During my research I encountered this problem by chance
Given a Multicriteria Optimization problem $\min_{x} \mathbf{f}(x) = \left( f_1(x), \ldots , f_p(x) \right)$ subject to $ x \in \mathcal{X}$
where $f_k: \mathbb{C}^N \to \mathbb{R}$ for $k = 1, \ldots , p$ and in general $\mathcal{X}\quad $ is given in a set of constraints.
The following method achieves all optimal Pareto points ( see Proper Efficiency in Nonconvex Multicriteria Programming E. U. Choo and D. R. Atkins for further details)
$\min_{x \in \mathcal{X}} \quad \max_{k = 1, \ldots p} \quad \lambda_k \left( f_k(x) - y_k^U \right) \quad (1) $
where $y_k^U$ is the optimal value when just the objective function $k$ is considered. Another method for finding the Pareto optimal points is the epsilon-constraint method (see M. Ehrgott. Multicriteria Optimization. Berlin, Springer, 2000. for further details) . In this case, the equivalent problem becomes
$\min_{x \in \mathcal{X}} \quad f_j(x) $ subject to $f_k(x) \leq \epsilon_k \quad k = 1, \ldots , p \quad k \neq j\quad (2)$
My question is
Given an optimal Pareto solution, $x^*$, obtained by (1) for a given set of $ \lambda_k \quad k = 1, \ldots , p $, how can be obtain the same solution, $x^*$, with (2) ? (i.e. Is there any mapping between $\epsilon_k$ and $\lambda_k$ for a given optimal solution?)
