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An Overview of Weighted and Unconstrained Scalarizing Functions

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Book cover Evolutionary Multi-Criterion Optimization (EMO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10173))

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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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Notes

  1. 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. 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. 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. 4.

    Although the French spelling Tchebycheff is the most preferred, the proper English transliteration is Chebyshev.

  5. 5.

    Available at http://computacion.cs.cinvestav.mx/~rhernandez.

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Acknowledgments

The authors gratefully acknowledge support from CONACyT project no. 221551 and PCCI140054 CONACYT/CONICYT project.

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Authors and Affiliations

  1. ESCOM, Instituto Politécnico Nacional, 07738, Ciudad de México, Mexico

    Miriam Pescador-Rojas

  2. Computer Science Department, CINVESTAV-IPN (Evolutionary Computation Group), 07360, Ciudad de México, Mexico

    Raquel Hernández Gómez & Carlos A. Coello Coello

  3. Universidad Técnica Federico Santa María, Avenida España, 1680, Valparaíso, Chile

    Elizabeth Montero, Nicolás Rojas-Morales & María-Cristina Riff

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  1. Miriam Pescador-Rojas
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  2. Raquel Hernández Gómez
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  3. Elizabeth Montero
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  4. Nicolás Rojas-Morales
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  5. María-Cristina Riff
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  6. Carlos A. Coello Coello
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Correspondence to Raquel Hernández Gómez .

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Editors and Affiliations

  1. University of Münster, Münster, Germany

    Heike Trautmann

  2. TU Dortmund University, Dortmund, Germany

    Günter Rudolph

  3. University of Wuppertal, Wuppertal, Germany

    Kathrin Klamroth

  4. CINVESTAV-IPN, Mexico City, Mexico

    Oliver Schütze

  5. Clemson University, Clemson, South Carolina, USA

    Margaret Wiecek

  6. University of Surrey, Guildford, United Kingdom

    Yaochu Jin

  7. University of Münster, Münster, Germany

    Christian Grimme

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Pescador-Rojas, M., Hernández Gómez, R., Montero, E., Rojas-Morales, N., Riff, MC., Coello Coello, C.A. (2017). An Overview of Weighted and Unconstrained Scalarizing Functions. In: Trautmann, H., et al. Evolutionary Multi-Criterion Optimization. EMO 2017. Lecture Notes in Computer Science(), vol 10173. Springer, Cham. https://doi.org/10.1007/978-3-319-54157-0_34

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