Spectrahedra and Hyperbolic Polynomials

半定规划是凸优化问题的一个分支，在理论和实践上都引起了人们极大的兴趣。典型的应用包括多项式优化或组合优化，如最大割问题。用邻点法可以在程序描述长度为多项式的时间内，以固定精度求解半定程序。一个基本的兴趣问题是，这是半定规划的可行集，即所谓的谱面集的特征。广义Lax猜想的内容就是这样一个假定的刻画。不同的作者在这个假设上进行了工作，并证明了一些特殊情况。该项目的目的是在这方面取得进一步的积极成果，并以最佳方式完全解决这一假设。

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.