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

07/12/2022

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by   Chuan He, et al.

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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 (ϵ,√(ϵ))-SOSP of this problem. Our method is not only implementable, but also achieves an iteration complexity of O(ϵ^-3/2), which matches the best known iteration complexity of second-order methods for finding an (ϵ,√(ϵ))-SOSP of unconstrained nonconvex optimization. The operation complexity of O(ϵ^-3/2min{n,ϵ^-1/4}), measured by the amount of fundamental operations, is also established for our method.