Global Convergence of Trust Region Algorithms Using Inexact Gradient Information.

摘要

Trust region algorithms are an important class of methods that can be used to solve unconstrained optimization problems. More has proven a global convergence result for a class of trust region methods where the gradient values are approximated rather than computed exactly, provided the approximations are consistent. We show that the assumption of consistency can be replaced by a simple condition on the relative error in the gradient approximation. This new condition has both practical and theoretical advantages. First, it provides a practical test for judging the adequacy of a given gradient approximation, and does not require new approximations to be computed for unsuccessful iterations. Second, it leads to stronger convergence results than obtained previously.