Recent Developments in HPR Methods for Large-Scale Convex Composite Quadratic Programming

Abstract

We present recent developments in Halpern Peaceman--Rachford (HPR) methods for solving large-scale convex composite quadratic programming problems. Within a unified HPR framework applied to the restricted Wolfe dual, we develop several algorithmic variants based on different choices of the semi-proximal operator $\mathcal{T}_1$ for the joint dual subproblem. Specifically, an sGS-based construction leads to a sequential $w$--$y$--$w$ update scheme, while a block-linearized construction eliminates cross-block coupling and enables parallel dual updates. To improve the latter, we further introduce a coupling-aware construction that exploits a second spectral moment to obtain a tighter row majorizer. We extend the framework to convex quadratically constrained quadratic programming through factor-based second-order cone reformulations. Additional computational enhancements include equality Gram scaling and asynchronous CPU polishing. We establish convergence results under fixed metrics and discuss practical adaptive strategies.