Citation: Zhang, X.; Qian, F.; Zhang,
L. Cluster-Based Regression Transfer
Learning for Dynamic
Multi-Objective Optimization.
Processes 2023, 11, 613. https://
doi.org/10.3390/pr11020613
Academic Editor: Anthony Rossiter
Received: 30 January 2023
Revised: 9 February 2023
Accepted: 15 February 2023
Published: 17 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Cluster-Based Regression Transfer Learning for Dynamic
Multi-Objective Optimization
Xi Zhang
1
, Feng Qian
1,∗
and Liping Zhang
2,∗
1
Key Laboratory of Smart Manufacturing in Energy Chemical Process, East China University of Science and
Technology, Shanghai 200237, China
2
Institute of Cotton Research, Shanxi Agricultural University, Yuncheng 044000, China
* Correspondence: fqian@ecust.edu.cn (F.Q.); lisazhang200601@163.com (L.Z.)
Abstract:
Many multi-objective optimization problems in the real world have conflicting objectives,
and these objectives change over time, known as dynamic multi-objective optimization problems
(DMOPs). In recent years, transfer learning has attracted growing attention to solve DMOPs, since
it is capable of leveraging historical information to guide the evolutionary search. However, there
is still much room for improvement in the transfer effect and the computational efficiency. In
this paper, we propose a cluster-based regression transfer learning-based dynamic multi-objective
evolutionary algorithm named CRTL-DMOEA. It consists of two components, which are the cluster-
based selection and cluster-based regression transfer. In particular, once a change occurs, we employ
a cluster-based selection mechanism to partition the previous Pareto optimal solutions and find
the clustering centroids, which are then fed into autoregression prediction model. Afterwards, to
improve the prediction accuracy, we build a strong regression transfer model based on TrAdaboost.R2
by taking advantage of the clustering centroids. Finally, a high-quality initial population for the new
environment is predicted with the regression transfer model. Through a comparison with some chosen
state-of-the-art algorithms, the experimental results demonstrate that the proposed CRTL-DMOEA is
capable of improving the performance of dynamic optimization on different test problems.
Keywords: dynamic multi-objective optimization; evolutionary algorithm; regression transfer;
transfer learning
1. Introduction
In the real world, many multi-objective optimization problems [
1
] have multiple
conflicting objectives that may change over time. Such problems are called dynamic multi-
objective optimization problems (DMOPs) [
2
]. In recent years, the research on solving
DMOPs has attracted more and more researchers and there have been lots of optimization
methods developed [
3
–
5
]. Multi-objective evolutionary algorithms (MOEAs) have been
widely applied to solve DMOPs in various areas, such as wireless sensor networks [
6
],
financial optimization problems [
7
], path planning [
8
] and so on. When applied to solve
DMOPs, traditional MOEAs [
9
–
12
] should be improved to adapt to the dynamisms, which
are capable of tracking the changing Pareto optimal fronts (POFs) and providing a diverse
set of Pareto optimal solutions (POSs) over time.
To solve DMOPs, there are various kinds of dynamic MOEAs (DMOEAs) in the
literature, which can be categorized as follows: diversity approaches [
13
–
15
], memory
mechanisms [
16
–
18
], and prediction-based methods [
19
–
21
]. Generally, the diversity ap-
proaches include increasing diversity [
22
], maintaining diversity [
15
], and multi-population
strategy [
23
]. More specifically, the environmental adaption of population diversity can
be addressed with increasing diversity by adding variety to the population after the de-
tection of a change, maintaining diversity by avoiding population convergence to track
the time-varying POS throughout the run, or dividing the population into some different
Processes 2023, 11, 613. https://doi.org/10.3390/pr11020613 https://www.mdpi.com/journal/processes
