Duality of convex relaxations for constrained variational problems

Giovanni Fantuzzi

arxiv: 1906.12136 · v1 · pith:ZDFZFAO7new · submitted 2019-06-28 · 🧮 math.OC · math.AP

Duality of convex relaxations for constrained variational problems

Giovanni Fantuzzi This is my paper

Pith reviewed 2026-05-25 13:56 UTC · model grok-4.3

classification 🧮 math.OC math.AP

keywords convex relaxationvariational problemsweak dualitystrong dualitysemidefinite programmingintegral functionalsprobability measuresoptimal control

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The pith

Two convex relaxation methods for bounding variational problems are weakly dual in general and strongly dual when admissible functions have bounded range and gradients.

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

The paper proves weak duality between a convex program over smooth functions subject to pointwise non-negativity constraints and a convex program over scaled probability measures, both used to bound the optimal value of a constrained variational problem whose objective is an integral functional. Weak duality ensures one program's optimal value always bounds the other's from the appropriate side. When the range and gradients of admissible functions are restricted to bounded sets, the duality becomes strong so the two infinite-dimensional programs attain the same optimal value. For polynomial data, each relaxation is approximated by a hierarchy of finite-dimensional semidefinite programs that remain strongly dual to each other under standard constraint qualifications, independent of the infinite-dimensional case.

Core claim

We prove weak duality between the convex program over sufficiently smooth functions with pointwise non-negativity constraints and the convex program over scaled probability measures. When the range and gradients of admissible functions are constrained to bounded sets, this duality is strong and the optimal values coincide. For variational problems with polynomial data, the optimal values of each convex relaxation can be approximated by solving weakly dual hierarchies of finite-dimensional semidefinite programs that are strongly dual under standard constraint qualification conditions irrespective of whether strong duality holds at the infinite-dimensional level.

What carries the argument

The duality relationship between the infinite-dimensional convex program over smooth functions with pointwise non-negativity constraints and the infinite-dimensional convex program over scaled probability measures.

If this is right

The two relaxation methods produce identical bounds on the optimal value of the variational problem whenever the boundedness condition holds.
Hierarchies of semidefinite programs approximating either relaxation converge to the same value.
Either formulation can be used interchangeably for numerical computation of the bound without changing the result.
Strong duality between the finite-dimensional SDP hierarchies holds regardless of whether it holds at the infinite-dimensional level.

Where Pith is reading between the lines

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

Practitioners facing a polynomial variational problem can select whichever formulation is easier to implement or numerically stable without sacrificing bound quality.
The result may extend to other pairs of relaxations in optimal control if analogous boundedness conditions can be identified.
For problems with unbounded functions, additional regularization or truncation might recover strong duality in practice.

Load-bearing premise

The range and gradients of admissible functions in the variational problem are constrained to bounded sets.

What would settle it

A constrained variational problem in which admissible functions may take unbounded values or gradients, yet the two infinite-dimensional convex programs attain different optimal values.

read the original abstract

We prove weak duality between two recent convex relaxation methods for bounding the optimal value of a constrained variational problem in which the objective is an integral functional. The first approach, proposed by Valmorbida et al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces the variational problem with a convex program over sufficiently smooth functions, subject to pointwise non-negativity constraints. The second approach, discussed by Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational problem into a convex program over scaled probability measures. We also prove that the duality between these infinite-dimensional convex programs is strong, meaning that their optimal values coincide, when the range and gradients of admissible functions in the variational problem are constrained to bounded sets. For variational problems with polynomial data, the optimal values of each convex relaxation can be approximated by solving weakly dual hierarchies of finite-dimensional semidefinite programs (SDPs). These are strongly dual under standard constraint qualification conditions irrespective of whether strong duality holds at the infinite-dimensional level. Thus, the two relaxation approaches are equivalent for the purposes of computations.

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.

