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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 [19] on non-exposed faces. We also solve… (More)

- TIM NETZER
- 2008

A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in [PS], is a very useful property. It often implies that the quadratic module is closed; furthermore it helps settling the… (More)

- Daniel Plaumann, Sebastian Krug, Salma Kuhlmann, Murray Marshall, Tim Netzer, Alexander Prestel +1 other
- 2008

Gewidmet meinen Eltern Marina und Michael Plaumann und meinem Onkel Peter Plaumann in tiefer Dankbarkeit Acknowledgements I would like to express my gratitude to my advisor Claus Scheiderer for the many ideas and insights into the topic that he shared with me, as well as for his constant encouragement. I am grateful for many interesting and helpful… (More)

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine-linear combinations of variable,; is positive semidefinite. :dotiv11ted by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinitely representable sets.… (More)

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Her-mite matrix. If… (More)

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define poly-hedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of… (More)

- TIM NETZER, Joseph A. Ball
- 2006

We prove the main result from Schmüdgen's 2003 article in a more elementary way. The result states that the question of whether a finitely generated preordering has the so-called strong moment property can be reduced to the same question for preorderings corresponding to fiber sets of bounded polynomials.

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford… (More)

- TIM NETZER
- 2009

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… (More)

- Tim Netzer
- 2008

Acknowledgements A lot of people helped me during the work on this thesis. First of all, I am greatly indebted to Alexander Prestel, Claus Scheiderer and Markus Schweighofer. I benefitted a lot from their constant logistic and mathematical support. I also want to thank Robert Denk, David Grimm and Daniel Plaumann for many interesting discussions on the… (More)