# Model-free stochastic linear quadratic control for discrete-time systems with multiplicative and additive noises via semidefinite programming

arXiv: https://arxiv.org/abs/2512.20911 · HTML: https://mlsanity.com/paper/2512.20911/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 4 of 4)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Model-free optimal control of discrete-time systems with additive and multiplicative noises](https://mlsanity.com/paper/2008.08734/) (2020)
- [Reinforcement Learning for Adaptive Optimal Stationary Control of Linear Stochastic Systems](https://mlsanity.com/paper/2107.07788/) (2021)
- [Robust Reinforcement Learning: A Case Study in Linear Quadratic Regulation](https://mlsanity.com/paper/2008.11592/) (2020)

## Abstract (arXiv)

This paper investigates a model-free solution to the stochastic linear quadratic regulation (LQR) problem for linear discrete-time systems with both multiplicative and additive noises. We formulate the stochastic LQR problem as a nonconvex optimization problem and rigorously analyze its dual problem structure. By exploiting the inherent convexity of the dual problem and analyzing Karush-Kuhn-Tucker conditions with respect to optimality in convex optimization, we establish an explicit relationship between the optimal point of the dual problem and the parameters of the associated Q-function. This theoretical insight, combined with the technique of the matrix direct sum, makes it possible to develop a novel model-free sample-efficient, non-iterative semidefinite programming algorithm that directly estimates optimal control gain without requiring an initial stabilizing controller, or noises measurability. The robustness of the model-free SDP method to errors is investigated. Our approach provides a new optimization-theoretic framework for understanding Q-learning algorithms while advancing the theoretical foundations of reinforcement learning in stochastic optimal control. Numerical validation on a pulse-width modulated inverter system demonstrates the algorithm's effectiveness, particularly in achieving a single-step non-iterative solution without hyper-parameter tuning.