Convex sets with semidefinite representation

Abstract

We provide a sufficient condition on a class of compact basic semialgebraic sets $${{\bf K} \subset \mathbb{R}^n}$$ for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g j that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed $${\epsilon > 0}$$ , there is a convex set $${{\bf K}_\epsilon}$$ such that $${{\rm co}({\bf K}) \subseteq {\bf K}_{\epsilon} \subseteq {\rm co}({\bf K}) + \epsilon {\bf B}}$$ (where B is the unit ball of $${\mathbb{R}^n}$$ ), and $${{\bf K}_\epsilon}$$ has an explicit SDr in terms of the g j ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L f associated with K and any linear $${f \in \mathbb{R}[X]}$$ is a sum of squares. We also provide an approximate SDr specific to the convex case.