# A New Two-dimensional Model-based Subspace Method for Large-scale Unconstrained Derivative-free Optimization: 2D-MoSub

arXiv: https://arxiv.org/abs/2309.14855 · HTML: https://mlsanity.com/paper/2309.14855/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Sep 2023.

## Built on by (most cited first, 2 of 2)

- [MpSub: A Momentum p-Dimensional Subspace Trust-Region Method for Derivative-Free Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2609.07666/) (2026)
- [Derivative-Free Structured Updates for Muon](https://mlsanity.com/paper/2609.17759/) (2026)

## Abstract (arXiv)

This paper proposes the method 2D-MoSub (2-dimensional model-based subspace method), which is a novel derivative-free optimization (DFO) method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems. Our method combines 2-dimensional quadratic interpolation models and trust-region techniques to iteratively update the points and explore the 2-dimensional subspace. Its framework includes initialization, constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. We introduce the framework and computational details of 2D-MoSub, and discuss the poisedness and quality of the interpolation set in the corresponding 2-dimensional subspace. We also analyze some properties of our method, including the model's approximation error with projection property and the algorithm's convergence. Numerical results demonstrate the effectiveness and efficiency of 2D-MoSub for solving a variety of unconstrained optimization problems.