We aim at completing the analysis in Fasano and Pesenti (2017) for quadratic hypersurfaces, where the geometric viewpoint suggested by the Polarity theory is considered, in order to recast basic properties of the Conjugate Gradient (CG) method (Hestenes and Stiefel, 1952) [1]. Here, the focus is on possibly exploiting theoretical advances on nonconvex quadratic hypersurfaces, in order to address guidelines for efficient optimization methods converging to second order stationary points, in large scale settings. We first recall some results from Fasano and Pesenti (2017), in order to fully analyze the relationship between the CG and the Polarity theory. Then, we specifically address, from a different perspective, the geometric insight of the pivot breakdown, which might occur when solving a nonsingular indefinite Newton’s equation applying the CG. Furthermore, we fully exploit some novel theoretical advances of the Polarity theory on nonconvex quadratic hypersurfaces not considered in Fasano and Pesenti (2017). Finally, we show that our approach describes a general framework, which also encompasses a class of CG–based methods, namely Planar CG–based methods. The framework we consider intends to emphasize a bridge between the geometry behind stationary points of nonconvex quadratic hypersurfaces and their efficient computation using Krylov–subspace methods.

我们旨在完成Fasano和Pesenti（2017）中关于二次超曲面的分析，其中考虑了极性理论提出的几何观点，以重塑共轭梯度（CG）方法的基本属性（Hestenes和Stiefel，1952年） [1]。在这里，重点是可能在非凸二次曲面上的理论开发，以便解决大规模设置中收敛到二阶固定点的有效优化方法的准则。我们首先回顾一下Fasano和Pesenti（2017）的一些结果，以便全面分析CG和极性理论之间的关系。然后，我们从不同的角度专门解决了枢轴击穿的几何见解，这在使用CG求解非奇异的牛顿方程时可能会发生。此外，我们充分利用了关于Fasano和Pesenti（2017）中未考虑的非凸二次超曲面的极性理论的一些新颖理论进展。最后，我们证明了我们的方法描述了一个通用框架，其中还包含一类基于CG的方法，即基于平面CG的方法。我们认为的框架旨在强调非凸二次曲面的静止点后面的几何形状与使用Krylov-子空间方法进行有效计算之间的桥梁。