# Dynamic Programming Through the Lens of Semismooth Newton-Type Methods (Extended Version)

arXiv: https://arxiv.org/abs/2203.08678 · HTML: https://mlsanity.com/paper/2203.08678/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2022.

## Built on by (most cited first, 7 of 7)

- [From Optimization to Control: Quasi Policy Iteration](https://mlsanity.com/paper/2311.11166/) (2023)
- [Operator Splitting for Convex Constrained Markov Decision Processes](https://mlsanity.com/paper/2412.14002/) (2024)
- [Rank-One Modified Value Iteration](https://mlsanity.com/paper/2505.01828/) (2025)
- [Warm-Start Actor-Critic: From Approximation Error to Sub-optimality Gap](https://mlsanity.com/paper/2306.11271/) (2023)
- [Beyond the Bellman Fixed Point: Geometry and Fast Policy Identification in Value Iteration](https://mlsanity.com/paper/2604.17457/) (2026)
- [Bridging the Gap between Newton-Raphson Method and Regularized Policy Iteration](https://mlsanity.com/paper/2310.07211/) (2023)
- [Control and Reinforcement Learning through the Lens of Optimization: An Algorithmic Perspective](https://mlsanity.com/paper/2512.07377/) (2025)

## Abstract (arXiv)

Policy iteration and value iteration are at the core of many (approximate) dynamic programming methods. For Markov Decision Processes with finite state and action spaces, we show that they are instances of semismooth Newton-type methods to solve the Bellman equation. In particular, we prove that policy iteration is equivalent to the exact semismooth Newton method and enjoys local quadratic convergence rate. This finding is corroborated by extensive numerical evidence in the fields of control and operations research, which confirms that policy iteration generally requires few iterations to achieve convergence even when the number of policies is vast. We then show that value iteration is an instance of the fixed-point iteration method. In this spirit, we develop a novel locally accelerated version of value iteration with global convergence guarantees and negligible extra computational costs.