The publisher regrets that one of the authors is missing in the PDF and web version of the article. Namely, Anton Bouter, who is affiliated with the Centrum Wiskunde & Informatica as described above in this erratum. In table 1, a space is missing after the character “<” in the line `Cervix -> Sigmoid' in column `Sparing criteria'. In table 3, the `%' signs in first column (DVI) should also be in subscript, similar to how it is written in the first column of Table 4. The supplementary material should state that needle contribution is limited to up to 40% of the total given dwell time (in applicator and needles). Lastly, the article should be seen as a research article and the authors would like to be cited as “L.R.M. Dickhoff and R.J. Scholman et al”.

The publisher would like to apologise for any inconvenience caused.@en

Even if a Multi-modal Multi-Objective Evolutionary Algorithm (MMOEA) is designed to find solutions well spread over all locally optimal approximation sets of a Multi-modal Multi-objective Optimization Problem (MMOP), there is a risk that the found set of solutions is not smoothly navigable because the solutions belong to various niches, reducing the insight for decision makers. To tackle this issue, a new MMOEAs is proposed: the Multi-Modal Bézier Evolutionary Algorithm (MM-BezEA), which produces approximation sets that cover individual niches and exhibit inherent decision-space smoothness as they are parameterized by Bézier curves. MM-BezEA combines the concepts behind the recently introduced BezEA and MO-HillVallEA to find all locally optimal approximation sets. When benchmarked against the MMOEAs MO_Ring_PSO_SCD and MO-HillVallEA on MMOPs with linear Pareto sets, MM-BezEA was found to perform best in terms of best hypervolume.

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The Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (RV-GOMEA) has previously been successfully used to achieve highly scalable optimization of various real-world problems in a gray-box optimization setting. Deformable Image Registration (DIR) is a multi-objective problem, aimed at finding the most likely non-rigid deformation of a given source image so that it matches a given target image. We specifically consider the case where the deformation model allows for finite-element-type modeling of tissue properties. This optimization problem is non-smooth, necessitating techniques like EAs to get good results. Though the objectives of DIR are non-separable, non-neighboring regions of the deformation grid are conditionally independent. We show that GOMEA allows to exploit such knowledge through the large-scale parallel application of variation steps, where each is only accepted when leading to an improvement, on a Graphics Processing Unit (GPU). On various 2-dimensional DIR problems, we find that this way, similar results can be achieved as when sequential processing is performed, while allowing for substantial speed-ups (up to a factor of 111) for the highest-dimensional problems (i.e., the highest deformation-grid resolution). This work opens the door to the extension of this type of DIR to larger (3-dimensional) deformation grids, and its application to other real-world problems.

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