A Multiresolution approach to solve large-scale optimization problems

TL;DR

This paper introduces a multilevel, multiresolution framework to accelerate large-scale convex optimization by solving a sequence of smaller, auxiliary problems, significantly reducing computational costs.

Contribution

It presents a novel multiresolution approach embedded in Harten's framework, enabling efficient large-scale optimization with theoretical accuracy guarantees.

Findings

Reduces computational cost for large-scale problems

Provides theoretical bounds relating approximation accuracy to solution quality

Demonstrates effectiveness through numerical experiments

Abstract

General purpose optimization techniques can be used to solve many problems in engineering computations, although their cost is often prohibitive when the number of degrees of freedom is very large. We describe a multilevel approach to speed up the computation of the solution of a large-scale optimization problem by a given optimization technique. By embedding the problem within Harten's Multiresolution Framework (MRF), we set up a procedure that leads to the desired solution, after the computation of a finite sequence of sub-optimal solutions, which solve auxiliary optimization problems involving a smaller number of variables. For convex optimization problems having smooth solutions, we prove that the distance between the optimal solution and each sub-optimal approximation is related to the accuracy of the interpolation technique used within the MRF and analyze its relation with the performance of the proposed algorithm. Several numerical experiments confirm that our technique provides a computationally efficient strategy that allows the end user to treat both the optimizer and the objective function as black boxes throughout the optimization process.