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...definite 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 non-exposed 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 projecti...

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... (SDP) representation. Projected spectrahedra form a much wider class than spectrahedra, and much research effort is currently spent on understanding their properties, e.g. [13], [12], [9], [5], [6], =-=[15]-=-, [14], [3]. In fact, Helton and Nie [5] have conjectured that every convex semi-algebraic set allows a lifted LMI representation. Obtaining explicit lifted LMI representations for concretely given co...

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...ion of a spectrahedron to a subspace of R n is often called a semidefinitely representable set. Helton and Nie [3] conjecture that every convex semialgebraic set is such a projection. See for example =-=[1, 2, 3, 4, 5, 6, 7]-=- for more detailed information on spectrahedra and their projections. We prove that the convex hull of finitely many projections of spectrahedra is again a projection of a spectrahedron. This generali...

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... under certain curvature conditions on the defining inequalities of a set. However, the Lasserre method can only work if all faces of the convex set are exposed, see Netzer, Plaumann and Schweighofer =-=[8]-=-. So there are basic closed semialgebraic convex sets for which the method fails. (iv) The convex hull of a finite union of semidefinite representable sets is again semidefinite representable. This is...

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...or solving GPMs. We refer to [12, 25, 27, 28, 38, 37] for moment and polynomial optimization problems. Semidefinite programs are also very useful in representing convex sets and convex hulls, like in =-=[11, 13, 14, 15, 23, 24, 30, 43]-=-, and in solving polynomial equations, like in [20, 21, 29]. In this paper, we study optimization problems with more general cones than Rd(K) and Pd(K). Let A ⊆ Nn be a finite set, and R[x]A := span{x...

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...5] for an in-depth survey of semidefinite programming. Unlike polyhedra, the class of spectrahedra is not closed under projection. One way to see this is that all faces of a spectrahedron are exposed =-=[70]-=-, but spectrahedral shadows can have non-exposed faces, as seen in Figure 1.1. Spectrahedral shadows, however, are closed under projection and duality [92]. As we can lift objective functions, linear ...