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.

翻译：本文研究寻找一般非凸锥优化问题的近似二阶稳定点（SOSP），该问题是在非线性等式约束和凸锥约束下最小化一个二阶可微函数。具体而言，我们提出一种基于牛顿-共轭梯度（Newton-CG）的障碍增广拉格朗日方法来寻找该问题的近似SOSP。在一些温和假设下，我们证明该方法以高概率找到一般非凸锥优化问题的$(\epsilon,\sqrt{\epsilon})$-SOSP时，其总内迭代复杂度为$\widetilde{\cal O}(\epsilon^{-11/2})$，运算复杂度为$\widetilde{\cal O}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})$。此外，在满足约束规范条件下，这些复杂度界分别提升至$\widetilde{\cal O}(\epsilon^{-7/2})$和$\widetilde{\cal O}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})$。据我们所知，这是首次对寻找一般非凸锥优化问题近似SOSP的复杂度进行研究。初步数值结果表明，所提方法在解的质量方面优于一阶方法。