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Preference representation using Gaussian functions on a hyperplane in evolutionary multi-objective optimization

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Abstract

Many-objective optimization has attracted much attention in evolutionary multi-objective optimization (EMO). This is because EMO algorithms developed so far often degrade their search ability for optimization problems with four or more objectives, which are frequently referred to as many-objective problems. One of promising approaches to handle many objectives is to incorporate the preference of a decision maker (DM) into EMO algorithms. With the preference, EMO algorithms can focus the search on regions preferred by the DM, resulting in solutions close to the Pareto front around the preferred regions. Although a number of preference-based EMO algorithms have been proposed, it is not trivial for the DM to reflect his/her actual preference in the search. We previously proposed to represent the preference of the DM using Gaussian functions on a hyperplane. The DM specifies the center and spread vectors of the Gaussian functions so as to represent his/her preference. The preference handling is integrated into the framework of NSGA-II. This paper extends our previous work so that obtained solutions follow the distribution of Gaussian functions specified. The performance of our proposed method is demonstrated mainly for benchmark problems and real-world applications with a few objectives in this paper. We also show the applicability of our method to many-objective problems.

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Notes

  1. It should be noted that optimization problems with more than three objectives are called many-objective problems in this paper as there is no clear definition on the terminology in the literature.

  2. For maximization problems, the following conditions must be satisfied for \(\mathbf{a}\) dominating \(\mathbf{b}\). \(\forall i: f_i(\mathbf{a}) \ge f_i(\mathbf{b})\ \mathrm {and\ } \exists j: f_j(\mathbf{a}) > f_j(\mathbf{b}).\)

  3. A solution is weakly Pareto optimal if there does not exist any other solutions for which all the objective functions are better (Miettinen 1999).

  4. R-NSGA-II allows only one solution to exist within a distance of \(\epsilon \) in the objective space.

  5. The total number of DTLZ problems depends on which paper to be referred to. For example, Deb et al. (2001) has nine DTLZ problems whereas Deb et al. (2002) has seven DTLZ problems in which DTLZ5 and DTLZ9 in Deb et al. (2001) are removed, and DTLZ6, DTLZ7, DTLZ8 in Deb et al. (2001) are described as DTLZ5, DTLZ6, DTLZ7, respectively. This paper uses the first seven DTLZ problems in Deb et al. (2001).

  6. The nadir vector is defined as a vector consisting of the worst value of each objective on the Pareto front.

  7. Larger values at the \(i\)th element in \(\mathbf{u}\) indicate that the \(i\)th objective is preferred higher.

  8. For maximization problems, the normalized \(\mathbf{u}\) is directly used as a center vector of Gaussian functions in P-NSGA-II. On the other hand, each element in \(\mathbf{u}\) is inversed and the normalized vector is used as the center vector for minimization problems.

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Authors and Affiliations

  1. Honda Research Institute Europe GmbH, Carl-Legien-Straße 30, 63073, Offenbach, Germany

    Kaname Narukawa, Markus Olhofer & Bernhard Sendhoff

  2. Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka, 599-8531, Japan

    Yu Setoguchi, Yuki Tanigaki & Hisao Ishibuchi

Authors
  1. Kaname Narukawa
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  2. Yu Setoguchi
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  3. Yuki Tanigaki
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  4. Markus Olhofer
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  5. Bernhard Sendhoff
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  6. Hisao Ishibuchi
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Correspondence to Kaname Narukawa.

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Communicated by V. Loia.

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Narukawa, K., Setoguchi, Y., Tanigaki, Y. et al. Preference representation using Gaussian functions on a hyperplane in evolutionary multi-objective optimization. Soft Comput 20, 2733–2757 (2016). https://doi.org/10.1007/s00500-015-1674-9

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