Higher order optimality conditions in nonsmooth vector optimization
Ivan Ginchev · Journal of Statistics and Management Systems · 2002
The vector optimization problem for the set-valued function is considered, where X 0 is a subset of a real Banach space X and Y is a real Banach space with a given partial order defined by a convex, closed and pointed cone with nonempty interior C. Introducing two infinite elements ±∞C and putting we transform this problem into the equivalent problem for the set-valued function , where p is the extension of F 0 with values +∞C on X\X 0. The nonsmoothness is understood that no regularity conditions for F are preliminary required. Efficient and weakly efficient optimality conditions are reviewed. Higher order lower directional derivatives for F are defined. In terms of these derivatives higher order necessary and sufficient optimality conditions are obtained. The results are illustrated by examples. The concept of an isolated minimizer is introduced. It is shown that the sufficìent conditions characterize (that is they are both sufficient and necessary) the isolated minimizer.