Geometry and Conditioning in Structured Conic Problems

二次规划允许人们通过现代优化方法（如内点算法）建模和解决大型工业问题。这些方法在解决大量问题时是有效和可靠的，然而，它们在相对较小但重要的病态实例集上失败，从而影响了技术的整体可靠性。这种行为的原因是深刻的，并且构成了现实复杂性中未解决的主要问题之一：没有已知的算法可以在多项式时间内解决真实数据的二次问题。该项目旨在发展对二次问题几何的深刻理解，旨在解决这一计算理论中的基本问题。

Conic programming allows one to model and solve large industrial problems via modern optimisation methods, such as interior-point algorithms. These methods are efficient and reliable in solving a vast number of problems, however, they fail on a relatively small but significant set of ill-posed instances, thus affecting the overall reliability of the technique. The reason for such behaviour is profound and constitutes one of the major unsolved problems in real complexity: there is no known algorithm that solves conic problems with real data in polynomial time. The project aims to develop a deep understanding of the geometry of conic problems, aiming for the resolution of this fundamental problem in computational theory.