# Merging Multigrid Optimization with SESOP

## TL;DR

This paper introduces a novel optimization method that combines SESOP and multigrid techniques, enhancing convergence efficiency for quadratic function optimization through a hybrid approach and optimal parameter tuning.

## Contribution

It presents a new combined framework of SESOP and multigrid optimization, including convergence analysis and parameter optimization for quadratic problems.

## Key findings

- Numerical experiments show improved optimization performance.
- Derived approximately optimal fixed parameters for SESOP-TG.
- Effective reduction in computational overhead for quadratic functions.

## Abstract

A merger of two optimization frameworks is introduced: SEquential Subspace OPtimization (SESOP) with MultiGrid (MG) optimization. At each iteration of the algorithm, the search direction implied by the coarse-grid correction process of MG is added to the low dimensional search-space of SESOP, which includes the preconditioned gradient and search directions involving the previous iterates, called {\em history}. Numerical experiments demonstrate the effectiveness of this approach. We then study the asymptotic convergence factor of the two-level version of SESOP-MG (dubbed SESOP-TG) for optimization of quadratic functions, and derive approximately optimal fixed parameters, which may reduce the computational overhead for such problems significantly.