Even if a Multi-modal Multi-Objective Evolutionary Algorithm (MMOEA) is designed to find all locally optimal approximation sets of a Multi-modal Multi-objective Optimization Problem (MMOP), there is a risk that the found approximation sets are not smoothly navigable because the solutions belong to various niches, which reduces the insight for decision makers. Moreover, when the multi-modality of MMOPs increases, this risk grows and the trackability of finding all locally optimal approximation sets decreases. One example where this issue occurs is that of High-Dose-Rate (HDR) brachytherapy for prostate cancer. In HDR brachytherapy a treatment plan is to be optimized that irradiates a tumour with a prescribed dose, whilst sparing all of the healthy organs surrounding the tumour. The radiation is administered through a radioactive source that is stopped at certain dwell positions within a set of hollow catheters that have been implanted into the patient. In a treatment plan, each of the dwell positions is given a specific dwell time for which the source is kept at that location in order to irradiate the surrounding tissue.
To tackle the navigability issues, two new MMOEAs are proposed: Multi-Modal Bézier Evolutionary Algorithm (MM-BezEA) and Set Bézier Evolutionary Algorithm (Set-BezEA). Both MMOEAs produce 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. Set-BezEA employs a novel multi-objective fitness function formulation to find limited numbers of diverse, locally optimal, approximation sets for MMOPs of high multi-modality.
Both algorithms, but especially MM-BezEA, are found to outperform the MMOEAs MO_Ring_PSO_SCD and MO-HillVallEA on MMOPs of moderate multi-modality with linear Pareto sets. Moreover, for MMOPs of high multi-modality, Set-BezEA is found to indeed be able to produce high-quality approximation sets, each pertaining to a single niche. Set-BezEA is also shown to be comparable to the current BRIGHT approach used in the Amsterdam UMC for the optimization of treatment plans for prostate cancer HDR brachytherapy, which opens the way for it to be introduced in the clinical practice in the future.