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The Directed Search Method for Unconstrained Parameter Dependent Multi-objective Optimization Problems

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NEO 2015

Part of the book series: Studies in Computational Intelligence ((SCI,volume 663))

Abstract

In this chapter we present the adaptions of the recently proposed Directed Search method to the context of unconstrained parameter dependent multi-objective optimization problems (PMOPs). The new method, called \(\lambda \)-DS, is capable of performing a movement both toward and along the solution set of a given differentiable PMOP. We first discuss the basic variants of the method that use gradient information and describe subsequently modifications that allow for a gradient free realization. Finally, we show that \(\lambda \)-DS can be used to understand the behavior of stochastic local search within PMOPs to a certain extent which might be interesting for the development of future local search engines, or evolutionary strategies, for the treatment of such problems. We underline all our statements with several numerical results indicating the strength of the novel approach.

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Notes

  1. 1.

    Named after the works of Karush [18] and Kuhn and Tucker [19].

  2. 2.

    If the rank of a matrix \(A\in \mathbb {R}^{m\times n}\), \(m\le n\), is m (i.e., maximal), its pseudo inverse is given by \(A^+ = A^T(AA^T)^{-1}\in \mathbb {R}^{n\times m}\).

  3. 3.

    Furthermore, we note that the original idea of NBI for a given MOP is not to maximize the distance from \(F(x_0)\) for a given point \(x_0\), but this is a straightforward adaption to the current context to steer the search in a given direction d.

  4. 4.

    We consider in our computations only the case \(l=1\).

  5. 5.

    Diverse neighborhood relationships can be established, in this work we induced it through the Euclidean distance.

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Acknowledgments

A. Sosa acknowledges support from the Conacyt to pursue his Ph.D. studies at the CINVESTAV-IPN. A. Lara acknowledges support from project SIP20162103. H. Trautmann acknowledges support from the European Center of Information Systems (ERCIS). All authors acknowledge support from DAAD project no. 57065955.

Author information

Authors and Affiliations

  1. Computer Science Department, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, 07360, Mexico City, Mexico

    Víctor Adrián Sosa Hernández & Oliver Schütze

  2. Escuela Superior de Física y Matemáticas del IPN, Mexico City, Mexico

    Adriana Lara

  3. Department of Information Systems, University of Münster, Leonardo-Campus 3, 48149, Munster, Germany

    Heike Trautmann

  4. Fakultät Für Informatik, Technische Universität Dortmund, 44221, Dortmund, Germany

    Günter Rudolph

Authors
  1. Víctor Adrián Sosa Hernández
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  2. Adriana Lara
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  3. Heike Trautmann
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  4. Günter Rudolph
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  5. Oliver Schütze
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Corresponding author

Correspondence to Víctor Adrián Sosa Hernández .

Editor information

Editors and Affiliations

  1. Departamento de Comput Cinvestav-IPN, Mexico, Mexico

    Oliver Schütze

  2. Posgrado en Ciencias de la Ingeniería, Instituto Tecnológico de Tijuana, Tijuana, Baja California, Mexico

    Leonardo Trujillo

  3. Université de Bordeaux, Bordeaux, France

    Pierrick Legrand

  4. Posgrado en Ciencias de la Ingeniería, Instituto Tecnológico de Tijuana, Tijuana, Mexico

    Yazmin Maldonado

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Adrián Sosa Hernández, V., Lara, A., Trautmann, H., Rudolph, G., Schütze, O. (2017). The Directed Search Method for Unconstrained Parameter Dependent Multi-objective Optimization Problems. In: Schütze, O., Trujillo, L., Legrand, P., Maldonado, Y. (eds) NEO 2015. Studies in Computational Intelligence, vol 663. Springer, Cham. https://doi.org/10.1007/978-3-319-44003-3_12

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  • DOI: https://doi.org/10.1007/978-3-319-44003-3_12

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