HIGH-ORDER APPROXIMATIONS AND GENERALIZED NECESSARY CONDITIONS FOR OPTIMALITY

In this paper we derive generalized necessary conditions for optimalit y for an optimization problem with equality and inequality constraints in a Banach space. The equality constraints are given in operator for m as Q = { x is an element of X : F(x) = 0} where F : X --> Y is an op erator between Banach spaces; the inequality constraints are given by smooth functionals or by closed convex sets. Models of this type are c ommon in the optimal control problem. The paper addresses the case whe n 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 ar e trivially satisfied, generating abnormal cases. A high-order general ization of the Lyusternik theorem derived earlier [U. Ledzewicz and H. Schattler, Nonlinear Anal., 34 (1998), pp. 793-815] is used to calcul ate high-order tangent cones to the equality constraint at points x Q where F'(x()) is not onto. Combining these with high-order approximat ing cones related to the other constraints of the problem (feasible co nes respectively cones of decrease) a high-order generalization of the Dubovitskii-Milyutin theorem is given and then applied to derive gene ralized necessary conditions for optimality. These conditions reduce t o classical conditions for normal cases, but they give new and nontriv ial conditions for abnormal cases.