发表机构
University of North Carolina at Chapel Hill; Southern Methodist University（北卡罗来纳大学教堂山分校; 南方卫理公会大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对光滑凸项加弱凸非光滑项的优化问题，提出 ADMM 及线性化 LADMM 分裂算法，证明收敛到方向稳定解，实验验证高效且解质量与基线相当。
AI 中文摘要
我们研究一类弱凸优化问题，其中目标函数是光滑凸项与可能非光滑的弱凸项之和。为利用这一结构，我们基于交替方向乘子法（ADMM）开发了一种分裂技术，将两个分量的最小化解耦为易于处理的子问题。由于与光滑项相关的更新可能需要内部迭代求解器，我们进一步对该项进行线性化，得到具有廉价一步更新的线性化 ADMM（LADMM）方案。在温和条件下，我们建立了 ADMM 和 LADMM 方法对方向稳定解的子序列收敛性，这些解等价于我们弱凸问题中的临界点和 Clarke 稳定解。在两个低维测试函数和一个高维对数正则化逻辑回归模型上的数值实验表明，所提出的方法计算效率高，并能产生与基线方法质量相当的解。
英文摘要
We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be nonsmooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization of the two components into tractable subproblems. Because the update associated with the smooth term may require an inner iterative solver, we further linearize this term, yielding a linearized ADMM (LADMM) scheme with an inexpensive one-step update. Under mild conditions, we establish the subsequence convergence of both ADMM and LADMM methods to directional stationary solutions, which are equivalent to critical points and Clarke stationary solutions for our weakly convex problem. Numerical experiments on two low-dimensional test functions and a high-dimensional logarithmic regularized logistic regression model demonstrate that the proposed approaches are computationally efficient and produce solutions of comparable quality to baseline methods.
Comments37 pages, 5 figures, 4 tables
