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In this paper we consider finding a second-order stationary point (SOSP) ofnonconvex equality constrained optimization when a nearly feasible point isknown. In particular, we first propose a new Newton-CG method for finding anapproximate SOSP of unconstrained optimization and show that it enjoys asubstantially better complexity than the Newton-CG method [56]. We then proposea Newton-CG based augmented Lagrangian (AL) method for finding an approximateSOSP of nonconvex equality constrained optimization, in which the proposedNewton-CG method is used as a subproblem solver. We show that under ageneralized linear independence constraint qualification (GLICQ), our AL methodenjoys a total inner iteration complexity of $\widetilde{\calO}(\epsilon^{-7/2})$ and an operation complexity of $\widetilde{\calO}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})$ for finding an$(\epsilon,\sqrt{\epsilon})$-SOSP of nonconvex equality constrainedoptimization with high probability, which are significantly better than theones achieved by the proximal AL method [60]. Besides, we show that it has atotal inner iteration complexity of $\widetilde{\cal O}(\epsilon^{-11/2})$ andan operation complexity of $\widetilde{\calO}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})$ when the GLICQ does not hold. Tothe best of our knowledge, all the complexity results obtained in this paperare new for finding an approximate SOSP of nonconvex equality constrainedoptimization with high probability. Preliminary numerical results alsodemonstrate the superiority of our proposed methods over the ones in [56,60].

A Newton-CG based augmented Lagrangian method for finding a second-order  stationary point of nonconvex equality constrained optimization with  complexity guarantees