Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality

Second-order sufficient conditions for local optimality have long been central to designing solution algorithms and justifying claims about their convergence. Here a far-reaching extension of such conditions, called variational sufficiency, is explored in territory beyond just classical nonlinear programming. Variational sufficiency is already known to support multiplier methods that are able, even without convexity, to achieve problem decomposition, but further insight has been needed into how it coordinates with other sufficient conditions. In the framework of this paper, it is shown to characterize local optimality in terms of having a convex–concave-type local saddle point of an augmented Lagrangian function. A stronger version of variational sufficiency is tied in turn to local strong convexity in the primal argument of that function and a property of augmented tilt stability that offers crucial aid to Lagrange multiplier methods at a fundamental level of analysis. Moreover, that strong version is translated here through second-order variational analysis into statements that can readily be compared to existing sufficient conditions in nonlinear programming, second-order cone programming, and other problem formulations which can incorporate nonsmooth objectives and regularization terms.