A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees

• URL: http://arxiv.org/abs/2207.05697v1
• Date: Tue, 12 Jul 2022 17:21:45 GMT
• Title: A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees
• Authors: Chuan He and Zhaosong Lu
• Abstract summary: We consider finding an approximate second-order stationary point (SOSP) that minimizes a twice differentiable function intersection. Our method is not only iteration, but also achieves $mincal O(epsilon/2) which matches the best known complexity.
• Score: 0.5922488908114022
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we consider finding an approximate second-order stationary point (SOSP) of nonconvex conic optimization that minimizes a twice differentiable function over the intersection of an affine subspace and a convex cone. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier method for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of this problem. Our method is not only implementable, but also achieves an iteration complexity of ${\cal O}(\epsilon^{-3/2})$, which matches the best known iteration complexity of second-order methods for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of unconstrained nonconvex optimization. The operation complexity of $\widetilde{\cal O}(\epsilon^{-3/2}\min\{n,\epsilon^{-1/4}\})$, measured by the amount of fundamental operations, is also established for our method.

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