A first-order multigrid method for bound-constrained convex optimization

TL;DR

This paper introduces a first-order multigrid method tailored for large-scale bound-constrained convex optimization problems, emphasizing efficiency and avoiding second derivative computations.

Contribution

It presents a novel multigrid approach using only first-order information, suitable for large-scale problems with bound constraints and possibly a single equality constraint.

Findings

Efficient multigrid method developed for bound-constrained convex optimization

Method avoids second derivatives, reducing computational cost

Demonstrates computational efficiency through analysis and experiments

Abstract

The aim of this paper is to design an efficient multigrid method for constrained convex optimization problems arising from discretization of some underlying infinite dimensional problems. Due to problem dependency of this approach, we only consider bound constraints with (possibly) a single equality constraint. As our aim is to target large-scale problems, we want to avoid computation of second derivatives of the objective function, thus excluding Newton like methods. We propose a smoothing operator that only uses first-order information and study the computational efficiency of the resulting method.