Reliability of convergence metric and hypervolume indicator for many-objective optimization

With the emergence and growth of Many-Objective Optimization algorithms, there has been an increased necessity to formulate new metrics that can perform quantitative assessment of the Pareto-Front returned as a solution from a Many-Objective Optimization algorithm. Out of the many evaluation metrics in use, convergence metric and hypervolume indicator have gained immense attention. This paper demonstrates how optimality obtained with respect to one or both of these metrics can be misleading at times. The demonstration is done in two-dimensional scenarios which suggests that the disadvantages of these metrics can be more pronounced when the applications are in higher dimensional space which not only has scalability issues but also where visualization of the space is not feasible. The paper is concluded stating the need for efficient evaluation metric which will accumulate information from the Pareto-Front in terms of convergence, diversity, number of solution (discarding outliers) and shape of the surface.