# Multilevel Geometric Optimization for Regularised Constrained Linear Inverse Problems

arXiv: https://arxiv.org/abs/2207.04934 · HTML: https://mlsanity.com/paper/2207.04934/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 43 references that are in the map.
- First posted on arXiv 11 Jul 2022.

## Builds on (most cited first, 10 of 10)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [Accelerated Bregman Proximal Gradient Methods for Relatively Smooth Convex Optimization](https://mlsanity.com/paper/1808.03045/) (2018)
- [Analysis of Optimization Algorithms via Integral Quadratic Constraints: Nonstrongly Convex Problems](https://mlsanity.com/paper/1705.03615/) (2017)
- [Image Labeling by Assignment](https://mlsanity.com/paper/1603.05285/) (2016)
- [A first-order multigrid method for bound-constrained convex optimization](https://mlsanity.com/paper/1602.03771/) (2016)
- [Performance Bounds For Co-/Sparse Box Constrained Signal Recovery](https://mlsanity.com/paper/1812.10471/) (2018)
- [Superiorization vs. Accelerated Convex Optimization: The Superiorized/Regularized Least-Squares Case](https://mlsanity.com/paper/1911.05498/) (2019)
- [A Multilevel Method for Self-Concordant Minimization](https://mlsanity.com/paper/2106.13690/) (2021)

## Abstract (arXiv)

We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are accurate but expensive to compute, while coarser models are less accurate but cheaper to compute. When working at the fine level, multilevel optimisation computes the search direction based on a coarser model which speeds up updates at the fine level. Moreover, exploiting geometry induced by the hierarchy the feasibility of the updates is preserved. In particular, our approach extends classical components of multigrid methods like restriction and prolongation to the Riemannian structure of our constraints.