全阶线性二次高斯控制中不存在虚假局部极小值
No Spurious Local Minima in Full-Order Linear Quadratic Gaussian Control
- University of California San Diego（加州大学圣地亚哥分校）
- Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明全阶LQG控制中所有局部极小值均为全局最优，次优驻点可通过增广策略转化为严格鞍点，揭示了良性景观。
AI中文摘要：
本文研究了直接状态空间参数化下线性二次高斯（LQG）控制的非凸优化景观。尽管LQG代价函数可能存在次优的驻点，但其是否允许次优局部极小值的问题一直悬而未决。我们对此问题给出否定回答：全阶LQG代价函数的每个局部极小值都是全局最优的。更一般地，在任意控制器阶数下，每个对应于不可控或不可观测实现的局部极小值都能达到所有任意阶稳定策略中的全局最优LQG代价。我们进一步证明，任何次优策略都可以通过解耦的稳定控制器状态进行增广，使得增广后的策略以概率1存在负曲率方向。因此，全阶LQG代价中的任何次优驻点都可以以概率1转化为严格鞍点。一个关键证明思想是将状态空间Hessian与由Youla参数化导出的频域全局最优性条件联系起来。这些结果揭示了尽管存在次优驻点，但局部极小值景观是良性的。
英文摘要：
This paper studies the nonconvex optimization landscape of Linear Quadratic Gaussian (LQG) control under direct state-space parameterization. Although the LQG cost may possess suboptimal stationary points, whether it admits suboptimal local minima has remained open. We answer this question negatively: Every local minimum of the full-order LQG cost is globally optimal. More generally, at any controller order, every local minimum corresponding to an uncontrollable or unobservable realization attains the globally optimal LQG cost over all stabilizing policies of arbitrary orders. We further show that any suboptimal policy can be augmented with decoupled stable controller states, so that the augmented policy admits a direction of negative curvature with probability one. Consequently, any suboptimal stationary point in full-order LQG cost can be converted into a strict saddle with probability one. One key proof idea connects the state-space Hessian to a frequency-domain global-optimality condition derived from the Youla parameterization. These results reveal a benign local-minimum landscape despite the presence of suboptimal stationary points.
