A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization

Abstract

A Newton-like method for unconstrained minimization is introduced in the present work. While the computer work per iteration of the best-known implementations may need several factorizations or may use rather expensive matrix decompositions, the proposed method uses a single cheap factorization per iteration. Convergence and complexity and results, even in the case in which the subproblems’ Hessians are far from being Hessians of the objective function, are presented. Moreover, when the Hessian is Lipschitz-continuous, the proposed method enjoys $$O(\varepsilon ^{-3/2})$$ evaluation complexity for first-order optimality and $$O(\varepsilon ^{-3})$$ for second-order optimality as other recently introduced Newton method for unconstrained optimization based on cubic regularization or special trust-region procedures. Fairly successful and fully reproducible numerical experiments are presented and the developed corresponding software is freely available.