second-order stationarity – Optimization Online

Curvature-oriented variance reduction methods for nonconvex stochastic optimization

Published: 2026/02/13, Updated: 2026/02/14

Constrained Nonlinear Optimization, Nonlinear Optimization cubic regularization, negative curvature direction, nonconvex constrained optimization, sample complexity, second-order stationarity

When pursuing an approximate second-order stationary point in nonconvex constrained stochastic optimization, is it possible to design a stochastic second-order method that achieves the same sample complexity order as in the unconstrained setting? To address this question in this paper, we first introduce Carme, a curvature-oriented variance reduction method designed for unconstrained nonconvex stochastic optimization. … Read more

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

Published: 2017/04/21, Updated: 2018/02/21

Serge Gratton

Clément W. Royer

Luis Nunes Vicente

Nonlinear Optimization, Unconstrained Optimization decoupling, first-order stationarity, global rates, second-order stationarity, trust-region methods, worst-case complexity

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 analyzes 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 … Read more