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18
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future
, 2012
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A note on the convex hull of finitely many projected spectrahedra
, 2009
"... A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton an ..."
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Cited by 7 (6 self)
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A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton and Nie [3], who prove the same result in the case of bounded sets.
Duality of nonexposed faces
"... Abstract – Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of nonexposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One o ..."
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Cited by 4 (3 self)
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Abstract – Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of nonexposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The second and third Galois connection is defined between the touching cones and the faces of each convex body separately. While the former is wellknown, we introduce the latter in this article for any convex set in any finite dimension. We demonstrate our results about conjugate faces with planar convex bodies and planar selfdual convex bodies, for which we also include constructions. Index Terms – convex body, polar, conjugate face, nonexposed point, singular face, dual, selfdual. AMS Subject Classification: 52A10, 52A20. 1 Nonexposed faces and dual convex bodies Duality of faces of a dual pair of closed convex cones was studied in [Ba2] with regard to the lattice of the inclusion ordering. This duality corresponds to the conjugate
On semidefinite representations of nonclosed sets
"... Abstract. Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinite representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent r ..."
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Abstract. Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinite representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinite representable set is shown to be semidefinite representable. More general, one can remove faces of a semidefinite representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way. 1.
LINEAR OPTIMIZATION WITH CONES OF MOMENTS AND NONNEGATIVE POLYNOMIALS
, 2013
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