A multifunction approach to extremum problems having infinite-dimensional image by F. Giannessi and G. Mastroeni Department of Mathematics, via Buonarroti 2, 56127 - PISA, Italy.

Abstract. The use of separation arguments in the definition of optimality conditions for a constrained extremum problem has stressed the importance of the analysis of the image space associated to the given problem, defined as the product space where the images of the objective and the constraining functions run.

Here, we analise the possibility of associating a finite-dimensional image space to a general infinite-dimensional extremum problem. This is done considering the constraints with an infinite-dimensional image, as multifunctions with values given by suitable subsets of a finite-dimensional space.

The existence of a selection, for the image multifunction, whose range has an empty intersection with a suitable subset of the image space, is a necessary and sufficient optimality condition for the given extremum problem. Under the hypothesis of continuity of the constraint and objective functions, the selection is defined as a weighted integral where the weights are closely related to the classic Lagrange multipliers. For this reason, these weights will be referred to as ”selection multipliers” and will be considered as an enlargement of the class of multipliers associated to the problem. In the hypothesis where the selection multipliers do not locally depend on the variable x of the problem, the classic results of Calculus of Variations can be recovered.

The main feature of the proposed approach is to postpone the introduction of infinite-dimensional arguments to the definition of the image of the problem in a finite-dimensional space. The infinite-dimensional nature of the problem arises in the analysis of the existence of a well-behaved selection of the image multifunction.