  Classical trust region methods were designed to solve problems in which
function and gradient information are exact. This paper considers the case when
there are bounded errors (or noise) in the above computations and proposes a
simple modification of the trust region method to cope with these errors. The
new algorithm only requires information about the size of the errors in the
function evaluations and incurs no additional computational expense. It is
shown that, when applied to a smooth (but not necessarily convex) objective
function, the iterates of the algorithm visit a neighborhood of stationarity
infinitely often, and that the rest of the sequence cannot stray too far away,
as measured by function values. Numerical results illustrate how the classical
trust region algorithm may fail in the presence of noise, and how the proposed
algorithm ensures steady progress towards stationarity in these cases.
