A Proximal Gradient Algorithm with Optimal Accelerated Parameter for Quadratic Programming

Quadratic Programming (QP) problems are fundamental in many scientific and engineering fields. While the accelerated proximal gradient (APG) algorithm is widely used, its convergence rate is highly sensitive to the choice of the accelerated parameter. In this paper, we demonstrate that by strategically selecting this parameter below its theoretical optimum, APG can achieve a superior linear convergence rate for QP problems compared to the proximal gradient algorithm. Building on this insight, we propose the APG algorithm with an Optimal accelerated parameter (APGO). Since the optimal value relies on the local condition number, we further develop an APG algorithm with adaptive optimal accelerated parameter (APGOA) that automatically tunes the parameter for practical implementation. Our numerical experiments on both QP and Lasso problems confirm the theoretical findings and demonstrate the significant efficiency gains of our proposed algorithms, underscoring their practical value for a wide range of large-scale optimization applications.