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