# A Dynamical Systems Perspective on Discrete Optimization

arXiv: https://arxiv.org/abs/2305.08536 · HTML: https://mlsanity.com/paper/2305.08536/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2023.

## Built on by (most cited first, 3 of 3)

- [Distributed Event-Based Learning via ADMM](https://mlsanity.com/paper/2405.10618/) (2024)
- [Controlling Participation in Federated Learning with Feedback](https://mlsanity.com/paper/2411.19242/) (2024)
- [A Systems-Theoretic View on the Convergence of Algorithms under Disturbances](https://mlsanity.com/paper/2512.17598/) (2025)

## Abstract (arXiv)

We discuss a dynamical systems perspective on discrete optimization. Departing from the fact that many combinatorial optimization problems can be reformulated as finding low energy spin configurations in corresponding Ising models, we derive a penalized rank-two relaxation of the Ising formulation. It turns out that the associated gradient flow dynamics exactly correspond to a type of hardware solvers termed oscillator-based Ising machines. We also analyze the advantage of adding angle penalties by leveraging random rounding techniques. Therefore, our work contributes to a rigorous understanding of oscillator-based Ising machines by drawing connections to the penalty method in constrained optimization and providing a rationale for the introduction of sub-harmonic injection locking. Furthermore, we characterize a class of coupling functions between oscillators, which ensures convergence to discrete solutions. This class of coupling functions avoids explicit penalty terms or rounding schemes, which are prevalent in other formulations.