非凸优化的联合结构识别与基于近端线搜索的牛顿加速

中文摘要 AI 辅助

我们考虑具有光滑（可能非凸）函数和可分凸多面体项的复合优化问题。此类问题具有简单的非光滑结构，但在应用中广泛出现。现有的牛顿型方法在方向计算过程中可能无法识别非光滑坐标，在线搜索过程中可能使坐标偏离全步长所达到的位置，或者依赖单独的一阶步骤进行识别。在本工作中，我们提出了一种统一的非精确近端牛顿方法，该方法通过设计一种近端线搜索，将结构识别与二阶加速相结合，以保留局部模型所揭示的非光滑结构。具体而言，在预测的工作集上，我们非精确地求解二次模型，但精确地强制执行其约束，允许坐标到达新的拐点而不会离开所选的仿射区间。该集合还包括小余量拐点坐标，以利用Hessian耦合效应。为了全局化方向，我们使用各向同性近端模型构建近端线搜索，其线性项的选择使得初始试验恰好恢复全步长。回溯增加近端曲率，直到充分下降成立，允许坐标保持在由全步长到达的拐点处，同时阻尼其他分量。生成每个试验点仅需要近端算子评估。我们证明了在严格互补性和非奇异简化Hessian下的有限回溯和有限活动集识别，以及在方向性Dennis-More条件和消失相对非精确性下的局部Q-超线性收敛。数值结果验证了所提出方法的有效性。

英文摘要

We consider composite optimization with a smooth, possibly nonconvex function and a separable convex polyhedral term. Such problems have a simple nonsmooth structure but arise widely in applications. Existing Newton-type methods may fail to identify nonsmooth coordinates during direction computation, move coordinates away from those reached by the full step during line search, or rely on separate first-order steps for identification. In this work, we propose a unified inexact proximal Newton method that couples structure identification with second-order acceleration through a proximal line search designed to retain the nonsmooth structure revealed at the local model. In particular, on a predicted working set, we solve a quadratic model inexactly but enforce its constraints exactly, allowing coordinates to reach new kinks without leaving the selected affine intervals. This set also includes small-margin kink coordinates to exploit the Hessian coupling effect. To globalize the direction, we construct a proximal line search using an isotropic proximal model whose linear term is chosen so that the initial trial exactly recovers the full step. Backtracking increases the proximal curvature until sufficient decrease holds, allowing coordinates to remain at kinks reached by the full step while damping other components. Generating each trial point requires only proximal operator evaluations. We prove finite backtracking and finite active-set identification under strict complementarity and a non-singular reduced Hessian, together with local Q-superlinear convergence under a directional Dennis-More condition and vanishing relative inexactness. Numerical results verify the effectiveness of the proposed method.

发表机构

• Imperial College London（帝国理工学院）

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