In this paper, we propose an active-set proximal-Newton algorithm for solving \(\ell _1\) regularized convex/nonconvex optimization problems subject to box constraints. Our algorithm first relies on the KKT error to estimate the active and free variables, and then smoothly combines the proximal gradient iteration and the Newton iteration to efficiently pursue the convergence of the active and free variables, respectively. We show the global convergence without the convexity of the objective function. For some structured convex problems, we further design a safe screening procedure that is able to identify/remove active variables, and can be integrated into the basic active-set proximal-Newton algorithm to accelerate the convergence. The algorithm is evaluated on various synthetic and real data, and the efficiency is demonstrated particularly on \(\ell _1\) regularized convex/nonconvex quadratic programs and logistic regression problems.

在本文中，我们提出了一个主动集近侧牛顿算法来求解\（\ ell _1 \）受框约束的正则化凸/非凸优化问题。我们的算法首先依靠KKT误差来估计有效变量和自由变量，然后平滑地组合近端梯度迭代和Newton迭代来分别有效地追求有效变量和自由变量的收敛。我们展示了没有目标函数凸性的全局收敛性。对于某些结构化凸问题，我们进一步设计了一种安全的筛选程序，该程序能够识别/删除活动变量，并且可以集成到基本活动集近侧牛顿算法中以加快收敛速度​​。该算法在各种综合和真实数据上进行了评估，并且在\（\ ell _1 \）上特别证明了效率 正则化凸/非凸二次规划和逻辑回归问题。