| Author: | Chen, Kaihuang |
| Title: | Accelerated first order methods for convex composite quadratic programming : theory, algorithms, and GPU implementation |
| Advisors: | Sun, Defeng (AMA) |
| Degree: | Ph.D. |
| Year: | 2026 |
| Department: | Department of Applied Mathematics |
| Pages: | xviii, 110 pages : color illustrations |
| Language: | English |
| Abstract: | Convex composite quadratic programming (CCQP) plays a central role in machine learning, finance, and engineering. This thesis develops efficient algorithms for large-scale CCQP problems, with an emphasis on accelerated first-order methods that exploit the massive parallelism of modern GPUs. On the theoretical side, we propose two Halpern Peaceman-Rachford (HPR) methods for solving different primal formulations of CCQP. Both enjoy an O(1/k) complexity bound in terms of the Karush-Kuhn-Tucker (KKT) residual and objective error. By introducing auxiliary slack variables and appropriate proximal operators, each iteration admits an explicit closed-form update, avoiding linear system solves and making the methods well-suited for GPU acceleration. More importantly, we introduce a dual HPR method designed for solving the restricted Wolfe dual of CCQP problems, which also enjoys a fast O(1/k) complexity bound in terms of the KKT residual and the objective error. One distinctive feature of the dual approach is that, instead of working with the primal formulations, it builds on the novel restricted Wolfe dual introduced in recent years. It also leverages the symmetric Gauss-Seidel technique to simplify subproblem updates without introducing auxiliary slack variables that typically lead to slow convergence. By restricting updates to the range space of the Hessian of the quadratic objective function, the dual approach employs proximal operators of smaller spectral norms to speed up the convergence. Shadow sequences are elaborately constructed to deal with the range space constraints. On the implementation side, we introduce three key enhancements. First, leveraging the O(1/k) complexity results in terms of the KKT residual of the HPR methods, we develop adaptive restart and penalty parameter updating strategies to accelerate practical convergence. Second, we design a GPU-oriented implementation that avoids linear solves and matrix inverse approximations, relying solely on matrix-vector multiplications and vector-level operations. Third, the solver operates without requiring the explicit matrix form of the quadratic objective, making it highly scalable to extremely large CCQP instances such as quadratic assignment and LASSO regression. In the numerical study, we first present case analyses comparing GPU versus CPU performance, HPR against preconditioned alternating direction method of multipliers (without acceleration), restricted Wolfe dual versus primal formulations, and the impact of algorithmic enhancements on solver efficiency and convergence. Extensive experiments demonstrate that the proposed methods achieve superior speed and scalability compared to state-of-the-art solvers. |
| Rights: | All rights reserved |
| Access: | open access |
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