PERTURBED OPTIMIZATION IN BANACH-SPACES .1. A GENERAL-THEORY BASED ONA WEAK DIRECTIONAL CONSTRAINT QUALIFICATION

Using a directional form of constraint qualification weaker than Robin son's, we derive an implicit function theorem for inclusions and use i t for first- and second-order sensitivity analyses of the value functi on in perturbed constrained optimization. We obtain Holder and Lipschi tz properties and, under a,to-gap condition, first-order expansions fo r exact and approximate solutions. As an application, differentiabilit y properties of metric projections in Hilbert spaces are obtained, usi ng a condition generalizing polyhedricity. We also present in the appe ndix a short proof of a generalization of the convex duality theorem i n Banach spaces.