ADMM无法达到$O(K^{-1})$的遍历KKT残差界
ADMM Fails to Achieve an $O(K^{-1})$ Ergodic KKT Residual Bound
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中文总结 AI 辅助
该研究证明经典ADMM无法达到$O(K^{-1})$的遍历KKT残差界，通过构造特定凸优化问题说明其KKT残差下界为$\u03a9(K^{-1/2})$，明确了该算法在一阶最优性度量上的性能局限。
中文摘要 AI 辅助
Karush-Kuhn-Tucker（KKT）残差是一阶最优性的基本度量，在误差界条件下，其与到KKT解集的距离仅相差常数因子。尽管目标误差和可行性违反已知存在$O(K^{-1})$的遍历速率，但本文证明经典ADMM的KKT残差通常无法满足统一的$O(K^{-1})$界。具体而言，本文构造了一类固定维数、依赖于时域的双块凸优化问题，在指定时域$K$下，其最后迭代及等权遍历平均的KKT残差均为$\u03a9(K^{-1/2})$，因此两种输出均无法满足统一的$O(K^{-1})$ KKT残差界。
英文摘要
The Karush--Kuhn--Tucker (KKT) residual is a fundamental measure of first-order optimality and, under an error bound condition, is comparable to the distance to the KKT solution set up to constant factors. Despite the $O(K^{-1})$ ergodic rates known for objective error and feasibility violations, we show that the KKT residual of classical ADMM cannot, in general, satisfy a uniform $O(K^{-1})$ bound. Specifically, we construct a fixed-dimensional, horizon-dependent family of two-block convex optimization problems for which the KKT residual is $Ω(K^{-1/2})$ at both the last iterate and the equal-weight ergodic average at the prescribed horizon $K$. Consequently, a uniform $O(K^{-1})$ KKT residual bound is impossible for either output.
