Generalized Dubovitskii-Milyutin Approach in Set-Valued Optimization
Akhtar Ali Khan · 2011
The primary objective of this paper is to employ the Dubovitskii-Milyutin approach to give first and second-order optimality conditions for set-valued optimization problems with more general set-valued constraints. In the particular case when some constraints are single-valued, we recover the case of set-valued optimization problems with many setvalued inequality constraints and many single-valued equality constraints. It is known that the classical Dubovitskii-Milyutin approach is not suitable for optimization problems with multi-equality constraints. The main reason for this deficiency is the fact that the separation arguments used in the classical Dubovitskii-Milyutin approach are applicable to an empty intersection of cones in which at most one cone can be closed. However, a proper formulation of multi-equality constraints leads to an empty intersection with more than one closed cones. To study optimization problems with multi-equality constraints, a generalized Dubovitskii-Milyutin theory has been developed. In this work we present an extension of the generalized Dubovitskii-Milyutin theory to the set-valued optimization problems. In this process, we also obtain new applications of this theory to nonsmooth optimization and to more general vector optimization problems. New second-order asymptotic derivatives of set-valued maps are introduced and used to give the optimality conditions.