Spectrahedra and Hyperbolic Polynomials

半定规划是凸优化的一个分支，在理论和实践上都引起了极大的兴趣。典型的应用包括多项式优化或组合优化，例如最大割问题。利用内点法可以在程序描述规模为多项式的时间内求解具有固定精度的半定规划。一个基本感兴趣的问题是刻画半定规划的可行集的集合，即所谓的多面体。广义Lax猜想的内容就是这样一个假定刻画。不同的作者对这一假设进行了研究，并证明了一些特例。该项目的目的是在这一方向上取得进一步的积极成果，并以最佳方式彻底解决这一假设。

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.