Improved local convergence results for augmented Lagrangian methods in $${\varvec{C}}^\mathbf{2}$$C2-cone reducible constrained optimization

Abstract

This paper deals with a class of cone-reducible constrained optimization problems which encompasses nonlinear programming, semidefinite programming, second-order cone programming, and any combination thereof. Using the second-order sufficient condition and a strict version of the Robinson constraint qualification, we provide a (semi-)local error bound which generalizes known results from the literature. Moreover, under the same assumptions, we prove that an augmented Lagrangian method is locally convergent with rate proportional to $$1/\rho _k$$, where $$\rho _k$$ is the penalty parameter, and that $$\{\rho _k\}$$ remains bounded.