发表机构
Sharif University of Technology（谢里夫理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于拉格朗日乘子的非凸优化操作 regime 分类学，建立结构定理并设计线性时间分类器，经数值实验验证，为相关算法设计与鲁棒性分析提供基础工具。
AI 中文摘要
本文基于KKT平稳点处拉格朗日乘子的符号，提出了带约束非凸优化的结构化分类学。利用对八大经典算法族（包括块坐标下降、ADMM、广义本德分解、逐次凸近似、内点法、镜像下降、弗兰克-沃尔夫算法及黎曼梯度下降）的统一博弈论解释，证明归一化乘子向量携带与算法无关的结构化指纹。该向量的四个无量纲形状特征将对偶空间划分为五个操作 regime：无约束、资源受限、饱和、强耦合及混合。本文建立刻画该划分的四个结构定理：在自然KKT对称下的不变性、在数据扰动下具有罗宾逊强正则性显式利普希茨边际的局部稳定性、余维一的 regime 转换，以及混合 regime 作为核心 regime 的勒贝格零边界的拓扑识别。提出一种线性时间分类器，其在正确性、迭代稳定性、样本复杂度及数据漂移下的在线跟踪方面具有可证保证。对104个混合整数非线性规划及下行波束成形实例的数值实验验证了理论预测。该框架为非凸优化中感知 regime 的算法设计与鲁棒性分析提供了基础工具。
英文摘要
This paper introduces a structural taxonomy for constrained non-convex optimization based on the signature of Lagrange multipliers at KKT stationary points. Leveraging a unified game-theoretic interpretation of eight classical algorithm families--including block coordinate descent, ADMM, generalized Benders decomposition, successive convex approximation, interior-point methods, mirror descent, Frank-Wolfe, and Riemannian gradient descent--we show that the normalized multiplier vector carries an algorithm-independent structural fingerprint. Four scale-free shape features of this vector partition the dual space into five operational regimes: Unconstrained, Resource-Limited, Saturation, Strongly-Coupled, and Hybrid. We establish four structural theorems characterizing the partition: invariance under natural KKT symmetries, local stability under data perturbation with explicit Lipschitz margins from Robinson's strong regularity, codimension-one regime transitions, and the topological identification of the Hybrid regime as the Lebesgue-null boundary of the core regimes. A linear-time classifier is proposed with provable guarantees on correctness, iteration stabilization, sample complexity, and online tracking under data drift. Numerical experiments on 104 mixed-integer nonlinear programs and a downlink beamforming instance validate the theoretical predictions. The framework provides a foundational tool for regime-aware algorithm design and robustness analysis in non-convex optimization.
Comments45 pages. SNMP
