A Topologically Consistent Visualization of High Dimensional Pareto-front for Multi-Criteria Decision Making

There are a good number of different algorithms to solve multi- and many-objective optimization problems and the final outcome of these algorithms is a set of trade-off solutions that are expected to span the entire Pareto-front. Visualization of a Pareto-front is vital for an initial decision-making task, as it provides a number of useful information, such as closeness of one solution to another, trade-off among conflicting objectives, localized shape of the Pareto-front vis-a-vis the entire front, and others. Two and three-dimensional Pareto-fronts are trivial to visualize and allow all the above analysis to be done comprehensively. However, for four or more objectives, visualization for extracting above decision-making information gets challenging and new and innovative methods are long overdue. Not only does a trivial visualization becomes difficult, the number of points needed to represent a higher-dimensional front increase exponentially. The existing high-dimensional visualization techniques, such as parallel coordinate plots, scatter plots, RadVis, etc., do not offer a clear and natural view of the Pareto-front in terms of trade-off and other vital localized information needed for a convenient decision-making task. In this paper, we propose a novel way to map a high-dimensional Pareto-front in two and three dimensions. The proposed method tries to capture some of the basic topological properties of the Pareto points and retain them in the mapped lower dimensional space. Therefore, the proposed method can produce faithful representation of the topological primitives of the high-dimensional data points in terms of the basic shape (and structure) of the Pareto-front, its boundary, and visual classification of the relative trade-offs of the solutions. As a proof-of-principle demonstration, we apply our proposed palette visualization method to a few problems.