  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.
