Occupation measure relaxations in variational problems: the role of convexity

TL;DR

This paper investigates the conditions under which occupation measure relaxations in calculus of variations and optimal control are equivalent to the original problems, emphasizing the role of convexity and extending results to non-convex settings.

Contribution

It provides simple, general sufficient conditions based on convexity for the equivalence of occupation measure relaxations to original problems, applicable in high dimensions and with various constraints.

Findings

Conditions for relaxation equivalence are based on convexity.

Occupation measure relaxation is at least as strong as the convex envelope.

Results extend to non-convex problems like magnetism and elasticity.

Abstract

This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming relaxation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in non-convex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity.