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In this paper we consider finding an approximate second-order stationarypoint (SOSP) of general nonconvex conic optimization that minimizes a twicedifferentiable function subject to nonlinear equality constraints and also aconvex conic constraint. In particular, we propose a Newton-conjugate gradient(Newton-CG) based barrier-augmented Lagrangian method for finding anapproximate SOSP of this problem. Under some mild assumptions, we show that ourmethod enjoys a total inner iteration complexity of $\widetilde{\calO}(\epsilon^{-11/2})$ and an operation complexity of $\widetilde{\calO}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})$ for finding an$(\epsilon,\sqrt{\epsilon})$-SOSP of general nonconvex conic optimization withhigh probability. Moreover, under a constraint qualification, these complexitybounds are improved to $\widetilde{\cal O}(\epsilon^{-7/2})$ and$\widetilde{\cal O}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})$, respectively.To the best of our knowledge, this is the first study on the complexity offinding an approximate SOSP of general nonconvex conic optimization.Preliminary numerical results are presented to demonstrate superiority of theproposed method over first-order methods in terms of solution quality.

A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization