# Trilevel and Multilevel Optimization using Monotone Operator Theory

arXiv: https://arxiv.org/abs/2105.09407 · HTML: https://mlsanity.com/paper/2105.09407/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 38 references that are in the map.
- First posted on arXiv 19 May 2021.

## Builds on (most cited first, 3 of 3)

- [A First Order Method for Solving Convex Bi-Level Optimization Problems](https://mlsanity.com/paper/1702.03999/) (2017)
- [Linear and strong convergence of algorithms involving averaged nonexpansive operators](https://mlsanity.com/paper/1402.5460/) (2014)
- [Convergence rate analysis for averaged fixed point iterations in the presence of H"older regularity](https://mlsanity.com/paper/1510.06823/) (2015)

## Built on by (most cited first, 6 of 6)

- [Efficient Dataset Distillation via Minimax Diffusion](https://mlsanity.com/paper/2311.15529/) (2023)
- [A Gradient Method for Multilevel Optimization](https://mlsanity.com/paper/2105.13954/) (2021)
- [A stochastic gradient method for trilevel optimization](https://mlsanity.com/paper/2505.06805/) (2025)
- [Non-Stationary Functional Bilevel Optimization](https://mlsanity.com/paper/2601.15363/) (2026)
- [Towards Differentiable Multilevel Optimization: A Gradient-Based Approach](https://mlsanity.com/paper/2410.11312/) (2024)
- [Mission-Aligned Learning-Informed Control of Autonomous Systems: Formulation and Foundations](https://mlsanity.com/paper/2507.04356/) (2025)

## Abstract (arXiv)

We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term.~Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.