Virtual reality high dimensional objective spaces for multi-objective optimization: An improved representation

This paper presents an approach for constructing improved visual representations of high dimensional objective spaces using virtual reality. These spaces arise from the solution of multi-objective optimization problems with more than 3 objective functions which lead to high dimensional Pareto fronts. The 3-D representations of m-dimensional Pareto fronts, or their approximations, are constructed via similarity structure mappings between the original objective spaces and the 3-D space. Alpha shapes are introduced for the representation and compared with previous approaches based on convex hulls. In addition, the mappings minimizing a measure of the amount of dissimilarity loss are obtained via genetic programming. This approach is preliminarily investigated using both theoretically derived high dimensional Pareto fronts for a test problem (DTLZ2) and practically obtained objective spaces for the 4 dimensional knapsack problem via multi-objective evolutionary algorithms like HLGA, NSGA, and VEGA. The improved representation captures more accurately the real nature of the m-dimensional objective spaces and the quality of the mappings obtained with genetic programming is equivalent to those computed with classical optimization algorithms.