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Convex sets with semidefinite representation

Authors:	Jean B Lasserre

Institution:	(1) LAAS-CNRS and Institute of Mathematics, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse cedex 4, France

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.

Keywords:	Convex sets Semidefinite representation Representation of positive polynomials Sum of squares
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