This paper investigates the efficient solution of penalized quadratic regressions in high-dimensional settings. A novel and efficient algorithm for ridge-penalized quadratic regression is proposed, leveraging the matrix structures of the regression with interactions. Additionally, an alternating direction method of multipliers (ADMM) framework is developed for penalized quadratic regression with general penalties, including both single and hybrid penalty functions. The approach simplifies the calculations to basic matrix-based operations, making it appealing in terms of both memory storage and computational complexity for solving penalized quadratic regressions in high-dimensional settings.

翻译：本文研究了高维情形下带罚二次回归的高效求解问题。针对岭罚二次回归，提出了一种新颖高效算法，该算法充分利用含交互项回归的矩阵结构。此外，针对一般罚函数（包括单一罚函数与混合罚函数）下的带罚二次回归，开发了基于交替方向乘子法（ADMM）的框架。该方法将计算简化为基本矩阵运算，在求解高维情形下的带罚二次回归时，在内存存储与计算复杂度方面均具有显著优势。