We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point is necessarily found when a limit point satisfies Robinson's condition. We conduct numerical experiments minimizing nonlinear functions subject to a copositive cone constraint. In order to do this, we consider a well known polyhedral approximation of this cone by means of refining the polyhedral constraints after each augmented Lagrangian iteration. We show that our strategy outperforms the standard approach of considering a close polyhedral approximation of the full copositive cone in every iteration.

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