We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton's method and linear conjugate gradient, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity results to be proved for convergence to points that satisfy approximate first-order and second-order optimality conditions. The complexity results match the best known results in the literature for second-order methods.
Technical Report, March 2018.
View A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization