# On the use of third-order models with fourth-order regularization for unconstrained optimization

In a recent paper, it was shown that, for the smooth unconstrained optimization problem, worst-case evaluation complexity $O(\epsilon^{-(p+1)/p})$ may be obtained by means of algorithms that employ sequential approximate minimizations of p-th order Taylor models plus (p + 1)-th order regularization terms. The aforementioned result, which assumes Lipschitz continuity of the p-th partial derivatives, generalizes the case p = 2, known since 2006, which has already motivated efficient implementations. The present paper addresses the issue of defining a reliable algorithm for the case p = 3. With that purpose, we propose a specific algorithm and we show numerical experiments.

2017