No Spurious Local Minima in Full-Order Linear Quadratic Gaussian Control
Abstract
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.
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BibTeX
@misc{zheng2026spurious,
title = {{No Spurious Local Minima in Full-Order Linear Quadratic Gaussian Control}},
author = {Yang Zheng and Yujie Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32875},
eprint = {2609.32875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32875},
}
RIS
TY - UNPB
TI - No Spurious Local Minima in Full-Order Linear Quadratic Gaussian Control
AU - Yang Zheng
AU - Yujie Tang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32875
UR - https://arxiv.org/abs/2609.32875
AB - 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.
ER -