Polarity and conjugacy for quadratic hypersurfaces: A unified framework with recent advances

Abstract

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.