# Self-Adaptive Ising Machines for Constrained Optimization

arXiv: https://arxiv.org/abs/2501.04971 · HTML: https://mlsanity.com/paper/2501.04971/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 32 references that are in the map.
- First posted on arXiv 9 Jan 2025.

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

- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Massively Parallel Probabilistic Computing with Sparse Ising Machines](https://mlsanity.com/paper/2110.02481/) (2021)
- [Efficient Optimization with Higher-Order Ising Machines](https://mlsanity.com/paper/2212.03426/) (2022)
- [CMOS + stochastic nanomagnets: heterogeneous computers for probabilistic inference and learning](https://mlsanity.com/paper/2304.05949/) (2023)

## Built on by (most cited first, 1 of 1)

- [Probabilistic Algorithms for Ising Machines from Optimization to Generative AI](https://mlsanity.com/paper/2610.03972/) (2026)

## Abstract (arXiv)

Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.