  In this paper we consider finding an approximate second-order stationary
point (SOSP) of general nonconvex conic optimization that minimizes a twice
differentiable function subject to nonlinear equality constraints and also a
convex conic constraint. In particular, we propose a Newton-conjugate gradient
(Newton-CG) based barrier-augmented Lagrangian method for finding an
approximate SOSP of this problem. Under some mild assumptions, we show that our
method enjoys a total inner iteration complexity of $\widetilde{\cal
O}(\epsilon^{-11/2})$ and an operation complexity of $\widetilde{\cal
O}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})$ for finding an
$(\epsilon,\sqrt{\epsilon})$-SOSP of general nonconvex conic optimization with
high probability. Moreover, under a constraint qualification, these complexity
bounds 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 of
finding an approximate SOSP of general nonconvex conic optimization.
Preliminary numerical results are presented to demonstrate superiority of the
proposed method over first-order methods in terms of solution quality.
