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A Newton-CG Based Barrier Method for Finding a Second-Order Stationary Point of Nonconvex Conic Optimization with Complexity Guarantees
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 based barrier method for finding an
-SOSP of this problem. Our method not only is implementable but also achieves an iteration complexity of , which matches the best known iteration complexity of second-order methods for finding an -SOSP of unconstrained nonconvex optimization. The operation complexity, consisting of Cholesky factorizations and
other fundamental operations, is also established for our method, where n is the problem dimension and
represents with logarithmic terms omitted.
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