Sequential optimality conditions have played a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions have been described in conic contexts, where many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the Augmented Lagrangian method satisfy the so-called Approximate Gradient Projection optimality condition, and, under an additional smoothness assumption, the so-called Complementary Approximate Karush-Kuhn-Tucker condition. The first result was unknown even for nonlinear programming while the second one was unknown, for instance, for semidefinite programming.
Submitted in 03/03/2020