# Black-box optimization for integer-variable problems using Ising machines and factorization machines

arXiv: https://arxiv.org/abs/2209.01016 · HTML: https://mlsanity.com/paper/2209.01016/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 1 Sep 2022.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Traffic flow optimization using a quantum annealer](https://mlsanity.com/paper/1708.01625/) (2017)
- [Control of automated guided vehicles without collision by quantum annealer and digital devices](https://mlsanity.com/paper/1812.01532/) (2018)
- [Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer](https://mlsanity.com/paper/1508.06182/) (2015)
- [Item Listing Optimization for E-commerce Websites based on Diversity](https://mlsanity.com/paper/1903.12478/) (2019)
- [A Review of Machine Learning Classification Using Quantum Annealing for Real-world Applications](https://mlsanity.com/paper/2106.02964/) (2021)

## Built on by (most cited first, 8 of 8)

- [Function Smoothing Regularization for Precision Factorization Machine Annealing in Continuous Variable Optimization Problems](https://mlsanity.com/paper/2407.04393/) (2024)
- [Real-Time Black-Box Optimization for Dynamic Discrete Environments Using Embedded Ising Machines](https://mlsanity.com/paper/2506.16924/) (2025)
- [Efficient Bit Labeling in Factorization Machines with Annealing for Traveling Salesman Problem](https://mlsanity.com/paper/2407.02091/) (2024)
- [Subsampling Factorization Machine Annealing](https://mlsanity.com/paper/2508.08778/) (2025)
- [Factorization Machine with Quadratic-Optimization Annealing for RNA Inverse Folding and Evaluation of Binary-Integer Encoding and Nucleotide Assignment](https://mlsanity.com/paper/2602.16643/) (2026)
- [A general learning scheme for classical and quantum Ising machines](https://mlsanity.com/paper/2310.18411/) (2023)
- [Initialization Method for Factorization Machine Based on Low-Rank Approximation for Constructing a Corrected Approximate Ising Model](https://mlsanity.com/paper/2410.12747/) (2024)
- [Stage-dependent integer-binary encoding in factorization-machine black-box optimization](https://mlsanity.com/paper/2606.23188/) (2026)

## Abstract (arXiv)

Black-box optimization has potential in numerous applications such as hyperparameter optimization in machine learning and optimization in design of experiments. Ising machines are useful for binary optimization problems because variables can be represented by a single binary variable of Ising machines. However, conventional approaches using an Ising machine cannot handle black-box optimization problems with non-binary values. To overcome this limitation, we propose an approach for integer-variable black-box optimization problems by using Ising/annealing machines and factorization machines in cooperation with three different integer-encoding methods. The performance of our approach is numerically evaluated with different encoding methods using a simple problem of calculating the energy of the hydrogen molecule in the most stable state. The proposed approach can calculate the energy using any of the integer-encoding methods. However, one-hot encoding is useful for problems with a small size.