On necessary conditions for infinite-dimensional extremum problems

In this paper, we carry on the analysis (introduced in [4] and developed in [2, 7]) of optimality conditions for extremum problems having infinite-dimensional image, in the case of unilateral constraints. This is done by associating to the feasible set a special multifunction. It turns out that the classic Lagrangian multiplier functions can be factorized into a constant term and a variable one; the former is the gradient of a separating hyperplane as introduced in [4, 5]; the latter plays the role of selector of the above multifunction. Finally, the need of enlarging the class of Lagrangian multiplier functions is discussed.