# On high-order multilevel optimization strategies

## TL;DR

This paper introduces a multilevel framework for high-order unconstrained optimization, reducing computational costs by leveraging problem hierarchies, with theoretical convergence guarantees and practical efficiency demonstrated through numerical experiments.

## Contribution

It extends high-order optimization methods with a multilevel approach, providing convergence analysis and demonstrating computational savings over traditional methods.

## Key findings

- The multilevel methods achieve convergence to stationary points.
- Numerical results show significant computational savings.
- Theoretical analysis confirms convergence and complexity bounds.

## Abstract

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q >= 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.