We consider a class of structured nonsmooth difference-of-convex minimization. We allow nonsmoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) d(irectional)-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear convergence results are based on direct generalizations of the assumptions of error bounds and separation of isocost surfaces proposed in the seminal work of Luo and Tseng (Ann Oper Res 46–47:157–178, 1993), as well as one additional assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions. An interesting by-product is to present a sharper characterization of the limit set of the basic algorithm proposed by Pang et al., which fits between d-stationarity and global optimality. We also discuss sufficient conditions under which these assumptions hold. Finally, we provide several realistic and nontrivial statistical learning models where all assumptions hold.

我们考虑一类结构化的非光滑凸差最小化。我们在目标函数的凹凸分量上都允许非光滑度，而在凹面部分则具有有限的最大结构。正如Pang等人最近的工作中所提倡的那样，我们的重点是计算（弱或标准）d（垂直）平稳点的算法。（Math Oper Res 42：95–118，2017年）。我们的线性收敛结果是基于罗和曾的开创性著作（Ann Oper Res 46–47：157–178，1993）中提出的误差边界和等值面分离的假设的直接概括，以及一个额外的假设关于某些平稳集合和优势区域的交点的局部线性规则性的关系。一个有趣的副产品是对Pang等人提出的基本算法的极限集进行更清晰的刻画，该极限集适合d平稳性和全局最优性。我们还将讨论这些假设成立的充分条件。最后，我们提供了所有假设都成立的几种现实且非平凡的统计学习模型。