The gap between a variational problem and its occupation measure relaxation

¹ CNRS, LAAS, 7 avenue du colonel Roche, 31400 Toulouse, France
² Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16626 Prague, Czech Republic
³ Department of Mathematics and Statistics, UiT The Arctic University of Norway, Norway

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 6 September 2023
Accepted: 1 March 2026

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 that, when the dimension of the codomain equals one, a relaxed occupation measure can be decomposed as a superposition of functions in Sobolev space W^1,∞(Ω). 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.