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JIMO

The problem of optimizing some given function
over the efficient set is one of the most interesting and important
concepts in multicriteria decision making. As the efficient set
is in general nonconvex, even for the case of linear
multicriteria programming problems, optimizing over the efficient
set belongs to a typical problem class of
multiextremal optimization problems, which can have local optima
different from global optima.

In this article, we consider the case where the multicriteria programming problem is linear. Characterizing the set of efficient solutions by some constraint of 'reverse convex' type in the space of criteria, we formulate the problem of minimizing a function $f$ over the efficient set as a global optimization problem with a special structure. For the resulting problem, a decomposition branch and bound based algorithm is then proposed, in which the branching procedure is performed in the criteria space. Convergence properties of the algorithm are discussed, and preliminary computational results are reported.

In this article, we consider the case where the multicriteria programming problem is linear. Characterizing the set of efficient solutions by some constraint of 'reverse convex' type in the space of criteria, we formulate the problem of minimizing a function $f$ over the efficient set as a global optimization problem with a special structure. For the resulting problem, a decomposition branch and bound based algorithm is then proposed, in which the branching procedure is performed in the criteria space. Convergence properties of the algorithm are discussed, and preliminary computational results are reported.

JIMO

In this article, we propose a method for finding the global
optimum of a class of nonlinear bilevel programming
problems. The main idea of this method is to construct iteratively a
sequence of points
either ending at an optimal solution of the equivalent problem with a
complementarity constraint, or
converging to an optimal solution. The construction of such a
sequence is performed by using a branch-and-bound scheme, together
with some relaxation techniques, which are successfully applied in
global optimization. Some illustrative examples and results on
computational experiments are reported.

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