C-index, spacing, and hypervolume

C-index, spacing, and hypervolume metrics used for comparing the Pareto results of multi-objective optimization algorithm

1. C-index: The C-index (or convergence metric) measures how well a set of Pareto-optimal solutions converge towards the true Pareto front. It is defined as the ratio of the number of solutions in the intersection of the Pareto front and the candidate set of solutions to the total number of solutions in the candidate set.
2. Spacing: The spacing metric measures the evenness of distribution of the Pareto-optimal solutions in the objective space. It is defined as the average distance between each Pareto-optimal solution and its nearest neighbor in the set.
3. Hypervolume: The hypervolume metric measures the volume of the objective space dominated by a set of Pareto-optimal solutions. It is defined as the volume of the portion of the objective space that is dominated by the set of solutions, relative to the volume of the entire objective space.

Please refer to this study to get the details of the formulations: https://www.sciencedirect.com/science/article/pii/S0378779623000093