Visualising Mutually Non-dominating Solution Sets in Many-objective Optimisation

As many-objective optimisation algorithms mature the problem owner is faced with visualising and understanding a set of mutually non-dominating solutions in a high dimensional space. We review existing methods and present new techniques to address this problem. We address a common problem with the well known heatmap visualisation, that the often arbitrary ordering of rows and columns renders the heatmap unclear, by using spectral seriation to rearrange the solutions and objectives and thus enhance the clarity of the heatmap. A multi-objective evolutionary optimiser is used to further enhance the simultaneous visualisation of solutions in objective and parameter space. Two methods for visualising multi-objective solution objectives in the plane are introduced. First, we use RadViz and exploit interpretations of barycentric coordinates for convex polygons and simplices to map a mutually non-dominating set to the interior of a regular convex polygon in the plane, providing an intuitive representation of the solutions and objectives. Second, we introduce a new measure of the similarity of solutions—the dominance distance—which captures the order relations between solutions. This metric provides an embedding in Euclidean space, which is shown to yield coherent visualisations in two dimensions. The methods are illustrated on standard test problems and data from a benchmark many-objective problem.