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An Expedition to Multimodal Multi-objective Optimization Landscapes

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Evolutionary Multi-Criterion Optimization (EMO 2017)

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Abstract

The research in evolutionary multi-objective optimization is largely missing a notion of functional landscapes, which could enable a visual understanding of multimodal multi-objective landscapes and their characteristics by connecting decision and objective space. This consequently leads to the negligence of decision space in most algorithmic approaches and an almost complete lack of Exploratory Landscape Analysis (ELA) tools. This paper dares a first step into this unexplored field based on gradient properties of the multi-objective landscape. For a first time, basins of attraction and superpositions of local optima are visualized and thereby made intuitively accessible. With this work, we hope to highlight the importance of detailed decision space analysis in multi-objective optimization and to stimulate further research in that direction.

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Notes

  1. 1.

    Note that we restricted ourselves to bi-objective mixed-sphere problems as we build upon the definitions and problems from [8], but in general our proposed visualization technique can also be applied to problems with more than two objectives.

  2. 2.

    A vector \(\mathbf {a} = (a_1, a_2, \ldots , a_n)^T\) dominates another vector \(\mathbf {b} = (b_1, b_2, \ldots , b_n)^T\), i.e., \(\mathbf {a} \prec \mathbf {b}\), if and only if \(\forall i \in \{1, \ldots , n\}: \; a_i \le b_i\) and \(\exists j \in \{1, \ldots , n\}: \; a_j < b_j\).

  3. 3.

    One can also use more coarse configurations without losing a lot of accuracy. For instance, we tried smaller grids (consisting of 1 000 by 1 000 points) and a smaller threshold (\(\delta = 10^{-3}\)) while detecting only minor differences in the resulting figures.

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Correspondence to Pascal Kerschke .

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Kerschke, P., Grimme, C. (2017). An Expedition to Multimodal Multi-objective Optimization Landscapes. 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_23

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  • DOI: https://doi.org/10.1007/978-3-319-54157-0_23

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