发表机构
Johann Radon Institute for Computational and Applied Mathematics; Dresden University of Technology, Institute of Numerical Mathematics（Johann Radon计算与应用数学研究所; 德累斯顿工业大学数值数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在抽象集值映射框架下，通过非临界拉格朗日乘子与两种平静型性质刻画误差界，将结果应用于复合优化问题，恢复并扩展了相关已知刻画。
AI 中文摘要
已知临界拉格朗日乘子会导致牛顿型方法收敛速度可能变慢，而给定乘子的非临界性与一种误差界相关，该误差界用于估计与扰动问题相关的原始-对偶对到原始解及未扰动问题乘子集的距离，对局部收敛分析具有决定性作用。我们在仅存在将参数映射为原始-对偶对的集值映射的抽象框架下研究该关系，通过基础乘子的非临界性及相关乘子映射的两种平静型性质刻画所关注的误差界，其中一种是模糊意义下的新型弱内部平静性。该刻画无需对基础问题施加任何结构假设，且这两种平静型条件不仅是充分的，也是必要的。之后，我们将这些结果应用于复合优化问题：当外部函数的次微分是多面体映射时，两种条件自动成立，这解释了多面体设定下非临界性与误差界等价的现象；当外部函数是$C^2$-可分解时，两种条件可通过受限乘子映射的平静性结合闭性条件得到保证，这恢复并扩展了$C^2$-锥可约约束系统的已知刻画。
英文摘要
Critical Lagrange multipliers are known to be responsible for the potentially slow convergence of Newton-type methods, and noncriticality of a given multiplier has been tied to an error bound which estimates the distance of a primal-dual pair associated with a perturbed problem to the primal solution and the multiplier set of the unperturbed one, and is decisive for a local convergence analysis. We investigate this relation in an abstract setting where merely a set-valued mapping assigning primal-dual pairs to a parameter is available, and characterize the error bound of interest in terms of noncriticality of the underlying multiplier and two calmness-type properties of the associated multiplier mapping, one of which is a novel weak inner calmness in the fuzzy sense. No structural assumptions on the underlying problem are needed for that, and the two calmness-type conditions are not only sufficient but also necessary. Afterwards, we specify these findings for composite optimization problems. Whenever the subdifferential of the outer function is a polyhedral mapping, both conditions hold automatically, which explains the equivalence of noncriticality and the error bound observed in the polyhedral setting. Whenever the outer function is $C^2$-decomposable, they are secured by calmness of a restricted multiplier mapping together with a closedness condition, which recovers and extends the known characterization for $C^2$-cone reducible constraint systems.
Comments30 pages