Desk Editor's Note private letter to a colleague

The paper proves weak duality between the Valmorbida function relaxation and the Korda measure relaxation for variational problems, with strong duality under bounded range/gradient assumptions, plus SDP equivalence for polynomials.

read the letter

The main contribution here is a direct proof that the two relaxations are dual, so their optimal values match under the stated conditions. Weak duality follows from standard Lagrangian arguments on the infinite-dimensional programs. Strong duality holds when admissible functions have bounded range and gradients, which brings in compactness to get attainment. For polynomial data the moment-SOS hierarchies then give matching finite SDPs that are strongly dual under ordinary Slater conditions, independent of the infinite-dimensional boundedness requirement. That equivalence for computation is the practical payoff. The argument uses standard convex duality and measure theory without circularity or hidden fitting steps. The boundedness condition for strong duality is a real restriction, but the paper states it plainly and does not claim it always holds. No derivation gaps appear in the outline, and the SDP reduction looks clean. This is incremental work inside an established line of convex relaxations rather than a new framework, but the unification is cleanly executed. A reader already working on these specific relaxations will find the duality result useful for understanding when the two approaches can be treated as interchangeable. It deserves a serious referee because the math is grounded and the computational implication is concrete.

Referee Report

0 major / 2 minor

Summary. The manuscript proves weak duality between the Valmorbida-style function relaxation (convex program over smooth functions with pointwise non-negativity) and the Korda-style measure relaxation (convex program over scaled probability measures) for constrained variational problems whose objective is an integral functional. It further establishes strong duality between these two infinite-dimensional convex programs when the range and gradients of admissible functions are constrained to bounded sets. For polynomial data, the paper constructs moment-SOS hierarchies of finite SDPs that approximate each relaxation and are strongly dual under standard Slater-type constraint qualifications, independent of the infinite-dimensional boundedness condition. The central conclusion is that the two relaxation approaches are equivalent for computational purposes.

Significance. The result provides a rigorous theoretical link between two distinct convex relaxation frameworks in the literature on variational problems and optimal control. By showing that their optimal values coincide under explicit boundedness assumptions and that their SDP approximations are interchangeable, the work supports the interchangeable use of either method in practice. The proofs rely on standard Lagrangian duality, measure theory, and compactness arguments rather than ad-hoc constructions, which is a strength.

minor comments (2)

[Abstract] The abstract cites Korda et al. as arXiv:1804.07565v1; the reference list should confirm whether a more recent version or journal publication exists and update accordingly for completeness.
[Introduction] Notation for the admissible function set and the measure space could be introduced with a single consolidated definition early in the paper to aid readability when moving between the function and measure formulations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. We are pleased that the connection between the two relaxation approaches is viewed as a useful contribution.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation establishes weak duality between the Valmorbida-style function relaxation and Korda-style measure relaxation via standard Lagrangian arguments applied to the infinite-dimensional convex programs, and proves strong duality under the explicit bounded-range-and-gradient assumption using compactness to obtain attainment. The moment-SOS SDP hierarchies for polynomial data are shown to be strongly dual under Slater-type constraint qualifications that are independent of the infinite-dimensional boundedness condition. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on external convex duality and measure-theoretic results rather than any renaming or ansatz imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard results from convex analysis and functional analysis; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Weak duality holds between a pair of infinite-dimensional convex programs under standard constraint qualifications
Invoked to establish the weak duality result between the function-based and measure-based relaxations.
domain assumption Strong duality holds when admissible functions have bounded range and gradients together with appropriate constraint qualifications
Required for the claim that the two infinite-dimensional programs attain the same optimal value.

pith-pipeline@v0.9.0 · 5728 in / 1431 out tokens · 37141 ms · 2026-05-25T13:56:06.092022+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. The convex programs on the righthand sides of (2.6) and (3.10) are weakly dual and D≤P. The duality is strong... if the sets Y and Z are bounded.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For variational problems with polynomial data, the optimal values of each convex relaxation can be approximated by solving weakly dual hierarchies of finite-dimensional semidefinite programs (SDPs).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 3 internal anchors

[1]

