The Fermat rule for set optimization problems with Lipschitzian set-valued mappings. (English) Zbl 1460.90141
Summary: In this paper, we consider set optimization problems where the solution concept is given by the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive the convexity and Lipschitzianity of suitable scalarizing functions under the assumption that the set-valued objective mapping has certain convexity and Lipschitzianity properties. Then, we obtain upper estimates of the limiting subdifferential of these functionals. These results, together with the properties of the scalarization functions, allow us to obtain a Fermat rule for set optimization problems with Lipschitzian data.
MSC:
90C26 | Nonconvex programming, global optimization |
65K10 | Numerical optimization and variational techniques |
90C30 | Nonlinear programming |
90C29 | Multi-objective and goal programming |