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In multicriteria optimization problems appart from the notion of Pareto optimality there is also the term weakly Pareto optimality. For that case, the definition is that

A feasible solution $\hat{x} \in \mathcal{X} \quad$ is called weakly efficient if there is no $x \in \mathcal{X} \quad$ such that $f(x) < f(\hat{x})$, i.e. $f_k(x) < f_k(\hat{x}) \quad \forall k = 1, \ldots , p$.

Notice that the Pareto optimal set is a subset of the weakly Pareto optimal set.

I wonder the practical importance of the weakly Pareto optimal set. Do you any situation where they are important?

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