A Framework for Input-Output Analysis of Wall-Bounded Shear Flows

M. Ahmadi, G. Valmorbida, D. Gayme, and A. Papachristodoulou, A framework for input-output analysis of wall-bounded shear ﬂows (2018), available at http://arxiv.org/abs/1802.04974

work page internal anchor Pith review Pith/arXiv arXiv 2018
[2]

Ahmadi, G

M. Ahmadi, G. Valmorbida, and A. Papachristodoulou, Input-output analysis of distributed parameter systems using convex optimization , Proc. 53 rd IEEE Conf. Decis. Control, 2014, pp. 4310–4315

work page 2014
[3]

54 th IEEE Conf

, A convex approach to hydrodynamic analysis , Proc. 54 th IEEE Conf. Decis. Control, 2015, pp. 7262–7267

work page 2015
[4]

, Dissipation inequalities for the analysis of a class of PDEs , Automatica 66 (2016), 163–171

work page 2016
[5]

Control Lett

, Safety veriﬁcation for distributed parameter systems using barrier functionals, Syst. Control Lett. 108 (2017), 33–39

work page 2017
[6]

Bertsimas and C

D. Bertsimas and C. Caramanis, Bounds on linear PDEs via semideﬁnite optimization , Math. Program. A 108 (2006), no. 1, 135–158

work page 2006
[7]

V. I. Bogachev, Measure Theory, Springer Berlin Heidelberg, 2007

work page 2007
[8]

Boyd and L

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004

work page 2004
[9]

Moments and convex optimization for analysis and control of nonlinear partial differential equations

M. Korda, D. Henrion, and J. B. Lasserre, Moments and convex optimization for analysis and control of nonlinear partial diﬀerential equations (2018), available at http://arxiv.org/abs/1804.07565

work page internal anchor Pith review Pith/arXiv arXiv 2018
[10]

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

work page 2015
[11]

Laurent, Sums of squares, moment matrices and optimization over polynomials (M

M. Laurent, Sums of squares, moment matrices and optimization over polynomials (M. Putinar and S. Sullivant, eds.), Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol. 149, Springer New York, 2009

work page 2009
[12]

Magron and C

V. Magron and C. Prieur, Optimal control of linear PDEs using occupation measures and SDP relaxations, IMA J. Math. Control Info. (2018), 1–16

work page 2018
[13]

S. Marx, T. Weisser, D. Henrion, and J. B. Lasserre,A moment approach for entropy solutions to nonlinear hyperbolic PDEs (2018), available at http://arxiv.org/abs/1807.02306

work page internal anchor Pith review Pith/arXiv arXiv 2018
[14]

Putinar, Positive polynomials on compact semi-algebraic sets , Indiana Univ

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

work page 1993
[15]

Sion, On general minimax theorems , Paciﬁc J

M. Sion, On general minimax theorems , Paciﬁc J. Math. 8 (1958), no. 1, 171–176

work page 1958
[16]

Valmorbida, M

G. Valmorbida, M. Ahmadi, and A. Papachristodoulou, Semi-deﬁnite programming and functional in- equalities for distributed parameter systems , Proc. 53 rd IEEE Conf. Decis. Control, 2014, pp. 4304–4309

work page 2014
[17]

54 th IEEE Conf

, Convex solutions to integral inequalities in two-dimensional domains , Proc. 54 th IEEE Conf. Decis. Control, 2015, pp. 7268–7273

work page 2015
[18]

, Stability analysis for a class of partial diﬀerential equations via semideﬁnite programming , IEEE Trans. Automat. Control 61 (2016), no. 6, 1649–1654

work page 2016
[19]

Valmorbida and A

G. Valmorbida and A. Papachristodoulou, Introducing INTSOSTOOLS : A SOSTOOLS plug-in for in- tegral inequalities, Proc. 2015 Eur. Control Conf., 2015, pp. 1231–1236

work page 2015

[1] [1]

