HIGHER-ORDER VARIATIONAL SETS, VARIATINAL DERIVATIVES AND HIGHER-ORDER NECESSARY CONDITIONS IN ABSTRACT MATHEMATICAL PROGRAMMING

We consider the problem of minimizing the value of a functionaI f (x) under the constraints g(x) E B, x E Q, where g is a mapping from a Banach space to a Banach space. We assume neither the differentiability in usual sense nor the convexity on f and g. The set B is an arbitrary subset of a Banach space which has a nonempty interior. A higher-order variational theory for the optimization problem is developed. Several kinds of concepts of higher-order variational sets are introduced in a Banach space, and concepts of higher-order variational derivatives of mappings are introduced too. We study fundamentaI properties of those variational sets and vari ationaI derivatives. We also give higher-order necessary optimality conditions for the problem in terms of the variational sets and derivatives. We apply the results of optimality condition to derive a second-order criterion of an optimaI solution in a nonlinear TrhPhvrH, ff nnnrnvimatinn nrnhTam