3D-RadVis Antenna: Visualization and performance measure for many-objective optimization

So far the focus of almost all multi- or many-objective performance metrics has been the convergence and distribution of solutions in the objective space (Pareto-surface). Pareto-surface metrics such as IGD, HV, and Spread are simple and provide knowledge about the overall performance of the solution set. However, these measures do not provide any insight into the distribution or spread of a solution set with respect to each objective. Further, in many-objective optimization, visualization of true Pareto fronts or obtained non-dominated solutions is difficult. A proper visualization tool must be able to show the location, range, shape, and distribution of obtained non dominated solutions (both Pareto-surface and objective-wise distribution). Existing commonly used visualization tools in many-objective optimization (e.g., parallel coordinates) fail to show the shape of the Pareto front or distribution of solutions along each objective. In this paper, we propose an extension of recently proposed visualization method called 3D-RadVis (we call it 3D-RadVis Antenna) to visualize the distribution of solutions along each objective. 3D-RadVis Antenna is capable of mapping M-dimensional objective space to a 3-dimensional radial coordinate plot while seeking to preserve the relative location of solutions, shape of the Pareto front, and distribution of solutions along each objective. Furthermore, 3D-RadVis Antenna can be used by decision-makers to visually navigate large many-objective solution sets, to observe the evolutionary process, to visualize the relative location of a solution, to evaluate trade-offs among objectives, and to select preferred solutions. Along with this visualization tool, we propose two novel performance measures, named objective-wise inverse generational distance (ObjIGD) and line distribution (Delta(Line)) to measure the convergence and distribution of solutions along each objective as well as the overall performance of approximate solutions. The effectiveness of the proposed methods are demonstrated on widely used many-objective benchmark problems containing a variety of Pareto fronts (linear, concave, convex, mixed, and disconnected). In addition, for a case study, we have demonstrated the capability of 3D-RadVis Antenna combined with the proposed performance measures for visual progress tracking of the NSGA-III algorithm through generations. Experimental results show that the proposed visualization method can effectively be used to compare and track the performance of many-objective algorithms. Moreover, the proposed measures can be used as reliable complementary measures along with other widely used performance measures to compare many-objective solution sets.