Yet another fast variant of Newton's method for nonconvex optimization.

This article proposes a class of second-order optimization algorithms, called AN2C (Adaptive Newton with Negative Curvature) and its variants, for minimizing smooth nonconvex functions by alternating between regularized Newton steps and negative curvature steps within iteration-dependent subspaces. The methods adaptively regularize the Hessian using the square root of the gradient norm and incorporate negative curvature information only when necessary, achieving a worst-case evaluation complexity of order \(\mathcal{O}(|\log \epsilon| \epsilon^{-3/2})\) to find an \(\epsilon\)-approximate first-order critical point, and \(\mathcal{O}(|\log \epsilon| \epsilon^{-3})\) to reach approximate second-order criticality. Two practical variants are detailed: full-space methods (AN2CE and AN2CER) that solve a single linear system per iteration, and Krylov-subspace methods (AN2CK) that use iterative Lanczos processes, both demonstrating competitive performance and reliability on standard test problems compared to established adaptive regularization and trust-region algorithms. The framework is fully adaptive, does not require knowledge of Hessian Lipschitz constants, and allows for flexible subspace choices, with numerical experiments supporting its efficiency and robustness across problem sizes.