In this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a Banach space. The equality constraints are given in operator form as Q = { x is an element of X : F(x) = 0} where F : X --> Y is an operator between Banach spaces; the inequality constraints are given by smooth functionals or by closed convex sets. Models of this type are common in the optimal control problem. The paper addresses the case when the Frechet-derivative F＇(x(*)) is not onto and hence the classical Lyusternik theorem does not apply to describe the tangent space to Q. In this case the classical Euler-Lagrange type necessary conditions are trivially satisfied, generating abnormal cases. A high-order generalization of the Lyusternik theorem derived earlier [U. Ledzewicz and H. Schattler, Nonlinear Anal., 34 (1998), pp. 793-815] is used to calculate high-order tangent cones to the equality constraint at points x Q where F＇(x(*)) is not onto. Combining these with high-order approximating cones related to the other constraints of the problem (feasible cones respectively cones of decrease) a high-order generalization of the Dubovitskii-Milyutin theorem is given and then applied to derive generalized necessary conditions for optimality. These conditions reduce to classical conditions for normal cases, but they give new and nontrivial conditions for abnormal cases.