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PDHCG-II: An Enhanced Version of PDHCG for Large-Scale Convex QP

Hongpei Li, Yicheng Huang, Huikang Liu, Dongdong Ge, Yinyu Ye

TL;DR

This work proposes PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs, which explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight.

Abstract

Quadratic programming (QP) is a fundamental optimization model with wide-ranging applications in decision-making and machine learning, yet efficiently solving large-scale instances remains a major computational challenge. Building upon the recently developed PDHCG framework, we propose PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs. The proposed method explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight. To further improve practical performance, PDHCG-II introduces a refined adaptive termination criterion for inner subproblems to prevent over-solving, together with an infeasibility detection mechanism for robust handling of ill-posed instances. Extensive numerical experiments demonstrate that PDHCG-II consistently achieves 2.5-5 times speedups over PDHCG on standard QP benchmarks. To facilitate reproducibility and broader adoption, we release a CUDA-C implementation of PDHCG-II as open-source software.

PDHCG-II: An Enhanced Version of PDHCG for Large-Scale Convex QP

TL;DR

This work proposes PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs, which explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight.

Abstract

Quadratic programming (QP) is a fundamental optimization model with wide-ranging applications in decision-making and machine learning, yet efficiently solving large-scale instances remains a major computational challenge. Building upon the recently developed PDHCG framework, we propose PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs. The proposed method explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight. To further improve practical performance, PDHCG-II introduces a refined adaptive termination criterion for inner subproblems to prevent over-solving, together with an infeasibility detection mechanism for robust handling of ill-posed instances. Extensive numerical experiments demonstrate that PDHCG-II consistently achieves 2.5-5 times speedups over PDHCG on standard QP benchmarks. To facilitate reproducibility and broader adoption, we release a CUDA-C implementation of PDHCG-II as open-source software.
Paper Structure (30 sections, 1 theorem, 32 equations, 4 tables)

This paper contains 30 sections, 1 theorem, 32 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Algorithm Framework
  3. The PDHG Iteration Scheme
  4. Primal Subproblem Solvers
  5. Structured Quadratic Objective.
  6. Algorithmic Enhancements
  7. Reflected Halpern Scheme
  8. Adaptive Primal--Dual Weight
  9. Adaptive Stopping Criterion for Primal Subproblems
  10. Optimality Conditions and Stopping Criteria
  11. KKT Conditions
  12. Termination Criteria
  13. Primal Residual
  14. Dual Residual
  15. Duality Gap
  16. ...and 15 more sections

Key Result

Theorem 5.1

The convex quadratic program eq:QP is unbounded if and only if eq:dualinfea holds for some $d \in \mathbb{R}^n$.

Theorems & Definitions (2)

  • Theorem 5.1
  • proof