Second-Order Necessary Optimality Conditions for Problems Without A Priori Normality Assumptions

In this article, we derive second-order necessary conditions of optimality for an abstract optimization problem with equality and inequality constraints and constraints in the form of an inclusion into a given closed set. An important feature is that our optimality conditions dispense with any a priori normality assumptions, such as Robinson’s constraint qualification, and remain informative even for abnormal points. Moreover, our optimality conditions take into account the second-order effect of the curvature of the set in the inclusion constraints.