TY - JOUR
T1 - Optimizing functionals using Differential Evolution
AU - Cantún-Avila, K. B.
AU - González-Sánchez, D.
AU - Díaz-Infante, S.
AU - Peñuñuri, F.
N1 - Publisher Copyright:
© 2020 Elsevier Ltd
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2021/1
Y1 - 2021/1
N2 - Metaheuristic algorithms are typically used for optimizing a function f:A→R, where A is a subset of RN. Nevertheless, many real-life problems require A to be a set of functions which makes f a functional. In this paper, we present a methodology to address the optimization of functionals by using the evolutionary algorithm known as Differential Evolution. Unlike traditional techniques where continuity and differentiability assumptions are required to solve some associated differential equations—like calculus of variations, Pontryagin's principle or dynamic programming, the optimization is carried out directly on the functional without the need of any of the assumptions mentioned before. Lagrangians involving derivatives are considered, these derivatives are computed implementing Automatic Differentiation with dual numbers. To the best of our knowledge, this is the first time that a metaheuristic optimization approach has been applied to directly optimize a broad variety of functionals. The effectiveness of our methodology is validated by solving two problems. The first problem is related to the implementation of quarantine and isolation in SARS epidemics and the second validation problem deals with the well-known brachistochrone curve problem. The results of both validation problems are in outstanding agreement with those obtained with the application of traditional techniques, specifically with the Forward–Backward-Sweep method in the first problem, and with the calculus of variations for the latter problem. We also found that interpolation may be employed to solve the large scale global optimization problems arisen in the optimization of functionals.
AB - Metaheuristic algorithms are typically used for optimizing a function f:A→R, where A is a subset of RN. Nevertheless, many real-life problems require A to be a set of functions which makes f a functional. In this paper, we present a methodology to address the optimization of functionals by using the evolutionary algorithm known as Differential Evolution. Unlike traditional techniques where continuity and differentiability assumptions are required to solve some associated differential equations—like calculus of variations, Pontryagin's principle or dynamic programming, the optimization is carried out directly on the functional without the need of any of the assumptions mentioned before. Lagrangians involving derivatives are considered, these derivatives are computed implementing Automatic Differentiation with dual numbers. To the best of our knowledge, this is the first time that a metaheuristic optimization approach has been applied to directly optimize a broad variety of functionals. The effectiveness of our methodology is validated by solving two problems. The first problem is related to the implementation of quarantine and isolation in SARS epidemics and the second validation problem deals with the well-known brachistochrone curve problem. The results of both validation problems are in outstanding agreement with those obtained with the application of traditional techniques, specifically with the Forward–Backward-Sweep method in the first problem, and with the calculus of variations for the latter problem. We also found that interpolation may be employed to solve the large scale global optimization problems arisen in the optimization of functionals.
KW - Automatic differentiation
KW - Cubic spline interpolation
KW - Differential Evolution
KW - Functional optimization
KW - Large-scale global optimization
UR - http://www.scopus.com/inward/record.url?scp=85095976852&partnerID=8YFLogxK
U2 - 10.1016/j.engappai.2020.104086
DO - 10.1016/j.engappai.2020.104086
M3 - Artículo
AN - SCOPUS:85095976852
SN - 0952-1976
VL - 97
JO - Engineering Applications of Artificial Intelligence
JF - Engineering Applications of Artificial Intelligence
M1 - 104086
ER -