Probability, Statistics, Computational Geometry, Control And Nonlinear Pdes , volume 4. World Scientific, 2020

    work page 2020

  12. [12]

    Moments and convex optimiza- tion for analysis and control of nonlinear pdes

    Milan Korda, Didier Henrion, and Jean-Bernard Lasserre. Moments and convex optimiza- tion for analysis and control of nonlinear pdes. In E. Zuazua E. Trelat, editor, Handbook of Numerical Analysis, volume 23, chapter 10, pages 339–366. Elsevier, 2022

    work page 2022

  13. [13]

    Nonlinear optimal control via occupation measures and LMI-relaxations

    Jean B Lasserre, Didier Henrion, Christophe Prieur, and Emmanuel Tr´ elat. Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM Journal on Control and Optimization, 47(4):1643–1666, 2008

    work page 2008

  14. [14]

    Moments, positive polynomials and their applications , volume 1

    Jean Bernard Lasserre. Moments, positive polynomials and their applications , volume 1. World Scientific, 2009. 49

    work page 2009

  15. [15]

    Relaxation of optimal control problems to equivalent convex programs

    RM Lewis and RB Vinter. Relaxation of optimal control problems to equivalent convex programs. Journal of Mathematical Analysis and Applications , 74(2):475–493, 1980

    work page 1980

  16. [16]

    Sets of finite perimeter and geometric variational problems: an intro- duction to Geometric Measure Theory

    Francesco Maggi. Sets of finite perimeter and geometric variational problems: an intro- duction to Geometric Measure Theory . Number 135 in Cambridge studies in advanced mathematics. Cambridge University Press, 2012

    work page 2012

  17. [17]

    A moment approach for entropy solutions to nonlinear hyperbolic pdes

    Swann Marx, Tillmann Weisser, Didier Henrion, and Jean Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic pdes. Mathematical Control and Related Fields, 10(1):113–140, 2020

    work page 2020

  18. [18]

    Geometric measure theory: a beginner’s guide

    Frank Morgan. Geometric measure theory: a beginner’s guide . Academic press, 2016

    work page 2016

  19. [19]

    Generalized curves and extremal points

    JE Rubio. Generalized curves and extremal points. SIAM Journal on Control, 13(1):28–47, 1975

    work page 1975

  20. [20]

    Extremal points and optimal control theory

    JE Rubio. Extremal points and optimal control theory. Annali di Matematica Pura ed Applicata, 109(1):165–176, 1976

    work page 1976

  21. [21]

    Real and Complex Analysis

    Walter Rudin. Real and Complex Analysis . McGraw-Hill, 1968

    work page 1968

  22. [22]

    Unrectifiable normal currents in Euclidean spaces

    Andrea Schioppa. Unrectifiable normal currents in euclidean spaces. arXiv:1608.01635

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Decomposizione di correnti normali

    Emanuele Tasso. Decomposizione di correnti normali. Master’s thesis, University of Pisa, 5 2015

    work page 2015

  24. [24]

    The equivalence of strong and weak formulations for certain problems in optimal control

    Richard B Vinter and Richard M Lewis. The equivalence of strong and weak formulations for certain problems in optimal control. SIAM Journal on Control and Optimization , 16(4):546–570, 1978

    work page 1978

  25. [25]

    Lectures on the calculus of variations and optimal control theory

    Laurence Chisholm Young. Lectures on the calculus of variations and optimal control theory. Saunders, Philadelphia, 1969

    work page 1969

  26. [26]

    Decomposition of normal currents

    Maciej Zworski. Decomposition of normal currents. Proceedings of the American Mathe- matical Society, 102(4):831–839, 1988. 50

    work page 1988