Explicitly Multi-Modal Benchmarks for Multi-Objective Optimization

In multi-objective optimization, designing good benchmark problems is an important issue for improving solvers. Although many benchmark problems have been proposed and some of them became de facto standards, designing multimodal problems that have a controllable landscape is still an open problem especially for high-dimensional cases. We thus propose the Benchmark with Explicit Multimodality (BEM), which lets the benchmark designer specify the basins of attraction using a graph structure known as the reachability graph. In this article, we focus on the mathematical formulation of the BEM. We will see that the BEM has preferable characteristics such as (i) realizing user-specified local Pareto set, (ii) allowing high-dimensional design spaces and (iii) possessing nonseparability.