Semidefinite representation of convex hulls of rational varieties

Didier Henrion (1, 2)

1 LAAS - Laboratoire d'analyse et d'architecture des systèmes
2 FEL CTU - Faculty of Electrical Engineering [Prague]

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.

Mots clés: semidefinite programming; algebraic varieties; linear matrix inequality (LMI)

Domaines: Optimisation et contrôle [math.OC]; Géométrie algébrique [math.AG]

Didier Henrion. Semidefinite representation of convex hulls of rational varieties. Acta Applicandae Mathematicae, 2011, 115 (3), p 319-327. ⟨hal-00352788v3⟩