Pareto Landscape: Visualising the Landscape of Multi-objective Optimisation Problems

Fitness landscape is a valuable framework to understand optimisation problems. In single-objective optimisation, by displaying fitness landscape in a 3D space with the “height” representing the fitness (objective function value) of solutions, one can easily comprehend a variety of problem characteristics (optimality, multi-modality, level of ruggedness, etc.) and spatial features of the search space (basin, ridge, funnel, etc.). However, such straightforward visualisation cannot be directly extended to the multi-objective optimisation case in which a solution corresponds to a vector of values on multiple objective functions. In this paper, we make an attempt to address this issue. Instead of objective function values, we use the Pareto dominance relation to stratify solutions, introducing a method we term Pareto landscape for visualising multi-objective problem landscape. We compare Pareto landscape with well-established fitness landscape visualisation methods, including cost landscape, gradient field heatmap and PLOT, and show that Pareto landscape can capture problem characteristics that the other methods cannot do. Lastly, we present the Pareto landscapes of commonly used benchmark problems (ZDT, DTLZ, WFG and BBOB) in the domain, and discuss their features and characteristics.