Scalar Lagrange multiplier rules for set-valued problems in infinite-dimensional spaces (Q1949579)

Optimization of set-valued maps has an increasing interest in the literature. In the last years, several theories have been proposed with different applications. In the paper, a set-valued optimization problem is considered as a vector optimization problem with a set-valued objective map. The aim of the paper is to state under what minima conditions still hold for infinite-dimensional spaces. The authors show (Example 4.3) that ``a direct extension to infinite-dimensional spaces is not possible. This example shows, even for the case of vector optimization in Hilbert spaces, that the scalar multiplier rule does not provide optimality conditions for weak minimizers''. Instead, the authors prove the necessity of assuming the directional compactness of maps in order for scalar multiplier rule still to hold. In Theorem 3.1, a general multiplier rule in terms of the contingent derivative is established. In Theorem 4.1 (the main result) it is proven, that ``adding a directional compactness hypothesis on the maps, a scalar multiplier rule can be established for the infinite-dimensional case''. Many counterexamples are given to prove that the assumptions of these results are minimal.