Pymoo/NSGA2 : How to interpreter MOO (multi objective optimization) output columns n_nds, eps, and indicator?

Question

I am doing a multi-objective optimization task using Pymoo.org, but I have a hard time understanding the last three columns of the output.

I assume n_nds is the number of nondominant solutions in each generation. However, I don't understand the eps and indicator. Particularly the indicator column. I have come across this page "Display" which is referring readers to A Running Performance Metric and Termination Criterion for Evaluating Evolutionary Multi- and Many-objective Optimization Algorithms.

I have studied half of the paper. However, I have not found any explanations in that regard.

I would appreciate your valuable comments.

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n_gen | n_eval | cv (min) | cv (avg) | n_nds | eps | indicator

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 1 |      42 |  0.00000E+00 |  0.00000E+00 |      14 |            - |            -
 2 |      84 |  0.00000E+00 |  0.00000E+00 |      19 |  0.024237527 |        ideal
 3 |     126 |  0.00000E+00 |  0.00000E+00 |       6 |  0.091298096 |        ideal
 4 |     168 |  0.00000E+00 |  0.00000E+00 |       8 |  0.023750728 |            f
 5 |     210 |  0.00000E+00 |  0.00000E+00 |       7 |  0.002902893 |            f
 6 |     252 |  0.00000E+00 |  0.00000E+00 |      10 |  0.032567624 |            f
 7 |     294 |  0.00000E+00 |  0.00000E+00 |      11 |  0.000912191 |            f
 8 |     336 |  0.00000E+00 |  0.00000E+00 |      12 |  0.076898816 |            f
 9 |     378 |  0.00000E+00 |  0.00000E+00 |      12 |  0.00000E+00 |            f
10 |     420 |  0.00000E+00 |  0.00000E+00 |      14 |  0.065060499 |        ideal

Answer 1

In the output of pymoo the columns are explained as follows:

1. n_nds: This stands for the "Number of Non-Dominated Solutions." In multi-objective optimization, solutions are categorized into two types: dominated and non-dominated. Non-dominated solutions are those that are not worse than any other solution in all objectives. n_nds gives you the count of solutions that are non-dominated in the Pareto front.

2. eps: The "Epsilon Indicator" is a metric that quantifies the quality of a Pareto front approximation. It measures how close the approximation is to the true Pareto front. A smaller value indicates a better approximation.

3. indicator columns: These typically represent various performance metrics or indicators that assess the quality of a multi-objective optimization algorithm's output. These indicators provide insights into aspects like convergence, diversity, and the spread of solutions in the Pareto front approximation.

These components are useful in evaluating the performance of multi-objective optimization algorithms and understanding the quality of the solutions they produce.