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An algorithm for solving smooth nonconvex optimization problems is proposedthat, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations todrive the norm of the gradient of the objective function below a prescribedpositive real number $\epsilon$ and can take $\mathcal{O}(\epsilon^{-3})$iterations to drive the leftmost eigenvalue of the Hessian of the objectiveabove $-\epsilon$. The proposed algorithm is a general framework that covers awide range of techniques including quadratically and cubically regularizedNewton methods, such as the Adaptive Regularisation using Cubics (ARC) methodand the recently proposed Trust-Region Algorithm with Contractions andExpansions (TRACE). The generality of our method is achieved through theintroduction of generic conditions that each trial step is required to satisfy,which in particular allow for inexact regularized Newton steps to be used.These conditions center around a new subproblem that can be approximatelysolved to obtain trial steps that satisfy the conditions. A new instance of theframework, distinct from ARC and TRACE, is described that may be viewed as ahybrid between quadratically and cubically regularized Newton methods.Numerical results demonstrate that our hybrid algorithm outperforms a cubliclyregularized Newton method.

An Inexact Regularized Newton Framework with a Worst-Case Iteration Complexity of $\mathcal{O}(\epsilon^{-3/2})$ for Nonconvex Optimization