# Optimizing Static Linear Feedback: Gradient Method

arXiv: https://arxiv.org/abs/2004.09875 · HTML: https://mlsanity.com/paper/2004.09875/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2020.

## Built on by (most cited first, 27 of 27)

- [Stochastic Policy Gradient Methods: Improved Sample Complexity for Fisher-non-degenerate Policies](https://mlsanity.com/paper/2302.01734/) (2023)
- [Derivative-Free Policy Optimization for Linear Risk-Sensitive and Robust Control Design: Implicit Regularization and Sample Complexity](https://mlsanity.com/paper/2101.01041/) (2021)
- [Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence](https://mlsanity.com/paper/2402.17722/) (2024)
- [Global Convergence of Direct Policy Search for State-Feedback H_∞ Robust Control: A Revisit of Nonsmooth Synthesis with Goldstein Subdifferential](https://mlsanity.com/paper/2210.11577/) (2022)
- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Model-Free μ Synthesis via Adversarial Reinforcement Learning](https://mlsanity.com/paper/2111.15537/) (2021)
- [From Gradient Flow on Population Loss to Learning with Stochastic Gradient Descent](https://mlsanity.com/paper/2210.06705/) (2022)
- [Learning Zero-Sum Linear Quadratic Games with Improved Sample Complexity and Last-Iterate Convergence](https://mlsanity.com/paper/2309.04272/) (2023)
- [Optimization Landscape of Policy Gradient Methods for Discrete-time Static Output Feedback](https://mlsanity.com/paper/2310.19022/) (2023)
- [Connected Superlevel Set in (Deep) Reinforcement Learning and its Application to Minimax Theorems](https://mlsanity.com/paper/2303.12981/) (2023)
- [Model-Free Learning for the Linear Quadratic Regulator over Rate-Limited Channels](https://mlsanity.com/paper/2401.01258/) (2024)
- [On the Complexity of Finite-Sum Smooth Optimization under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/2402.02569/) (2024)
- [Linear Dynamics meets Linear MDPs: Closed-Form Optimal Policies via Reinforcement Learning](https://mlsanity.com/paper/2508.17185/) (2025)
- [On the Optimization Landscape of Dynamic Output Feedback: A Case Study for Linear Quadratic Regulator](https://mlsanity.com/paper/2209.05042/) (2022)
- [Asynchronous Parallel Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/2407.03233/) (2024)
- [Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems](https://mlsanity.com/paper/2503.23641/) (2025)
- [On the Robustness of Derivative-free Methods for Linear Quadratic Regulator](https://mlsanity.com/paper/2506.12596/) (2025)
- [Policy Gradient Method for LQG Control via Input-Output-History Representation: Convergence to O(ε)-Stationary Points](https://mlsanity.com/paper/2510.19141/) (2025)
- [Model-Free Output Feedback Stabilization via Policy Gradient Methods](https://mlsanity.com/paper/2601.19284/) (2026)
- [Gradient Dominance in the Linear Quadratic Regulator: A Unified Analysis for Continuous-Time and Discrete-Time Systems](https://mlsanity.com/paper/2602.22577/) (2026)
- [Accelerated Optimization Landscape of Linear-Quadratic Regulator](https://mlsanity.com/paper/2307.03590/) (2023)
- [Model-Free μ-Synthesis: A Nonsmooth Optimization Perspective](https://mlsanity.com/paper/2402.11654/) (2024)
- [Convergence of Policy Gradient for Stochastic Linear-Quadratic Control Problem in Infinite Horizon](https://mlsanity.com/paper/2404.11382/) (2024)
- [Some remarks on gradient dominance and LQR policy optimization](https://mlsanity.com/paper/2507.10452/) (2025)
- [Actor-Critic Learning for Risk-Constrained Linear Quadratic Regulation](https://mlsanity.com/paper/2510.22267/) (2025)
- [Learning Suffers More Than the Policy Class Under Partial Observability: A Closed-Form Analysis](https://mlsanity.com/paper/2608.07228/) (2026)
- [(Cheap) Stochastic Policy Gradient Converges with High Probability for Linear Quadratic Regulator](https://mlsanity.com/paper/2609.19712/) (2026)

## Abstract (arXiv)

The linear quadratic regulator is the fundamental problem of optimal control. Its state feedback version was set and solved in the early 1960s. However the static output feedback problem has no explicit-form solution. It is suggested to look at both of them from another point of view as matrix optimization problems, where the variable is a feedback matrix gain. The properties of such a function are investigated, it turns out to be smooth, but not convex, with possible non-connected domain. Nevertheless, the gradient method for it with the special step-size choice converges to the optimal solution in the state feedback case and to a stationary point in the output feedback case. The results can be extended for the general framework of unconstrained optimization and for reduced gradient method for minimization with equality-type constraints.