Newton-type multilevel optimization method

Abstract

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impressive performance of multilevel optimization methods is an empirical observation, and no theoretical explanation has so far been proposed. In order to address this issue, we study the convergence properties of a multilevel method that is motivated by second-order methods. We take the first step toward establishing how the structure of an optimization problem is related to the convergence rate of multilevel algorithms.

Non‐parametric measure approximations for constrained multi‐objective optimisation under uncertainty

“…As for the constrained case, definition (6) shows that for a to dominate b, either b must be in the failure domain, or a must be in the admissible domain and dominate b according to the classical Pareto dominance rule. The integral in (8) can be split up depending on b belonging to the failure or admissible domain, and an additional dominance constraint can be added in the latest case. Hence, the domination probability reads:…”

“…In this context, some parsimonious techniques rely on the assumption that there exist either different levels of accuracy for computing the Quantities of Interest. Namely, multi-level techniques 8,9 usually involve several mesh refinements for numerical simulation, and multi-fidelity approaches [10][11][12] use different modelling strategies. Recent works have also proposed adaptive approaches for tackling problems where accuracy can be tuned online.…”