ABSTRATC:
Conditions associated with local optimality, whether necessary or sufficient, have traditionally been approached through techniques of generalized differentiation. On the first-order level, this has been a long-standing success, although serious challenges remain for equilibrium constraints and the like. On the second-order level, a difficulty areses with the complex concepts of generalized second derivatives and the sometimes-inadequate calculus for determining them.
In fact, sufficient second-order conditions of a practical sort, which are the most important aid for numerical methodology are largely lacking outside of classical frameworks like nonlinear programming. In nonlinear programming, well known conditions like so-called strong second-order optimality are tied to a convexity-concavity property of an augmented Lagrangian function. It turns out that this pattern can be developed in vastly larger territory by exploiting the recently introduced concept of variational convexity.