By Brenda L. Miller
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Extra resources for Art Fantastix 16. The Art of Max Bertolini
Such problems are the so called multiobjective extremal problems. Also for such problems (and even more general ones, think of set valued or stochastic functions which may be involved in extremal problems) optimality conditions have been derived. Loridan  in 1984 was the first who successfully made use of a special multiobjective variational principle, discovered by himself, to get optimality conditions. He  has presented a vector-valued variational principle for the finite-dimensional case using a scalarization and Ekeland’s original result.
As a by-product we obtain the corrected formulations of corresponding results for the exact solutions. The main part of this section is devoted to the mentioned relations between the (approximate) solutions of the original multicriteria fractional problem and the transformed one. Moreover, we discuss possibilities to solve the transformed problem by a three-level dialogue approach following ideas of the book . , the problem subject to 38 MULTIPLE CRITERIA OPTIMIZATION where and i = 1, …, m.
It is an old idea to try to complement a given optimization problem with minimal value I) by a dual problem with supremal value S, remember the dual variational principles of Dirichlet and Thompson (cf. g. O. Friedrichs  or simply the pair of dual programs in linear optimization. The reasons for the introduction of a useful dual problem are the following: The dual problem has (under additional conditions) the same optimal value as the given “primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics.
Art Fantastix 16. The Art of Max Bertolini by Brenda L. Miller