A Modified Truncated Newton Method Which Uses Negative Curvature Directions for Large Scale Unconstrained Problems

Summary

In this work we consider large scale unconstrained minimization problems assuming that the objective function is twice continuously differentiable. Our aim is to define efficient and globally convergent algorithms which can handle problems where the dimension is large. The distinguishing feature of the methods considered in this work, is to ensure, under suitable assumptions, the global and superlinear convergence to stationary points where the Hessian matrix is positive semidefinite. This result is obtained by means of nonmonotone stabilization strategy based on a curvilinear line-search which uses a Newton-type direction along with a negative curvature direction. In order to be able to solve large scale problems, we use an iterative truncated algorithm for computing an approximation of both directions. The error introduced by this truncated scheme does not affect the second order global convergence and the superlinear rate of convergence.