Newton-CG Methods for Nonconvex Unconstrained Optimization with Hölder Continuous Hessian

In this paper, we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with Hölder 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 Hölder 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. Funding: C. He was partially financially supported by the Wallenberg AI, Autonomous Systems and Software Program funded by the Knut and Alice Wallenberg Foundation. H. Huang was partially financially supported by the National Science Foundation [Award IIS-2347592]. Z. Lu was partially financially supported by the National Science Foundation [Award IIS-2211491], the Office of Naval Research [Award N00014-24-1-2702], and the Air Force Office of Scientific Research [Award FA9550-24-1-0343].