A Framework for Input-Output Analysis of Wall-Bounded Shear Flows

M. Ahmadi, G. Valmorbida, D. Gayme, and A. Papachristodoulou, A framework for input-output analysis of wall-bounded shear ﬂows (2018), available at http://arxiv.org/abs/1802.04974

work page internal anchor Pith review Pith/arXiv arXiv 2018

[2] [2]

Ahmadi, G

M. Ahmadi, G. Valmorbida, and A. Papachristodoulou, Input-output analysis of distributed parameter systems using convex optimization , Proc. 53 rd IEEE Conf. Decis. Control, 2014, pp. 4310–4315

work page 2014

[3] [3]

54 th IEEE Conf

, A convex approach to hydrodynamic analysis , Proc. 54 th IEEE Conf. Decis. Control, 2015, pp. 7262–7267

work page 2015

[4] [4]

, Dissipation inequalities for the analysis of a class of PDEs , Automatica 66 (2016), 163–171

work page 2016

[5] [5]

Control Lett

, Safety veriﬁcation for distributed parameter systems using barrier functionals, Syst. Control Lett. 108 (2017), 33–39

work page 2017

[6] [6]

Bertsimas and C

D. Bertsimas and C. Caramanis, Bounds on linear PDEs via semideﬁnite optimization , Math. Program. A 108 (2006), no. 1, 135–158

work page 2006

[7] [7]

V. I. Bogachev, Measure Theory, Springer Berlin Heidelberg, 2007

work page 2007

[8] [8]

Boyd and L

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004

work page 2004

[9] [9]

Moments and convex optimization for analysis and control of nonlinear partial differential equations

M. Korda, D. Henrion, and J. B. Lasserre, Moments and convex optimization for analysis and control of nonlinear partial diﬀerential equations (2018), available at http://arxiv.org/abs/1804.07565

work page internal anchor Pith review Pith/arXiv arXiv 2018

[10] [10]

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

work page 2015

[11] [11]

Laurent, Sums of squares, moment matrices and optimization over polynomials (M

M. Laurent, Sums of squares, moment matrices and optimization over polynomials (M. Putinar and S. Sullivant, eds.), Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol. 149, Springer New York, 2009

work page 2009

[12] [12]

Magron and C

V. Magron and C. Prieur, Optimal control of linear PDEs using occupation measures and SDP relaxations, IMA J. Math. Control Info. (2018), 1–16

work page 2018

[13] [13]

S. Marx, T. Weisser, D. Henrion, and J. B. Lasserre,A moment approach for entropy solutions to nonlinear hyperbolic PDEs (2018), available at http://arxiv.org/abs/1807.02306

work page internal anchor Pith review Pith/arXiv arXiv 2018

[14] [14]

Putinar, Positive polynomials on compact semi-algebraic sets , Indiana Univ

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

work page 1993

[15] [15]

Sion, On general minimax theorems , Paciﬁc J

M. Sion, On general minimax theorems , Paciﬁc J. Math. 8 (1958), no. 1, 171–176

work page 1958

[16] [16]

Valmorbida, M

G. Valmorbida, M. Ahmadi, and A. Papachristodoulou, Semi-deﬁnite programming and functional in- equalities for distributed parameter systems , Proc. 53 rd IEEE Conf. Decis. Control, 2014, pp. 4304–4309

work page 2014

[17] [17]

54 th IEEE Conf

, Convex solutions to integral inequalities in two-dimensional domains , Proc. 54 th IEEE Conf. Decis. Control, 2015, pp. 7268–7273

work page 2015

[18] [18]

, Stability analysis for a class of partial diﬀerential equations via semideﬁnite programming , IEEE Trans. Automat. Control 61 (2016), no. 6, 1649–1654

work page 2016

[19] [19]

Valmorbida and A

G. Valmorbida and A. Papachristodoulou, Introducing INTSOSTOOLS : A SOSTOOLS plug-in for in- tegral inequalities, Proc. 2015 Eur. Control Conf., 2015, pp. 1231–1236

work page 2015