Effective methods for spectrahedra in real and convex algebraic geometry

等高面体是半定规划的可行集，同时被广泛认为是一种非常通用的几何结构，它构成了(实)代数几何和凸优化之间的中心纽带。近年来，半定规划和多面体在处理各种类型的实代数问题方面取得了巨大的进展，但多面体的有效处理仍然是一个颇具挑战性的问题。在最新技术的基础上，当前项目的目标是促进对镜面的理解和处理。特别是，我们将探索有效的方法来解决变形虫(代数簇的对数映象)的计算问题，实代数几何中的镜面逼近和松弛问题，以及多项式、镜面和完全正性的优化，以及处理镜面的稳定性问题。

Spectrahedra are the feasible sets of semi definite programming and are meanwhile widely recognized as a very versatile geometric structure which forms a central link between (real) algebraic geometry and convex optimization. While recent years have provided tremendous progress in the use of semidefinite programming and spectrahedra in approaching real-algebraic problems of various types, effective handling of spectrahedra is still a rather challenging issue. Building upon the state of the art, the goal of the current project is to advance the understanding and handling of spectrahedra. In particular, we shall explore effective methods for tackling computational problems of amoebas (the logarithmic images of algebraic varieties), spectrahedral approaches and relaxations in real algebraic geometry and optimization of polynomials, spectrahedra and complete positivity, as well as stability issues in handling spectrahedra.