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On the Effect of Scalarising Norm Choice in a ParEGO implementation Read chapter Read first chapter

2017 | OriginalPaper | Chapter

An Overview of Weighted and Unconstrained Scalarizing Functions

Authors : Miriam Pescador-Rojas, Raquel Hernández Gómez, Elizabeth Montero, Nicolás Rojas-Morales, María-Cristina Riff, Carlos A. Coello Coello

Published in: Evolutionary Multi-Criterion Optimization

Publisher: Springer International Publishing

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Abstract

Scalarizing functions play a crucial role in multi-objective evolutionary algorithms (MOEAs) based on decomposition and the R2 indicator, since they guide the population towards nearly optimal solutions, assigning a fitness value to an individual according to a predefined target direction in objective space. This paper presents a general review of weighted scalarizing functions without constraints, which have been proposed not only within evolutionary multi-objective optimization but also in the mathematical programming literature. We also investigate their scalability up to 10 objectives, using the test problems of Lamé Superspheres on the MOEA/D and MOMBI-II frameworks. For this purpose, the best suited scalarizing functions and their model parameters are determined through the evolutionary calibrator EVOCA. Our experimental results reveal that some of these scalarizing functions are quite robust and suitable for handling many-objective optimization problems.

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previous chapter Predator-Prey Techniques for Solving Multiobjective Scheduling Problems for Unrelated Parallel Machines
next chapter Multi-objective Representation Setups for Deformation-Based Design Optimization
Footnotes
1
A solution \(\mathbf {x} \in \mathcal {S}\) dominates a solution \(\mathbf {y} \in \mathcal {S}\) (\(\mathbf {x} \prec \mathbf {y}\)), if and only if \(\forall i \in \left\{ 1,\ldots ,m\right\} \), \(f_{i}(\mathbf {x}) \le f_{i}(\mathbf {y}) \) and \(\exists j \in \left\{ 1,\ldots ,m\right\} \), \(f_{j}(\mathbf {x}) < f_{j}(\mathbf {y})\).
 
2
\(POF :=\{\mathbf {F}(\mathbf {x}) \in \mathbb {R}^m :\mathbf {x} \in \mathcal {S}, \not {\exists } \mathbf {y} \in \mathcal {S}, \mathbf {y} \prec \mathbf {x} \}.\)
 
3
Let be \(\mathbf {x}, \mathbf {y} \in \mathcal {S}\). It is said that \(\mathbf {x}\) is Pareto optimal if there is no \(\mathbf {y}\) such that \(\mathbf {y} \prec \mathbf {x}\). \(\mathbf {x}\) is weakly Pareto optimal if there is no \(\mathbf {y}\) such that \(\forall i \in \left\{ 1,\ldots ,m\right\} \), \(f_{i}(\mathbf {y}) < f_{i}(\mathbf {x})\).
 
4
Although the French spelling Tchebycheff is the most preferred, the proper English transliteration is Chebyshev.
 
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Metadata
Title
An Overview of Weighted and Unconstrained Scalarizing Functions
Authors
Miriam Pescador-Rojas
Raquel Hernández Gómez
Elizabeth Montero
Nicolás Rojas-Morales
María-Cristina Riff
Carlos A. Coello Coello
Copyright Year
2017
Book
Evolutionary Multi-Criterion Optimization

Print ISBN: 978-3-319-54156-3

Electronic ISBN: 978-3-319-54157-0

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-54157-0_34

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