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

In a recent paper (Birgin et al. in Math Program 163(1):359-368, 2017), 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=3With that purpose, we propose a specific algorithm and we show numerical experiments.