Necessary and sufficient conditions for some general optimization problems (Q799929)

Let X, Y be real Banach spaces, let Y be ordered by a closed convex cone M. Let \(f_ 0\) be a real valued function defined on an open set D, \(D\subset X\), and suppose F is a mapping from D into Y. The following optimization problem is considered: (I) \(f_ 0(x)\to\inf \) subject to F(x)\(\in M\). A general second order sufficient optimality condition and some necessary optimality conditions for the problem (I) are derived under some additional assumptions concerning \(f_ 0\) and F. Some conditions, which are known from the literature can be obtained as special cases from the general results presented in the paper.