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## Multicriteria Optimization methods connection: from Chebyshev to epsilon-constraint

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?)

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