Semidefinite representation of convex hulls of rational varieties

TL;DR

This paper demonstrates that the convex hulls of certain rationally parameterized algebraic varieties can be represented using semidefinite programming, expanding the class of problems solvable with convex optimization techniques.

Contribution

It proves semidefinite representability of convex hulls for specific classes of rational varieties using positive semidefinite moment matrices and sum-of-squares decompositions.

Findings

Convex hulls of curves are semidefinite representable.

Hypersurfaces parameterized by quadratics are semidefinite representable.

Hypersurfaces parameterized by bivariate quartics are semidefinite representable.

Abstract

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.