PERTURBED OPTIMIZATION IN BANACH-SPACES .2. A THEORY-BASED ON A STRONG DIRECTIONAL CONSTRAINT QUALIFICATION

We study the sensitivity of the optimal value and optimal solutions of perturbed optimization problems in two cases. The first one is when m ultipliers exist but only the weak (and not the strong) second-order s ufficient optimality condition is satisfied. The second case is when n o Lagrange multipliers exist; To deal with these pathological cases, w e are led to introduce a directional constraint qualification stronger than in part I of this paper, which reduces to the latter in the impo rtant case of equality-inequality constrained problems. We give sharp upper estimates of the cost based on paths varying as the square root of the perturbation parameter and, under a no-gap condition, obtain th e first term of the expansion for the cost. When multipliers exist we study the expansion of approximate solutions as well. We show in the a ppendix that the strong directional constraint qualification is satisf ied for a large class of problems, including regular problems in the s ense of Robinson.