增广原始-对偶动力学在采样约束下的均衡偏差与收敛性

中文摘要 AI 辅助

本文研究了当约束值从样本中估计时，增广原始-对偶动力学的稳定性与收敛性。无偏的约束观测可能产生有偏的增广乘子信号，从而移动平均动力学的均衡点。对于分量不等式，我们给出了保持Karush-Kuhn-Tucker（KKT）均衡的充要条件，并构造了一个具有局部指数稳定均衡但违反互补性的凸例子。为解决此偏差，在形成增广乘子信号之前，对约束值进行递归估计。对于光滑凸锥问题，联合能量分析确立了原始、对偶和估计状态的有界性、估计误差的消失性，以及在全局正则性和有界条件二阶矩下原始-对偶迭代几乎必然收敛到单一KKT点。该结果允许非唯一解和乘子，同时保持每次迭代的样本数固定。数值研究验证了预测的均衡偏差，并考察了具有非唯一KKT点和非线性约束的收敛性。

英文摘要

This work studies the stability and convergence of augmented primal-dual dynamics when constraint values are estimated from samples. Unbiased constraint observations can produce a biased augmented multiplier signal, shifting the equilibria of the mean dynamics. For componentwise inequalities, we give a necessary and sufficient condition for preserving the Karush-Kuhn-Tucker (KKT) equilibria and construct a convex example with a locally exponentially stable equilibrium that violates complementarity. To address this bias, constraint values are estimated recursively before forming the augmented multiplier signal. For smooth convex conic problems, a joint energy analysis establishes boundedness of the primal, dual, and estimation states, vanishing estimation error, and almost sure convergence of the primal-dual iterates to a single KKT point under global regularity and bounded conditional second moments. The result allows nonunique solutions and multipliers while keeping the number of samples per iteration fixed. Numerical studies illustrate the predicted equilibrium bias and examine convergence with nonunique KKT points and nonlinear constraints.

发表机构

• The Hong Kong Polytechnic University（香港理工大学）
• Huazhong University of Science and Technology（华中科技大学）

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