Multicriteria Optimization methods connection: from Chebyshev to epsilon-constraint - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:37:30Z http://mathoverflow.net/feeds/question/74583 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74583/multicriteria-optimization-methods-connection-from-chebyshev-to-epsilon-constrai Multicriteria Optimization methods connection: from Chebyshev to epsilon-constraint mikitov 2011-09-05T13:54:24Z 2011-09-06T12:28:48Z <p>During my research I encountered this problem by chance</p> <p>Given a Multicriteria Optimization problem $\min_{x} \mathbf{f}(x) = \left( f_1(x), \ldots , f_p(x) \right)$ subject to $x \in \mathcal{X}$</p> <p>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. </p> <p>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)</p> <p>$\min_{x \in \mathcal{X}} \quad \max_{k = 1, \ldots p} \quad \lambda_k \left( f_k(x) - y_k^U \right) \quad (1)$ </p> <p>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</p> <p>$\min_{x \in \mathcal{X}} \quad f_j(x)$ subject to $f_k(x) \leq \epsilon_k \quad k = 1, \ldots , p \quad k \neq j\quad (2)$ </p> <p>My question is</p> <blockquote> <p>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?)</p> </blockquote>