Describing convex semialgebraic sets by linear matrix inequalities
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Exposed Faces of Semidefinitely Representable Sets
SIAM Journal on Optimization
Theta Bodies for Polynomial Ideals
SIAM Journal on Optimization
Polynomial Matrix Inequality and Semidefinite Representation
Mathematics of Operations Research
Semidefinite Representation of Convex Hulls of Rational Varieties
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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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.