Spectrahedra and Hyperbolic Polynomials

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Semidefinite programming is a branch of convex optimization that has aroused great interest both theoretically and practically. Typical applications include polynomial optimization or combinatorial optimization, such as the Max-Cut Problem. With the help of interior-point method one can solve a semidefinite program for fixed precision in a time that is polynomial in the program description size. A question of fundamental interest is that of characterizing the sets which are the feasible sets of semidefinite programming, namely the so-called spectrahedra. The content of the generalized Lax conjecture is such a presumed characterization. Different authors have worked on this assumption and proved some special cases. The aim of this project is to achieve further positive results in this direction and optimally to solve the assumption completely.