# Duality of convex relaxations for constrained variational problems

## TL;DR

This paper establishes the weak duality and strong duality between two convex relaxation methods for constrained variational problems, showing their equivalence in computational applications, especially with polynomial data.

## Contribution

It proves the duality relationship and equivalence of two recent convex relaxation approaches for variational problems, including their finite-dimensional SDP approximations.

## Key findings

- Weak duality between the two convex relaxation methods.
- Strong duality when admissible functions are bounded.
- Equivalence of the relaxation approaches for polynomial data.

## 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.