\(\)

A second-order algorithm is proposed for minimizing smooth nonconvex

functions that alternates between regularized Newton and negative

curvature steps. In most cases, the Hessian matrix is regularized with

the square root of the current gradient and an additional term taking

moderate negative curvature into account, a negative curvature step

being taken only exceptionnally. As a consequence, the proposed

method only requires the solution of a single linear system

at nearly all iterations. We establish that at most

$\mathcal{O}\left( |\log\epsilon|\,\epsilon^{-3/2}\right)$

evaluations of the problem's objective function and derivatives are needed

for this algorithm to obtain an $\epsilon$-approximate first-order

minimizer, and at most

$\mathcal{O}\left(|\log\epsilon|\,\epsilon^{-3}\right)$

to obtain a second-order one. Initial numerical experiments with two

variants of the new method are finally presented.

## Article

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