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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]. 

In this paper, we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone, whereas the other is locally Lipschitz continuous. We propose primal-dual (PD) extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of [Formula: see text] and [Formula: see text], measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an ε-residual solution of strongly and nonstrongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity [Formula: see text]. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an ε-KKT or ε-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under local Lipschitz continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods. Funding: This work was partially supported by the National Science Foundation [Grant IIS-2211491], the Office of Naval Research [Grant N00014-24-1-2702], and the Air Force Office of Scientific Research [Grant FA9550-24-1-0343].