发表机构
School of Mathematics and Statistics, Beijing Institute of Technology; Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology（北京理工大学数学与统计学院; 北京理工大学计算机辅助设计与图形学北京市重点实验室/信息安全数学理论与计算北京市重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对凸复合优化问题，引入约化原始-对偶KKT映射，建立了其度量次正则性与原始、对偶局部误差界的等价关系，并统一应用于多类模型。
AI 中文摘要
我们研究了凸复合优化问题及其Fenchel对偶问题的原始和对偶误差界。对于原始问题，我们使用近端梯度残差；而对于一般非光滑的对偶问题，我们使用次微分残差。我们引入了一个约化的原始-对偶KKT映射，在通常的可解性和强对偶性假设下，其零点集与原始和对偶解集的笛卡尔积重合。我们建立了约化KKT映射的度量次正则性与原始和对偶局部误差界之间的双向蕴含关系。如果复合矩阵具有满行秩且光滑梯度是局部Lipschitz连续的，则KKT度量次正则性与原始局部误差界等价。如果光滑项是强凸的，则它与对偶局部误差界等价。在满行秩和强凸性下，KKT度量次正则性等价于两个误差界同时成立。我们还建立了Luo-Tseng误差界与局部误差界之间的关系。该框架随后被应用于二次-多面体模型、具有一般凸正则项的强凸光滑模型以及超出多面体性和强凸性的正则化最小二乘模型，为原始、对偶和KKT误差界性质提供了统一的处理。
英文摘要
We study primal and dual error bounds for convex composite optimization problems and their Fenchel duals. For the primal problem, we use a proximal-gradient residual, while for the generally nonsmooth dual problem we use a subdifferential residual. We introduce a reduced primal--dual KKT mapping whose zero set coincides with the Cartesian product of the primal and dual solution sets under the standing solvability and strong-duality assumptions. We establish implications in both directions between metric subregularity of the reduced KKT mapping and the primal and dual local error bounds. If the composite matrix has full row rank and the smooth gradient is locally Lipschitz continuous, KKT metric subregularity is equivalent to the primal local error bound. If the smooth term is strongly convex, it is equivalent to the dual local error bound. Under full row rank and strong convexity, KKT metric subregularity is equivalent to the simultaneous validity of both error bounds. We also establish relations between the Luo--Tseng and local error bounds. The framework is then applied to quadratic--polyhedral models, strongly convex smooth models with general convex regularizers, and regularized least-squares models beyond polyhedrality and strong convexity, providing a unified treatment of primal, dual, and KKT error-bound properties.
