发表机构
inspire AG（inspire公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出鞍点PID（SPPID）动力学，统一PID控制框架求解约束优化，等价于增广拉格朗日的预条件原始-对偶梯度流，并证明其在凸及强凸问题中的收敛性。
AI 中文摘要
本文通过反馈控制的视角研究约束优化问题。基于拉格朗日乘子作为反馈控制器的解释，我们提出了鞍点PID（SPPID）动力学：一个用于连续时间鞍点动力学的统一比例-积分-微分（PID）框架。所提出的动力学对等式约束采用PID控制，对不等式约束采用抗积分饱和的PI控制。我们证明SPPID等价于增广拉格朗日函数的预条件原始-对偶梯度流。这种等价性揭示了每个反馈分量的独特作用：积分作用强制约束满足，比例作用引入增广拉格朗日结构，而微分作用通过状态相关的黎曼度量改变原始动力学的几何结构。对于凸问题，我们建立了收敛到KKT集以及全局渐近稳定性的条件。对于等式约束问题，我们在标准假设下建立了非线性等式约束的局部指数收敛性，并表明投影梯度流是无限微分增益极限。对于具有仿射等式或不等式约束的强凸问题，我们利用收缩理论建立了全局指数收敛性。最后，我们提供了各种数值示例来说明SPPID的实用性。
英文摘要
This paper studies constrained optimization problems through the lens of feedback control. Building on the interpretation of Lagrange multipliers as feedback controllers, we propose the \emph{saddle-point PID (SPPID) dynamics}: a unified proportional--integral--derivative (PID) framework for continuous-time saddle-point dynamics. The proposed dynamics employ PID control for equality constraints and anti-windup PI control for inequality constraints. We show that SPPID is equivalent to a preconditioned primal--dual gradient flow of the augmented Lagrangian. This equivalence reveals the distinct role of each feedback component: integral action enforces constraint satisfaction, proportional action induces the augmented Lagrangian structure, and derivative action modifies the geometry of the primal dynamics via a state-dependent Riemannian metric. For convex problems, we establish convergence to the KKT set together with conditions for global asymptotic stability. For equality-constrained problems, we establish local exponential convergence for nonlinear equality constraints under standard assumptions, and show that projected gradient flow emerges as the infinite derivative-gain limit. For strongly convex problems with affine equality or inequality constraints, we establish global exponential convergence by leveraging contraction theory. Finally, we provide various numerical examples to illustrate the utility of SPPID.
Comments37 pages, 11 figures
