Obtaining Smoothly Navigable Approximation Sets in Bi-Objective Multi-Modal Optimization

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, reducing 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. To tackle these issues, two new MMOEAs are proposed: Multi-Modal B\'ezier Evolutionary Algorithm (MM-BezEA) and Set B\'ezier Evolutionary Algorithm (Set-BezEA). Both MMOEAs produce approximation sets that cover individual niches and exhibit inherent decision-space smoothness as they are parameterized by B\'ezier 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.