Why does NSGAII (alg for Multiobjective Optimisation) always selects BOTH boundary points in crowding-distance-assignment part of algorithm?

Question

Why does NSGA II (Multiobjective Optimisation) always selects BOTH boundary points in crowding-distance-assignment part of algorithm ? I understand that in each iteration it choose the solution with the best value of one of values of multiobjective function, but why does it choose solution with the worst value also ? For me it seems that this algorithms tries to extend Pareto front (or actively searched space of solutions) as possible.

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.4257

pseudocode page 5

293 citations so NSGA II is very popular algorithm for Multiobjective Optimisation, so I think that my question isn't too specific.

Answer 1

You have to understand that the worst value for one objective (in a two objectives case) yield the best value in the second objective by definition of the Pareto front.

This list of undominated solutions are sorted according to the first objective (minimization)

(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)

The first value is the best on the first objective but the worst on the second, but still it is undominated. Keeping this solution helps to, maybe, get along the evolution a solution outside the current Pareto front. The true Pareto front of the problem is unknown and may extend further beyond the extrema found so far.

Moreover, a practionner may want to have a set of undominated solutions to choose from his final solution and may decide that he cannot aford compromise on one objective. The extreme solution is thus the best he can take based on his decision criterion.

The three (and more) objective cases are less evident but still true.