In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method [56]. We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of \(\widetilde{\cal O}(\epsilon^{-7/2})\) and an operation complexity of \(\widetilde{\cal O}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})\) for finding an \((\epsilon,\sqrt{\epsilon})\)-SOSP of nonconvex equality constrained optimization with high probability, which are significantly better than the ones achieved by the proximal AL method [60]. Besides, we show that it has a total inner iteration complexity of \(\widetilde{\cal O}(\epsilon^{-11/2})\) and an operation complexity of \(\widetilde{\cal O}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})\) when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability. Preliminary numerical results also demonstrate the superiority of our proposed methods over the ones in [56,60].
Citation
To appear in SIAM Journal on Optimization