Second-order optimality conditions play an important role in continuous optimization. In this paper, we present and discuss new constraint qualifications to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single Lagrange multiplier, instead of the whole set of Lagrange multipliers, and they are consistent with second-order algorithms where, usually, at each iteration, one only has access to a single approximate Lagrange multiplier. For each condition, we characterize the weakest second-order constraint qualification that guarantees its fulfillment at local minimizers, while proposing new weak conditions implying them. Relations with other constraint qualifications stated in the literature are discussed.
View On Constraint Qualifications for Second-Order Optimality Conditions Depending on a Single Lagrange Multiplier.