A decoupled first/second-order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds

In order to be provably convergent towards a second-order stationary point, optimization methods applied to nonconvex problems must necessarily exploit both first and second-order information. However, as revealed by recent complexity analyses of some of these methods, the overall effort to reach second-order points is significantly larger when compared to the one of approaching first-order ones. On the other hand, there are other algorithmic schemes, initially designed with first-order convergence in mind, that do not appear to maintain the same first-order performance when modified to take second-order information into account. In this paper, we propose a technique that separately computes first and second-order steps, and that globally converges to second-order stationary points: it consists in better connecting the steps to be taken and the stationarity criteria, potentially guaranteeing larger steps and decreases in the objective. Our approach is shown to lead to an improvement of the corresponding complexity bound with respect to the first-order optimality tolerance, while having a positive impact on the practical behavior. Although the applicability of our ideas is wider, we focus the presentation on trust-region methods with and without derivatives, and motivate in both cases the interest of our strategy.