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arXiv · 2311.13094

Newton-CG methods for nonconvex unconstrained optimization with H\"older continuous Hessian

Chuan He·Heng Huang·Zhaosong Lu
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Abstract

In this paper we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with H\"older continuous Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG) method for finding an approximate first- and second-order stationary point of this problem, assuming the associated the H\"older parameters are explicitly known. Then we develop a parameter-free Newton-CG method without requiring any prior knowledge of these parameters. To the best of our knowledge, this method is the first parameter-free second-order method achieving the best-known iteration and operation complexity for finding an approximate first- and second-order stationary point of this problem. Finally, we present preliminary numerical results to demonstrate the superior practical performance of our parameter-free Newton-CG method over a well-known regularized Newton method.

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Chuan He, Heng Huang, Zhaosong Lu. 2023-11-22. Newton-CG methods for nonconvex unconstrained optimization with H\"older continuous Hessian. https://arxiv.org/abs/2311.13094

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